The ledger

What the figures prove

This collection claims that a picture can be an argument rather than an illustration of one. That is a strong claim, so the generators check it while they draw — and this page is every claim they check, collected by listening to the assertions as the drawings are made. Nothing on it was typed in.

• point 1 of the nine is the same distance from the centre ×9
• A lies on the circle
• a trisector and the far side are not parallel
• and its second and third
• and so do the other two
• and that distance is half the radius of the circle through the corners
• and the centroid sits twice as far from the orthocentre as from the circumcentre
• and the same construction on the other trisectors is not
• and the same on the other side
• B lies on the circle
• every apex on the arc gives the same angle
• every triangle in the sweep puts its nine points on one circle
• every trisector triangle in the sweep is equilateral
• one angle, not several
• opposite angles add to a straight angle
• the angle on a diameter is a right angle
• the apex is on the major arc
• the apex lies on the circle
• the centre's angle is twice the apex's
• the chord is a diameter
• the circumcentre, the centroid and the orthocentre lie on one line
• the exterior angle is twice the base angle
• the inner triangle's first two sides are equal
• the nine-point centre is the midpoint of that line
• the quadrilateral's angles add to 360°
• the shape sweep is between 6 and 60 steps on a side
• the splitters at A and B are not parallel
• the splitters at B and C are not parallel
• the splitters at C and A are not parallel
• the sweep covers a decent share of the shape space
• the sweep found enough non-degenerate triangles to mean something
• the sweep runs over between 20 and 2000 triangles
• the three angles are the angles of a triangle
• the three angles are those of a triangle and none is a sliver
• the triangle has three corners and some area
• the two halves are the whole angle
• the view is one the family draws
• triangle OPA is isosceles
• triangle OPB is isosceles

• the two curves cross at a right angle at (0.450, 0.583) ×24
• the ellipse with semi-axis 1.35 has the same foci ×7
• the hyperbola with semi-axis 0.4 has the same foci ×6
• every point of the 0.55 curve keeps the ratio ×3
• a circle needs a cut that produces one
• a ellipse needs a cut that produces one
• a hyperbola needs a cut that produces one
• a parabola needs a cut that produces one
• and each point is as far from the focus as from the directrix
• and on the hyperbola
• and reaches it going forwards, not backwards
• and the two distances differ by the same amount wherever it strikes
• and the two legs together are the same length wherever it bounces
• and travels towards it rather than away from it
• between 3 and 60 rays are traced
• between one and four curves
• each curve is a name and six finite coefficients
• each ratio is between nothing and 2.4
• every ellipse's semi-axis is longer than the focal distance
• every hyperbola's semi-axis is shorter than the focal distance
• every pair of curves was checked at its crossing
• every parallel ray reflects to the focus
• the crossing lies on the ellipse
• the curve named a hyperbola really is one
• the curve named a parabola really is one
• the curve named an ellipse really is one
• the cutting plane is tipped by less than a quarter turn
• the discriminant and the count of crossings with a large circle agree that a hyperbola is a hyperbola
• the discriminant and the count of crossings with a large circle agree that a parabola is a parabola
• the discriminant and the count of crossings with a large circle agree that an ellipse is an ellipse
• the ellipse has a longer axis and a shorter one
• the foci are between 0.2 and 3 from the centre
• the hyperbola has both semi-axes positive
• the incoming ray has a length to normalise
• the mirror is an ellipse, a parabola or a hyperbola
• the normal to the curve has a length to normalise
• the outgoing ray has a length to normalise
• the point is on the upper half of the curve
• the radius to the first focus has a length to normalise
• the radius to the second has a length to normalise
• the ratio is between nothing and 2.4
• the reflected ray lies on the line through the near focus
• the reflected ray passes through the other focus
• the tangent direction has a length to normalise
• the tangent makes the same angle with both focal radii
• the three values give three different kinds of curve
• the view is one the family draws
• the window is between 1 and 8 wide

• the smallest square has side gcd(34, 13) ×4
• the squares tile the 34 by 13 rectangle ×4
• the leftover strip is still there after every pass
• the squares peeled at each pass are the continued fraction of the ratio

• the first 6 odd numbers sum to 6² ×4
• the L-shaped shells fill the square

• one square per Fibonacci number
• the convergents alternate about φ
• the last convergent is close to φ
• the long side is the next Fibonacci number
• the number of convergents plotted is between 2 and 20
• the number of whirling squares is between 1 and 12
• the rectangle is nearly golden
• the short side is the one before it
• the whirling squares tile their rectangle

• the circles round vertex 0 subtend exactly a full turn at it ×7
• ring 1: and the circle it surrounds from outside ×2
• ring 1: and touches the containing circle from inside ×2
• ring 1: each circle touches the next ×2
• a circle missing the centre inverts to a circle
• a circle orthogonal to the mirror is carried to itself
• a circle through the centre inverts to a line
• a point inside goes outside and a point outside comes in
• a point on the mirror circle is left exactly where it is
• and a line missing the centre inverts to a circle
• and a point of the circle lands on it
• and conjugates the imaginary part
• and every edge of the triangulation is a tangency of two circles
• and every point of the straight line lands on its circle
• and its curvature is a whole number
• and one for each of the eight ways of choosing inside or outside
• and so do their images, because a circle's image is a circle
• and the two short gaps overshoot
• and touches each of them on the page
• between two and four points are inverted, none of them at the centre
• both images are circles here
• centre and all
• circles whose vertices are not joined do not overlap
• each drawn circle really touches each given one
• each image gap is the original gap divided by the two distances from the centre
• each new circle satisfies Descartes's relation with its three parents
• every curvature in the packing is a whole number
• every pair of the seed circles touches
• every point of the circle lands on its image
• every point of the circle through the centre lands on its image
• every walk round an interior vertex closes on the circles already placed
• four points on one circle have a real cross-ratio
• four points, in order round the circle and none of them nearly coincident
• in both parts
• neither boundary passes through the inversion centre
• no chain circle passes through the inversion centre
• no point being inverted sits at a mirror's centre
• no point being inverted sits at the centre
• off the circle the identity becomes a strict inequality
• off the circle the three images are not collinear
• on a circle the products satisfy Ptolemy's identity exactly
• one inversion leaves the real part of the cross-ratio alone
• so the two short image gaps add to the long one
• the angle between the two curves is the same after the map as before
• the chord's end sits on the mirror
• the circle's image is a circle
• the concentric ring closes exactly when the radius ratio is the sine
• the constructed point is the one inside the mirror
• the drawn ring is genuinely lopsided rather than a disguised concentric one
• the first mirror has a sensible radius
• the four points are not concyclic, so their cross-ratio has a real imaginary part to conjugate
• the fourth point is pushed off the circle by less than half a radius
• the gasket is drawn to between one and six generations
• the image of the centre is not the centre of the image
• the image of the crossing point is on both image circles
• the inner circle has a sensible radius
• the inversion centre is inside the inner circle and off its centre
• the line inverted does not pass through the centre
• the mirror circle has a sensible radius
• the ring has between three and twelve circles
• the ring round the centre closes up
• the second mirror is offset by a sensible amount
• the seed satisfies Descartes's relation
• the tangent construction lands on the same point the formula gives
• the three centres are not collinear
• the three given circles lie outside one another
• the three images fall on one straight line
• the two circles really cross
• the two distances from the centre multiply to the square of the radius
• the view is one the family draws
• three boundary radii, repeated round the outside
• three circles in general position admit exactly eight tangent circles
• three given circles, each with a sensible radius
• two inversions put the cross-ratio back exactly
• which passes through the centre of the mirror

• the 3-gon encloses less than a circle of the same perimeter ×7
• the 3-gon is drawn at the same perimeter as the rest ×7
• regular 3-gon does not beat the circle ×5
• 2:1 rectangle does not beat the circle
• an L does not beat the circle
• and is still short of the circle's
• and it is the circle
• and its area is a quarter of the perimeter, squared
• and the area is strictly larger
• and the circle's quotient is exactly one
• even the best rectangle falls short of the circle
• every extra side encloses strictly more
• exactly one shape in the list reaches one, and it is the circle
• reflecting the dent leaves the perimeter exactly as it was
• Reuleaux triangle does not beat the circle
• so the quotient rises
• the circle does not beat the circle
• the circle of that perimeter encloses more than any of the polygons drawn
• the dent is between nothing and 160 units deep
• the largest rectangle in the sweep is the square
• the polygons drawn have between 3 and 60 sides
• the shared perimeter is between 60 and 900 units
• the sweep takes between 20 and 2000 samples
• the view is one the family draws
• whose two sides are equal

• a line with three points on it yields a strictly smaller distance
• each pair of points lies on one connecting line only
• no configuration searched avoids ordinary lines altogether
• some connecting line holds exactly two of the points
• some point is off some connecting line
• some set of this size is not all in one line
• the closest point-and-line pair uses a line with exactly two points
• the connecting lines between them account for every pair of points
• the distance the argument produces is strictly the smaller one
• the grid searched is three or four points across
• the point set is one this family knows
• the points are drawn between 12 and 60 units apart
• the search runs to between four and seven points
• the view is one the family draws

• the order-4 axes carry 9 turns between them ×7
• a turn of order 3 sits on an axis of order 3 ×5
• and each such axis carries a turn of order 4 ×4
• 3 triangles at a vertex leave a gap ×3
• 5-cell: and the edge count agrees with that degree ×3
• 5-cell: every vertex is the same distance from the centre ×3
• 5-cell: every vertex meets the same number of edges ×3
• 3 hexagons at a vertex cannot leave a gap
• 3 pentagons at a vertex leave a gap
• 3 squares at a vertex leave a gap
• 4 pentagons at a vertex cannot leave a gap
• 4 squares at a vertex cannot leave a gap
• 6 triangles at a vertex cannot leave a gap
• a star polygon has between 5 and 13 points
• and 32 − 90 + 60 = 2, the Euler number of any sphere
• and at the depth found every edge is the same length
• and every one of them is the same triangle
• and sends every corner to a corner
• and steps by at least 2, less than half of p
• and steps by between 2 and 6
• and the angle at each point is π − 2πq/p
• as a surface it has thirty-two corners
• cuboctahedron: and lies on a circle, so it is regular
• cuboctahedron: and the walk round a corner visits every face at it once
• cuboctahedron: each face has equal sides
• cuboctahedron: Euler's formula still holds
• cuboctahedron: every edge is the same length
• cuboctahedron: every vertex is surrounded by the same faces in the same order
• each candidate is a turn rather than a reflection
• each polytope is one this family builds: 5-cell, tesseract, 16-cell, 24-cell
• each solid is one this family cuts: truncated-tetrahedron, truncated-cube, truncated-octahedron, cuboctahedron, icosidodecahedron, truncated-icosahedron
• every turn has an axis to be about
• exactly five vertex figures close up
• icosidodecahedron: and lies on a circle, so it is regular
• icosidodecahedron: and the walk round a corner visits every face at it once
• icosidodecahedron: each face has equal sides
• icosidodecahedron: Euler's formula still holds
• icosidodecahedron: every edge is the same length
• icosidodecahedron: every vertex is surrounded by the same faces in the same order
• no neighbour lies straight along the corner's own axis
• read as twelve pentagrams meeting five at a point, the number is −6
• tesseract: and the edge count agrees with that degree
• tesseract: every vertex is the same distance from the centre
• tesseract: every vertex meets the same number of edges
• the axes account for every turn, the one that does nothing included
• the corner polygon's edge starts shorter than the remaining edge and ends longer
• the faces at a corner close into a ring
• the number of distinct turns is twice the number of edges
• the path visits every point exactly once
• the path winds about the centre q times
• the projected solid has a measurable extent
• the solid is one of the five
• the spiky surface is sixty triangles
• the star has between 5 and 13 points
• the step and the number of points share no factor, so the path closes once
• the turn in four dimensions is at most half a circle
• the view is one the family draws
• truncated-cube: and lies on a circle, so it is regular
• truncated-cube: and the walk round a corner visits every face at it once
• truncated-cube: each face has equal sides
• truncated-cube: Euler's formula still holds
• truncated-cube: every edge is the same length
• truncated-cube: every vertex is surrounded by the same faces in the same order
• truncated-icosahedron: and lies on a circle, so it is regular
• truncated-icosahedron: and the walk round a corner visits every face at it once
• truncated-icosahedron: each face has equal sides
• truncated-icosahedron: Euler's formula still holds
• truncated-icosahedron: every edge is the same length
• truncated-icosahedron: every vertex is surrounded by the same faces in the same order
• truncated-octahedron: and lies on a circle, so it is regular
• truncated-octahedron: and the walk round a corner visits every face at it once
• truncated-octahedron: each face has equal sides
• truncated-octahedron: Euler's formula still holds
• truncated-octahedron: every edge is the same length
• truncated-octahedron: every vertex is surrounded by the same faces in the same order
• truncated-tetrahedron: and lies on a circle, so it is regular
• truncated-tetrahedron: and the walk round a corner visits every face at it once
• truncated-tetrahedron: each face has equal sides
• truncated-tetrahedron: Euler's formula still holds
• truncated-tetrahedron: every edge is the same length
• truncated-tetrahedron: every vertex is surrounded by the same faces in the same order

• a larger exponent gives a ball that contains the smaller one
• a marked third side is inside the range drawn
• a triple whose squares add gives a right angle
• a triple whose squares do not add gives an angle that is not right
• and below one it is not, which is why p < 1 gives no distance at all
• and every one the search found is in the tree
• and half the rectangle — same base, apex on the parallel through the altitude
• and has not appeared before in the tree
• and is primitive
• at nine dimensions the inner sphere reaches the cube's faces
• at ten it is outside them
• consecutive knots are one unit apart
• each fixed ball drawn behind is between 1 and 12
• each leg at the corner is a positive length
• each leg is between nothing and a quarter turn
• each leg of the triangle is a positive length
• each side is a positive length
• each side of the box is a positive length
• every node of the tree is a right triangle in whole numbers
• every point drawn is at distance one in this norm
• four triangles and the tilted square fill the left square
• four triangles and the two upright squares fill the right square
• so the square and the rectangle are equal
• the altitude cuts the big square into exactly these two rectangles
• the angle at C is a right angle
• the angle at the corner is a right angle
• the ball meets the diagonal at 2^(-1/p) in each coordinate
• the box has three positive sides
• the constructed triangle's hypotenuse is the root of the sum of the squares
• the corner is square exactly when the squares add
• the dimensions drawn run from 1 to between 4 and 40
• the drawn third side is c
• the eight triangles are all the same
• the exponent is between 0.4 and 12 — below 1 it is drawn as a warning
• the exponent of the norm is between 0.4 and 12
• the first right face is half the product of its legs
• the fixed balls behind it are between 1 and 12
• the floor diagonal meets the vertical edge at a right angle
• the given triangle and the constructed right triangle are congruent
• the hyperbolic excess in c² is a²b²/3 at small size
• the inner circle touches each corner circle
• the largest triangle drawn has legs under a quarter turn
• the legs are positive lengths
• the long leg is b
• the proof is one the family draws
• the rope carries one knot per unit of its length
• the rope's three sides are whole numbers of knots, each under sixty
• the second right triangle has the floor diagonal and the height as its legs
• the sensitivity at the right angle is c/ab radians per unit
• the short leg is a
• the slanted face's area squared is the sum of the three right faces' areas squared
• the space diagonal squared is the sum of the three squares
• the sphere shortens the hypotenuse and the hyperbolic plane lengthens it
• the spherical hypotenuse is shorter than the flat one
• the spherical hypotenuse satisfies cos c = cos a cos b
• the spherical shortfall in c² is a²b²/3 at small size
• the square on the hypotenuse is c²
• the square on the leg has the leg's area squared
• the three faces at the corner meet at right angles
• the three lengths close into a triangle
• the tilted square equals the two upright squares
• the tilted square is c²
• the tree finds exactly as many primitive triples under 200 as the search does
• the tree is checked against a search up to a hypotenuse between 50 and 2000
• the tree is drawn to between 1 and 3 generations
• the tree of triples is drawn to between 1 and 3 generations
• the triangle is half the square — same base, apex on the opposite side
• the two floor edges meet at a right angle
• the two triangles are the same triangle turned
• with p at least one, the midpoint of any two points of the ball is inside it

• the 3-sided curve is built at the common width ×4
• the 3-sided Reuleaux curve has one width at every angle ×4
• the integral of h round a 3-sided curve is π times the width ×3
• a Reuleaux polygon needs an odd number of sides
• all the shapes have one width
• and is convex
• and it holds for the solid of revolution, measured
• and it is the same in every direction
• and its radius of curvature stays positive, so it is convex
• and lands between the ball and the conjectured minimum
• and the circle's is a quarter of π times it
• and the drawn boundary really is that long
• and the drawn points confirm it, direction by direction
• and the largest is the edge times (√6 − 1)/√2, across two opposite curved edges
• and the Reuleaux triangle the least
• and the smooth family gets only part of the way to the triangle
• and the support function itself is not flat, so the flat sum says something
• and the sweep runs past the point where convexity fails
• and they do so in every direction, not only the one drawn
• at the width the function was given
• between one and four smooth amplitudes, each at most twelve
• between thirty and a hundred and sixty samples per angle
• between twenty and two hundred samples
• between two and five odd side counts, each at most eleven
• Blaschke's relation holds for the ball, which is the check that it is stated right
• convexity fails exactly where h + h″ first vanishes
• every arc bulges away from the centre
• every harmonic in the support function is odd
• every shape in the table has the same width
• ground and plank stay one width apart
• so the solid is not of constant width
• the area falls the whole way, so the minimum is on the constraint boundary
• the boundary is sampled between five hundred and twenty thousand times
• the built curve has constant width
• the circle encloses the most
• the circle is convex
• the four corners are equally spaced
• the generating curve has the stated width
• the harmonic is odd, between three and seven
• the least is about nine tenths of the most
• the plotted sum is flat at the width
• the smallest width is the edge itself
• the smooth curve has the common width
• the spun triangle encloses less than the ball
• the triangle's area is (π − √3)/2 times the square of the width
• the two support distances in opposite directions add to the width
• the view is one the family draws
• the width does not depend on how far it has rolled
• the width is positive
• the width is the arc radius
• though it misses by only a few per cent

• the rings account for the whole disc
• the triangle of base 2πr and height r has the disc's area

• the centre of cell 0 is nearest to site 0 ×20
• a planar triangulation of n points has at most 3n − 6 edges
• a point is under the chord exactly when it is inside the interval
• a site has a cell exactly when its lifted point is a corner of the lower hull
• a site inside the circle is nearer to both ends than the ends are to each other
• a spanning tree has one edge fewer than it has sites
• and at least enough to be connected
• and so the refuted edge is not in the shortest tree after all
• at least one pair of circles actually crosses
• at least one site has no cell, which is what the picture is about
• between 2 and 60 sites, each a point in or near the unit square
• by the largest count drawn, the pairs outnumber the triangulation's edges several times over
• each graph in the chain has no more edges than the next
• each panel is on the side of the threshold the figure claims
• every Delaunay triangle spans a face of the lower hull
• every edge of the minimum spanning tree is an edge of the triangulation
• every Gabriel edge is a Delaunay edge
• every Lloyd step lowers the total squared distance to the nearest site
• every lower face projects to a Delaunay triangle
• every nearest neighbour edge is a shortest tree edge
• every relative neighbourhood edge is a Gabriel edge
• every shortest tree edge is a relative neighbourhood edge
• every site keeps a cell
• every upper face's circumcircle contains every other site
• exactly one of the two triangulations is the Delaunay one
• more than half of the whole fall happens in the first two rounds
• most interior cells settle at six sides, by more than double the next count
• no point beats the centroid
• no triangle's circumcircle holds another site
• the bisector is where the two distances tie
• the cells account for the whole box, with nothing double-covered
• the cells still account for the whole box
• the chain is not a chain of equalities
• the circle on a tree edge as diameter holds no other site
• the circumcircle verdict and the lifted-hull verdict agree
• the cost falls at every round
• the edge chosen to be refuted really does have a site inside its circle
• the farthest-point triangulation uses only sites on the convex hull
• the gap between chord and parabola is −(x − a)(x − b)
• the highlighted cell is one of the sites
• the iteration has actually moved, rather than starting at its own fixed point
• the lift finds as many triangles as the circumcircle test does
• the middle cell is absent at the small weight and present at the large one
• the middle of each cell has that cell's site nearest in the power distance
• the moment about the site is the moment about the centroid plus area times the gap squared
• the settled cells are far closer to equal in area than the scatter's
• the settled diagram holds a larger share of six-sided interior cells
• the tested point is inside the interval, which is what makes the pair illegal
• the triangulation is not empty
• where two circles cross, the two power distances agree

• the point is on the circle
• the point's height is the wave's height there
• the two coordinates are a point of the unit circle

• a member fits inside the tube exactly when the convergence is uniform
• between two and six member numbers, each a whole number up to 400
• each column of the picture settles on the limit
• no point of the interval beats the stated largest gap
• one to three sequences this family draws
• refining the grid finds a larger gap, because the largest is reached nowhere
• the grid finds the largest gap, because it is reached at a point
• the largest gap for x / n falls with n
• the largest gap for xⁿ does not fall with n
• the largest gap for xⁿ never drops below where it started
• the largest gap shrinks exactly when the convergence is uniform
• the members are given in increasing order
• the search for a member inside the tube runs to between 20 and 2000
• the sequence is one this family draws
• the sup distance is plotted for between 6 and 80 values of n
• the tube has a half-width between 0.02 and 0.6
• the view is one the family draws

• A has three to eight corners
• A's corners are all on its own hull, so the set is convex
• and a function that is not convex fails the same test, so the test has content
• and all but a handful of borderline cases are one or the other
• and every corner of the second strictly on the other
• and never more than three
• and positive against the target, which is the refutation
• and the other has more than one
• and the second is outside it
• and with every three assumed, the whole family has a common point
• B has three to eight corners
• B's corners are all on its own hull, so the set is convex
• between ninety and thirty-six hundred directions swept
• between one and six targets
• between six and twenty points in the scatter
• between twenty and four thousand random quadruples
• between two and eight values are averaged
• between two and seven points are averaged
• between two and six touch points, all inside the interval
• both ends of the chord are inside the interval drawn
• every corner of the first set is strictly on one side
• every point is inside the interval drawn
• every point of the hull lies in the hull of three of the points
• every three of the family have a common point
• every two of the family have a common point
• every value is positive and inside the interval drawn
• no line in any of the 720 directions separates the two sets
• no vector is both a non-negative combination and refuted by a direction
• one positive weight per point
• so a refuting direction exists for the second
• some chord of this function dips below it
• the average of the values is at least the value at the average
• the average of the values is at most the value at the average
• the chord runs left to right
• the chord stays above the curve at every sampled point
• the chord stays below the curve at every sampled point
• the conjugate of x² is p²/4, found by search
• the convex function has exactly one local minimum on the interval
• the crescent's hull swallows its own mouth, so the crescent is not convex
• the curve stays above the tangent everywhere, not only nearby
• the direction is non-positive against every generator
• the first target is inside the cone
• the function drawn is a convex one
• the function is one this family draws
• the gap the line leaves is exactly the distance between the sets
• the geometric mean is at most the arithmetic mean
• the great majority of quadruples were non-degenerate
• the mean of the logarithms is at most the logarithm of the mean
• the mean of the logarithms names the geometric mean
• the scatter has a two-dimensional hull
• the search examined some pair of ends
• the search runs over between 20 and 200 candidate ends
• the small set lies outside the crescent
• the transform is drawn for a convex function
• the two generators are independent
• the two means agree exactly when every value is the same
• the two parts' hulls share the point the dependence produces
• the two sets are disjoint
• the view is one the family draws
• the weakened hypothesis is on or off
• the weights add to one
• transforming twice returns the original function
• two functions, side by side
• two generating vectors
• while the sweep does find a wall when one exists, so it is not simply blind
• with only the pairs assumed, the whole family can have nothing in common

• so over 0.7702 exactly the factor is 2 ×4
• a base that is not e does not have slope equal to height
• and at base e the slope is the height
• and is what the closed form gives over that interval
• and the gap left is smaller than the last term added
• and the limit of the staircase is e itself
• and the place where that area is exactly 1 is e
• and the ratio falls towards 1 every step
• both factors are above 1 and their product fits
• both intervals lie inside the integrated range
• each solution starts at a positive height of at most 4
• every partial sum is below e
• more splits is closer to e
• no number of splits reaches e
• so by the end it is under one per cent
• so the two areas add to the area of the product
• the approximation is always an underestimate here
• the area is taken from 1 up to somewhere between 1 and 12
• the area under 1/x from 1 is the logarithm
• the base is a positive number of at most 20
• the curve integrated from the equation is the exponential
• the factor is the same wherever along the curve the interval starts
• the gap closes like one twelfth of 1/n, measured at the last row
• the growth rate is between a twentieth and three in size
• the interval over which the height changes by that factor is inside the window drawn
• the region from a to ab has the same area as the region from 1 to b
• the region runs from 1 to somewhere between 1 and 12
• the table runs to between 4 and 20
• the table shown gets e right to three decimals
• the tangent drawn is the curve's own slope
• the two factors are both above 1 and their product is at most 12
• the view is one the family draws
• the year is split between 2 and 6 ways, each a whole number up to a million

• the 0-th line of the period-2 train sits on the envelope ×94
• harmonic 1 has decayed by exp(−1²t) at t = 0.005 ×18
• sine 1 against sine 2 integrates to nothing ×12
• the 1-term curve is sampled fast enough to be the function rather than an alias ×5
• sine 1 against sine 1 integrates to π ×4
• the lines of the period-2 train are 1/T apart ×4
• the projection onto sine 1 is the harmonic this family draws ×3
• the projection onto sine 2 cancels to nothing ×2
• a longer period puts the lines closer together
• and its finest oscillation is more than three pixels wide
• and the slopes grow faster than the amplitudes shrink, which is the whole condition
• and the terms it leaves out are below a twentieth of what the window shows
• between 5 and 121 harmonics carry the profile
• between 5 and 61 harmonics
• between two and four harmonics, each between 1 and 12
• between two and four periods, each positive
• between two and six times, none of them negative
• by the last frame the profile is a single sine to within two percent
• each window is between a hundred-thousandth and two wide
• each window is drawn from between 200 and 4000 samples
• each window is sampled fast enough for the terms it draws
• every term count is a whole number between 1 and 200
• magnifying ten times does not flatten the curve ten times
• more terms is closer, away from the jump
• the amplitudes shrink
• the coefficients of the sawtooth wave fall like 1/m^1
• the coefficients of the square wave fall like 1/m^1
• the coefficients of the triangle wave fall like 1/m^2
• the construction is the one with no tangents
• the curve drawn is the sum of the harmonics
• the deepest zoom still has something to look at
• the fine detail fades faster than the coarse shape
• the frequencies multiply by an odd whole number
• the pulse fits inside every period drawn
• the spectrum has something in it
• the steepest chord gets steeper as the spacing shrinks
• the view is one the family draws
• the wave is one the family draws

• piece 1 is 0.50 of what was left ×13
• the first 1 terms add to the closed form ×12
• stage 0 has two to the 0 intervals ×8
• and their lengths add to (2/3) to the 0 ×7
• block 1 reaches a half ×4
• and between them they are all 32 strings of 5 bits
• and by the last drawn stage it is already tiny
• and by this stage they already cover most of the interval
• and ends at one
• and it is still coming down towards the limit
• and lists none of them twice
• and no address uses the digit one
• and points of different classes are different points
• and the remainder is the ratio to the power of the number of pieces
• and they add to what the series says
• and what has gone is the geometric series it looks like
• and what is left is exactly the tail
• between 12 and 40 rationals are covered
• between 3 and 7 ternary digits are listed
• between 3 and 9 stages are drawn
• between three and nine translates are drawn
• between three and ten stages are built
• between two and six classes are drawn
• consecutive partial sums differ by the next term
• each address names the left end of its own interval
• each block leans further out than the one below it
• each class is sampled at between 6 and 30 points
• each rational is inside its own interval
• every partial sum falls short of the limit
• it starts at nought
• no candidate length puts the union between one and three
• no point is in two classes at once
• no term is used twice
• no two addresses read the same in binary
• no two seeds is a small rational away from another, which would be a typo
• so the translates of the selection do not overlap
• the answer is caught between an even and an odd partial sum
• the bound on each block is exactly a half
• the covered rationals leave no wide gap, so the covering meets every part of the line
• the covering budget is between a hundredth and a half
• the drawn lean is the sum of the drawn steps
• the drawn running total is the partial sum
• the error is smaller than the last term used
• the first middle removed is between a third and an eighth
• the function is constant across each removed interval
• the function never decreases
• the gap to ln n has settled on γ
• the last term is small
• the length left is the product of what each stage keeps
• the lengths add to less than the budget
• the list holds the rationals it was asked for
• the longest surviving interval shrinks at every stage
• the middle-thirds set of the same depth has far less left
• the pieces and what is left fill the square exactly
• the plotted partial sums are the series
• the ratio is strictly between nothing and one
• the rearranged total settles near the number it was aimed at
• the removed intervals so far have the length the construction gives
• the signs are the ones the series has
• the smallest interval is still wide enough to be represented
• the surviving length is bounded away from nothing
• the target is visibly away from the sum the same terms give in order
• the total is not small
• the union is no longer than the sum of the pieces
• the view is one the family draws
• there is one address for each surviving interval
• two different translates never carry the same point of a class twice
• what is left and what has gone add to the whole interval

• at 1 steps the length is still 2 ×7
• the staircase of 2 steps is exactly 2 long ×3
• and by this stage it is already small
• and is nearly nothing by the end
• and its length is nowhere near the curve's
• and the fat one keeps a length in the limit
• each panel is at least as close as the one before it
• each staircase runs closer to the curve than the last
• middle quarter, halving: no cell lies wholly inside the set, so the lower sum is nought
• middle quarter, halving: refining the partition never raises the upper sum
• middle quarter, halving: the upper sum never falls below the set's own length
• middle thirds: no cell lies wholly inside the set, so the lower sum is nought
• middle thirds: refining the partition never raises the upper sum
• middle thirds: the upper sum never falls below the set's own length
• more bars means less error
• no width, no area
• so in the limit the middle-thirds set has no length at all
• so the first upper sum is on its way down to nothing
• the accumulation only ever climbs, because f is positive
• the area between the two shrinks every time the steps are halved
• the drawn partition has between 24 and 240 cells
• the fat construction's first middle is between a third and an eighth
• the finest one is within a twentieth of the curve everywhere
• the function is one the figure knows
• the middle-thirds approximation's length is two-thirds to the power of the stage
• the panels really are in increasing order of rectangles
• the printed total is the area of the bars drawn
• the quoted exact area is the area under this curve
• the set is approximated at between 5 and 14 stages
• the slope of the accumulated area is the height of the curve
• the staircase is built against quarter or diagonal
• the staircase of 1 step is exactly 2 long
• the step counts are between 2 and 5 whole numbers up to 512
• the sweep runs over between three and eight partition sizes
• the sweep runs up to between 4 and 512 steps
• the view is one the family draws
• while the second has already settled on the set's own length

• and the difference quotient closes on u'v + uv'
• and the gap as a fraction of the window falls with it
• and the rough curve's do not settle on anything
• between 1 and 4 mirrored pairs are marked
• between 1 and 4 pairs are marked
• between 2 and 4 windows are drawn
• between 4 and 14 spacings are measured
• both sides grow, so the picture is four pieces and not a signed sum
• both sides of the rectangle are positive lengths
• by the last halving the corner is under a twentieth of the increment
• each magnification is between 2 and 20
• each secant is closer than the last
• each step magnifies by between 2 and 20
• every secant from the left has slope −1
• every secant from the right has slope 1
• the bisection really inverts the function
• the corner's share of the increment falls at every halving
• the curve is one the figure knows
• the first window is at most 4 wide
• the four pieces fill the grown rectangle
• the function increases across the window it is drawn in
• the function is increasing where it is marked, so it has an inverse there
• the function is one the family knows
• the function is one the figure knows
• the gap to the tangent falls by the square of the magnification
• the increment is the two strips and the corner
• the last secant is near the tangent
• the marked point is inside the window
• the pair is one the figure knows
• the rectangle is drawn inside the domain of both functions
• the secants are drawn at a list of 2 to 24 positive offsets
• the smooth curve's chords close on its tangent
• the step is a positive length under 1.6
• the step the rectangle grows by is a positive length under 1.6
• the two sides disagree, so there is no limit
• the two slopes at a matched pair multiply to one
• the view is one the family draws

• the polynomial passes through node 1 ×21
• the degree-1 sum matches to order 1 at the expansion point ×7
• the degree-2 sum about 1.2 matches the function to order 2 there ×6
• as x shrinks, the point settles at 1/(n+2) for degree 1 ×4
• the degree-2 sum about -1.4 matches the function to order 2 there ×3
• between 3 and 41 nodes ×2
• a degree between 1 and 20
• an amplification factor is never below one, since the nodes reproduce themselves
• and by the largest count drawn it is orders of magnitude worse
• and it dwarfs anything in the middle of the interval
• between one and five degrees, each between 1 and 8
• between one and four degrees
• between one and four singularities, each a point of the plane
• between one and six centres
• between three and fourteen node counts, each between 3 and 25
• Chebyshev nodes keep the error small across the whole interval
• each new centre lies inside the previous disc, so its coefficients are known there
• each node is the projection of its angle
• equally spaced nodes never amplify less than Chebyshev ones
• every term count is a whole number between 1 and 40
• inside the new interval more terms help
• inside the radius, more terms help
• ln(1+x) is centred where it is defined
• ln(1+x) is expanded inside its own radius
• no centre sits on a singularity
• no singularity lies strictly inside a disc
• outside it they hurt
• outside it, more terms hurt
• the bound is a bound
• the centre is inside the drawn domain
• the Chebyshev end gaps are several times smaller than the middle one
• the equally spaced constant grows with every step
• the error never exceeds the bound Lagrange's form puts on it
• the evenly spaced nodes have one gap throughout
• the function is one the figure knows
• the function is one whose derivatives the figure knows
• the interval drawn is inside the function's radius
• the interval drawn is not empty
• the interval is between 0.4 and 8 wide
• the mode draws the functions whose derivative is monotone
• the mode of the taylor family is one of sums, radius, remainder, xi, centre, discs, interp, lebesgue, nodes
• the nodes are equally spaced or Chebyshev
• the point moves steadily in one direction as x grows
• the radius is the distance to the nearest singularity
• the smallest gap is at one of the ends
• the tail of the series is the difference between the function and its partial sum
• the unknown point lies strictly between nought and x
• with equally spaced nodes the worst error is out near the ends

• and the reassembled rectangle has the same area
• the four tiles are the square
• the larger length is larger
• the length really solves the equation
• the missing corner is (b/2)²
• the negative root solves it too
• the picture is about positive lengths
• the positive root is the one a length can be
• the square tile is x²
• the strips are bx however they are cut
• the three pieces and the corner make a square of side x + b/2
• the two pieces are what is left of a² after b² is removed
• which is completing the square, as arithmetic
• which is the identity

• the root at step 0 first returns to 1 after 1 multiplications ×26
• root 0 raised to the 5 is 1 ×24
• the coefficient of x^0 in the product is the one in xⁿ − 1 ×16
• and that number is the Möbius function of 1 ×12
• the primitive roots of order 1 add to a real number ×12
• the factor belonging to the roots of order 1 has degree φ(1) ×8
• the curve the image of the circle of radius 0.4 never passes exactly through the origin ×7
• the image of radius 0.4 turns once for each root inside it ×7
• the image of the circle of radius 0.4 closes up after a whole number of turns ×7
• the walk along the image of the circle of radius 0.4 is fine enough to see which way it turned ×7
• the circle of radius 0.4 has 0 roots inside and turns 0 times ×4
• the curve the image of radius 0.4 never passes exactly through the origin ×4
• the image of radius 0.4 closes up after a whole number of turns ×4
• the walk along the image of radius 0.4 is fine enough to see which way it turned ×4
• far out, the image turns once for each power of z, which is 3 times ×2
• a polynomial of degree 3 has 3 roots
• all 12 roots add to zero
• and close in it does not go round the origin at all
• and has no imaginary part left
• and it crosses between them
• and multiply to the constant term, with a sign that follows the degree
• and the angles add
• and the degrees of the factors add to n, because every root has exactly one order
• and their imaginary parts cancel
• at a place where the discriminant is passing through zero
• each root found really is a root
• each step of the division gives a whole number
• every root found really is a root
• every root is on the unit circle
• in both coordinates
• the division leaves no remainder
• the equation is zⁿ = 1 for n between 4 and 20
• the factors multiply out to a polynomial of the right degree
• the lengths multiply
• the number of powers drawn is between 2 and 24
• the number of powers taken is between 1 and 24
• the number of roots of unity is between 2 and 24
• the polygon drawn has between 3 and 16 corners
• the polynomial has a degree
• the polynomial has a degree between 1 and 6
• the radii drawn are between 2 and 5 positive numbers no larger than 4
• the roots add to minus the next coefficient over the leading one
• the roots sum to zero
• the sign of the discriminant says how many roots are real
• the sums take all three of the values that function takes
• the sweep meets both cases
• the sweep takes between 20 and 800 samples
• the swept family is a depressed cubic with a negative linear term
• the table of sums runs to between 6 and 16
• the table runs to between 6 and 16
• the view is one the family draws
• with nothing imaginary left over

• a shared root is found exactly when the determinant says there is one
• all 2 roots of the eliminated polynomial are accounted for
• both polynomials have a degree of at least one
• each step of the division lowers the degree
• every root of the eliminated polynomial is the abscissa of a real crossing
• every zero of the swept determinant is a parameter at which a root is shared
• the determinant equals the second polynomial evaluated at every root of the first
• the determinant vanishes exactly when the two share a factor
• the eliminated polynomial is the resultant at every sampled x
• the fixed polynomial is an integer quadratic
• the pair of curves is one this family knows
• the resultant of a polynomial and its slope vanishes exactly at a repeated root
• the sweep runs upward over a range of 1 to 20
• the view is one the family draws
• two pairs of integer polynomials, each of degree 1 to 3
• two to four integer cubics

• a subgroup of 1 divides the group of 8 ×8
• S3 has the number of elements it should ×3
• the elements reachable in 1 steps are exactly those at distance 1 or less ×3
• e leaves 16 colourings alone, one per cycle coloured freely ×2
• exactly 8 of the 24 relabellings preserve every distance ×2
• m₁ leaves 8 colourings alone, one per cycle coloured freely ×2
• m₂ leaves 4 colourings alone, one per cycle coloured freely ×2
• m₃ leaves 8 colourings alone, one per cycle coloured freely ×2
• m₄ leaves 4 colourings alone, one per cycle coloured freely ×2
• r leaves 2 colourings alone, one per cycle coloured freely ×2
• r² leaves 4 colourings alone, one per cycle coloured freely ×2
• r³ leaves 2 colourings alone, one per cycle coloured freely ×2
• the classes hold all 16 colourings between them and share none ×2
• a subgroup whose blocks differ has an element that shows it
• A5 has the number of elements it should
• and no element is in two of them
• and none of them is the identity, which would generate nothing
• and the second subgroup's are not — which is what makes it the control
• and the two maps are genuinely different graphs
• and the whole group is reached
• between one and four groups this family knows
• between one and four permutation groups this family knows
• between one and three generators are named
• both maps are of the same group
• column e does too
• column m₁ does too
• column m₂ does too
• column m₃ does too
• column m₄ does too
• column r does too
• column r² does too
• column r³ does too
• e preserves every distance between corners
• each step's size divides the last
• every block is the size of the subgroup
• every class has a size dividing the group's
• every composition of two motions is again one of the motions
• every element sits in exactly one shell
• every generator names an element of the group
• every named element is in the group
• m₁ preserves every distance between corners
• m₂ preserves every distance between corners
• m₃ preserves every distance between corners
• m₄ preserves every distance between corners
• m₅ preserves every distance between corners
• m₆ preserves every distance between corners
• multiplying on the left carries every edge to an edge
• no two of the motions do the same thing to the corners
• r preserves every distance between corners
• r² preserves every distance between corners
• r³ preserves every distance between corners
• r⁴ preserves every distance between corners
• r⁵ preserves every distance between corners
• row e contains every element exactly once
• row m₁ contains every element exactly once
• row m₂ contains every element exactly once
• row m₃ contains every element exactly once
• row m₄ contains every element exactly once
• row r contains every element exactly once
• row r² contains every element exactly once
• row r³ contains every element exactly once
• so the subgroup's size times the number of blocks is the group's size
• some group in the table does not reach the identity
• some pair of motions gives a different result in each order
• the average number left alone is the number of classes
• the block a product lands in does not depend on which elements were picked
• the blocks between them cover the group exactly once
• the classes are drawn for at most 64 colourings
• the commutators generate a subgroup of the group they came from
• the conjugacy classes account for every element exactly once
• the corners are coloured in between 2 and 4 colours
• the derived subgroup is carried into itself by every conjugation
• the elements are closed under composition
• the even symmetries of five letters have no normal subgroup in between
• the exhaustive search is drawn for at most 5 corners
• the first has as many blocks as its index
• the first subgroup is named by element labels
• the first subgroup's left and right blocks are the same blocks
• the generators are distinct
• the group is one this family knows
• the group is the dihedral one or the cyclic one
• the identity is a class of its own
• the m₁ and m₂ generate the whole group
• the named elements are closed under composition
• the permutation group is one this family knows
• the permutations are of between two and five letters
• the polygon has between 3 and 8 corners
• the r and m₁ generate the whole group
• the r generate the whole group
• the second is too
• the series never grows
• the subgroup is named by between one and eight element labels
• the trivial subgroup and the whole group are both among them
• the two subgroups nobody has to look for are among the ones found
• the view is one the family draws
• the word e lands on e
• the word m₁ lands on m₁
• the word m₁·m₂ lands on r³
• the word m₁·m₂·m₁ lands on m₄
• the word m₁·m₂·m₁·m₂ lands on r²
• the word m₁·r lands on m₄
• the word m₁·r lands on m₅
• the word m₁·r lands on m₆
• the word m₁·r·m₁ lands on r⁴
• the word m₁·r·m₁ lands on r⁵
• the word m₁·r·r lands on m₄
• the word m₁·r·r lands on m₅
• the word m₂ lands on m₂
• the word m₂·m₁ lands on r
• the word m₂·m₁·m₂ lands on m₃
• the word r lands on r
• the word r·m₁ lands on m₂
• the word r·r lands on r²
• the word r·r·m₁ lands on m₃
• the word r·r·r lands on r³
• the word r·r·r·m₁ lands on m₄
• the word r·r·r·r lands on r⁴
• the word r·r·r·r·r lands on r⁵
• they differ in something a reader can see — the number of steps across, or the number of arrows
• two generating sets are compared

• and the 5th power agrees with 5 multiplications ×2
• a map that keeps both dimensions has a non-zero determinant
• a map that loses a dimension has determinant zero
• a power of the matrix is between 2 and 24
• a real 2×2 map has two eigenvalues counted with sign
• a skew matrix exponentiates to a rotation, first entry
• a symmetric matrix's eigen-directions are perpendicular
• a touching pair and a free pair were both found
• adding a multiple of one column to another changes nothing
• an ellipse of area past four times the determinant does hold a non-zero point
• an odd number of grid lines between 3 and 9
• an odd number of grid lines between 3 and 9, so one runs through the point
• an unsymmetric one's are not, which is what the panel is for
• and agrees with the expansion along a row
• and at a root the shifted map crushes something to nothing
• and does not already point along the answer
• and each direction really is left alone
• and every pair running to the other finishes does cross
• and every point at all lands on one line
• and exponentiating the two eigenvalues gives the same matrix
• and it keeps area exactly
• and its size is the size of that number
• and its two eigen-directions are genuinely different
• and moves the permanent
• and multiply to the determinant
• and reaches it on the smallest patch drawn
• and so are their images
• and so does the triple product of the columns
• and the angle shrinks by the ratio of the two eigenvalues
• and the direction it occurs in is an eigen-direction, to within one sample
• and the identity leaves the unit square alone
• and the shortest is 1/√λ for the larger
• and the shortest is the second
• and there is a direction it sends to the origin
• and where it has got to is what that rate predicts
• AᵀA is symmetric and has two real eigenvalues
• Av is λv in the first coordinate
• Av is λv in the second
• both eigenvalues are positive, so the level set is a closed curve
• each matrix in a row is four finite entries under 8 in size
• each root really is a root
• every choice of struck row gives the same number
• every corner of the image stays inside its panel
• every pair was classified once
• every point drawn satisfies xᵀAx = 1
• every point of that line lands on the origin
• every start reaches every finish
• every step brings the arrow nearer the dominant direction
• every tree found has one edge fewer than it has vertices
• fourth
• in both coordinates
• one dimension survives and one is lost, and they add to the two started with
• one eigenvalue is strictly the largest
• P D P⁻¹ is the map it started from
• scaling one column scales the area by the same factor
• second
• so relative to the patch it falls by two
• so the determinant counts the pairs that never meet
• swapping two columns turns the sign over
• the area factor of e^A is e to the trace of A
• the area factor of the image closes on |det J| as the patch shrinks
• the area multiplier is zero
• the area of the drawn parallelogram is ad − bc
• the arrow it starts from is a pair of numbers with a length
• the arrow takes at least three steps before it arrives
• the basis is four finite entries no larger than four
• the basis spans an area rather than a line
• the camera angles are within a half turn
• the change of basis is invertible
• the column is scaled by between one and three
• the counting window is between 2 and 8 across
• the crossing pairs are matched exactly by the pairs running to the other finishes
• the curved map is one the family knows
• the determinant is the number of spanning trees, counted
• the determinant of the product is the product of the determinants
• the drawn area is the determinant
• the ellipse has two positive radii no larger than four
• the expansion along a row agrees with the sum over permutations
• the first matrix of a product is four finite entries under 8 in size
• the flow runs for a positive time of at most 4
• the flow runs for between 0 and 4 units of time
• the flow starts from between three and twelve points
• the flow's own arrival points are the columns of the series' answer
• the form has two real eigenvalues
• the form is positive in every direction
• the four terms of the expansion add to ad − bc
• the graph is one the family knows
• the image line has a length to normalise
• the image of the unit square has the determinant's area
• the iteration runs between 2 and 40 steps
• the lattice is drawn far enough out to fill the counting window
• the lattice is drawn out to between 3 and 12 steps
• the longest image measured on the ellipse is the first stretch
• the longest radius measured on the curve is 1/√λ for the smaller eigenvalue
• the map does not crush the arrow to nothing at any step
• the map does not flatten the circle to a segment
• the map has two real eigenvalues
• the map has two real eigenvalues to build a basis from
• the map is one the family knows
• the maps drawn have different ranks
• the matrix drawn here collapses the plane onto a line
• the matrix is four finite entries no larger than four
• the matrix is four finite entries under 8 in size and is not all zero
• the matrix is three rows of three finite entries no larger than four
• the matrix is three rows of three whole numbers under ten
• the measured Jacobian is the analytic one
• the multiple is a non-zero whole number no larger than four
• the patch is at most 1.2 across
• the paths counted by hand agree with the binomial
• the permanent of the pattern counts the ways to pick one entry per row and column
• the point is a pair of coordinates
• the points in a window are its area over the determinant, up to the boundary
• the residual falls by four at every halving of the patch
• the roots add to the trace
• the row operation adds one row to a different one
• the row operation leaves the determinant exactly where it was
• the second is too
• the series has converged
• the shifted determinant and λ² − (trace)λ + determinant agree
• the signed sum is the determinant
• the smallest ellipse holding a lattice point has area at most four times the determinant
• the starting arrow has a length
• the starting points are between three and twelve pairs of numbers
• the struck row is one of the vertices
• the three columns are not flat, or there is no solid to draw
• the two directions in the circle are perpendicular
• the two stretches are equal, so every direction is stretched alike
• the two stretches multiply to the area factor
• the unsigned sum is at least as large as the signed one
• the view is one the family draws
• the λ window is a positive width up to 20
• third
• two different roots give two independent directions
• two equal columns leave no area at all
• two finishing points, on a small grid
• two starting points, on a small grid
• what survives and what is lost add to two

• basis vectors 1 and 2 are perpendicular ×12
• the polynomials of degree 0 and 1 are exactly perpendicular ×10
• the polynomial of degree 0 is monic ×5
• basis vector 1 has length one ×4
• the degree-0 result is the Legendre polynomial scaled to be monic ×4
• vector 1 is rebuilt from its own row of the table ×4
• moving coefficient 0 by -0.25 makes the total worse ×3
• moving coefficient 0 by 0.25 makes the total worse ×3
• the 1th vector uses no direction built after it ×3
• the residual is perpendicular to the column of x^0 ×3
• vector 1 is two finite coordinates no larger than twelve ×2
• a non-zero vector has positive weighted length
• a rational is two whole numbers
• a squared length is never negative
• a·b is |a||b| cos θ
• a·b is the shadow's length times |b|
• and is not perpendicular in the ordinary sense unless the weights agree
• and it is linear in its first slot
• and so has the second
• and to the second
• between one and three stages
• between three and forty points
• between two and five columns
• between two and four vectors go in
• both weights are positive and no larger than twenty-five
• degree between one and five
• each column is three finite coordinates
• every neighbouring point of the plane is further from the target
• every point is a finite pair
• every point of the drawn curve has weighted length one
• every vector has the same number of finite coordinates
• every ε is between nought and one
• no division by nought
• no sampled value falls below the vertex
• on the smallest ε drawn, the classical order loses orthogonality outright
• removing a shadow never lengthens a vector
• the camera angles are within a half turn
• the classical order is never the better of the two
• the degree is 1, 2 or 3
• the dropped line meets b at a right angle
• the first basis vector has been scaled to length one
• the first vector is not zero
• the inequality holds for the weighted product too
• the least value is zero exactly when the two vectors are parallel
• the mode of the projection family is one of shadow, signs, schwarz, gram, qr, least, columns, weight, orthloss, legendre
• the modified order stays near the arithmetic's own precision
• the normal equations have a solution
• the nudged fit has one coefficient per column
• the nudged line is worse than the fitted one
• the pair still spans the same area, so it spans the same plane
• the parabola's least value is |a|² − (a·b)²/|b|²
• the plotted span is between 0.2 and 6
• the quadratic cannot have two distinct real roots
• the residual is perpendicular to the first column
• the second drawn vector is perpendicular in the weighted sense
• the sign of a·b follows the angle
• the target is three finite coordinates
• the target's squared length splits into the projection's and the residual's
• the two columns are independent, or they span a line rather than a plane
• the two vectors are not parallel, or there is nothing left after the subtraction
• the vector being bounded is two finite coordinates no larger than twelve
• the vector it is measured against is two finite coordinates no larger than twelve
• the vector projected onto is not zero
• the vectors are independent, so nothing vanishes at its own step
• the weighted product does not care about the order
• there are at least as many coordinates as vectors
• there are more points than coefficients, or nothing is being fitted
• two columns span the plane
• two vectors go in
• what is left after the subtraction is perpendicular to the first vector

• the units 1 and 2 lie on exactly one line ×21
• 24 of the 64 ordered pairs of units fail to commute
• 3 and 5 are each a sum of three squares and 15 is not, so no three-square identity exists
• a labelling exists on which every line reads in its own multiplication order
• a left multiplication and a right multiplication commute
• a product of whole octonions is whole
• a product of whole quaternions is whole
• a quaternion and its negative perform the same rotation
• all seven units are placed
• and every non-zero octonion still has an inverse
• and every one of them has length one
• and every one of them has squared length two
• and every point of a fibre lies over the same point of the sphere
• and every root sees the same profile of neighbours as every other
• and in the ring with halves it is closer than one, which is a division algorithm
• and its determinant is one, so it turns rather than reflects
• and most of every circle is still drawn, so the breaks read as crossings
• and no two circles touch
• and none of them has two arguments the same, which is alternativity
• and one hundred and twenty-eight have every coordinate a half
• and one is its negative
• and the five classes account for every root
• and the other six are straight
• and the quotient that achieves it is one of the halves
• and the rotations they perform number twelve, two units to each
• and the trace of its square gives the same two angles again
• and the twenty-four sit inside the hundred and twenty five times over
• and the two-square identity is the same statement in the plane
• and their real parts take nine values
• and they are twenty-four different quaternions
• and they are two hundred and forty different vectors
• and tilts within a right angle
• between two and eight circles are drawn
• conjugating a pure quaternion leaves it pure
• doubling the complex numbers gives Hamilton's own multiplication
• each circle is sampled at between 60 and 400 points
• each composite is a rotation rather than a reflection
• each unit has eight others at distance one, which is the 24-cell's vertex figure
• every icosian is counted once
• every number up to the bound is a sum of four squares
• every pair of circles is linked exactly once
• every pair of roots meets at an inner product of −2, −1, 0, 1 or 2
• every point of a fibre is on the unit three-sphere
• exactly one of the seven lines is the drawn circle
• fifty-six roots stand at sixty degrees to any given one
• i j k = −1, which is the rule the rest follows from
• in the whole-coordinate ring the quotient is a full unit away, so no remainder is smaller than the divisor
• its centre is exactly plus and minus one
• negating both quaternions gives the same rotation
• negating only one of them does not
• no drawn circle comes near the point the projection removes
• no two vertices project onto one another
• one full turn of the object leaves the quaternion at −1
• one hundred and twenty-six stand at a right angle
• one of them is the identity
• reflecting one root in another lands on a root
• sixty-four straddle the two halves
• some strand passes behind another, which is what makes the picture a diagram
• some triples of units fail to associate
• the associator changes sign when two of its arguments are swapped
• the base points sit at a latitude strictly inside the poles
• the columns of the map are orthonormal, so lengths are kept
• the eight squares of the product add to the product of the two sums of eight squares
• the first quaternion is four whole numbers no larger than twenty
• the four squares of the product add to the product of the two sums of four squares
• the highlighted line is one of the seven, or none
• the highlighted product is ij, ji, squares, or none
• the icosians number one hundred and twenty
• the left quaternion is four numbers
• the left quaternion is meant to have length one
• the lift really does send i to the base point
• the multiplication is associative on every triple of units
• the numbers that really need four squares are the ones Legendre's condition names
• the octonions are not even associative
• the plane has seven lines
• the quaternion has length one
• the quaternions do not commute
• the reals commute
• the right quaternion is four numbers
• the right quaternion is meant to have length one
• the rotation after one full turn is the identity again
• the second quaternion is four whole numbers no larger than twenty
• the solid has ninety-six edges
• the sums of squares are checked between 20 and 400 far
• the sweep is taken in 8 to 24 steps, a multiple of four
• the system is symmetric under negation
• the tower stops at dimension 1, 2, 4 or 8
• the trace of the matrix is twice the sum of the cosines of the two angles
• the turn is between a tenth and a half of a full circle
• the two blocks are orthogonal
• the two blocks are shaded or not
• the two orders give different quaternions
• the two rotations move at least one axis a long way apart
• the units are closed under multiplication
• the view is one the family draws
• the view is tilted between minus ninety and ninety degrees
• the view turns by a real angle
• there are twenty-four units
• there are two hundred and forty roots
• they are closed under multiplication
• thirty of them are half-turns
• twenty-four in the last four
• twenty-four of them are the Hurwitz units
• twenty-four roots live in the first four coordinates
• two different axes from i, j, k
• two full turns bring it back to 1
• two-dimensional inputs give a two-dimensional answer, which is why the plane closes
• which are themselves closed under multiplication
• which is all of them, counted twice by two different rules

• the homomorphisms to the cyclic group of order 2 number 2 ×5
• the class of shape 3+1 has 8 members by the formula too ×4
• the class of shape 2+1+1 has 6 members by the formula too ×3
• all 24 permutations are listed ×2
• all 24 permutations of 4 places are listed ×2
• the class of shape 1+1+1+1 has 1 members by the formula too ×2
• the class of shape 4 has 6 members by the formula too ×2
• a single closed tour moves the tiles in one cycle
• a tour of 4 squares cycles 3 tiles
• adding a detour changes the crossing count by an even number
• and at least one target admits the sign
• and exactly two of the squares are unit squares
• and its sign is what a cycle of that length has
• and no face turn changes the joint parity of the two permutations
• and one of the two is the sign
• and the half they generate is the even permutations
• and the other is the odd side
• and the sign is constant on it, which is why the sign is a function of the shape
• and the squares of their dimensions add to the size of the group
• and they are the same size
• because a single exchange is odd
• between five and two hundred scrambles
• between twenty and two thousand scrambles
• each scramble is between five and two hundred turns
• each scramble is between ten and four hundred moves
• each step of the tour moves the blank one square
• every board one legal move from a reachable one has the same parity
• every one of the eighteen turns was tried at every position
• every reachable arrangement has the same sign-and-blank-distance parity
• exactly half of the 720 arrangements can be reached
• exactly half the permutations are even
• exactly two of them are one-dimensional
• exchanging two tiles puts the board on the other value of the invariant
• no face turn flips the edges as a whole
• no face turn twists the corners as a whole
• no legal move leaves the component it starts in
• one component is the even side
• one corner twisted breaks its own invariant
• one corner twisted breaks only its own invariant
• one edge flipped breaks its own invariant
• one edge flipped breaks only its own invariant
• swapping two places flips the sign of every permutation
• swapping two tiles leaves the puzzle unsolvable
• the blank comes back to where it started
• the board is between nine and sixteen squares
• the board is between two and twelve squares
• the board is small enough to exhaust — at most six squares
• the characters are orthonormal under the group's own average
• the class of shape 1+1+1+1+1 has 1 members by the formula too
• the classes account for every permutation exactly once
• the commutators generate exactly half the group
• the commutators themselves are inside the subgroup they generate
• the cyclic groups tested have orders between two and eight
• the detour is between 1 and 3 extra loops
• the even classes together are exactly half the group
• the group is on three, four or five places
• the identity's column holds each representation's dimension
• the invariant is the same at both ends of the tour
• the long decomposition is the same permutation
• the matrix is 3×3 with whole entries no larger than 20
• the number of swaps has the parity of the crossing count
• the permutation sends each of its 4 places somewhere different
• the reachable side has twelve arrangements
• the route is between three and nine squares of the board
• the second decomposition is 1 to 3 swaps longer, in pairs
• the short decomposition really is this permutation
• the sign of the arrangement alone does change, so it is not by itself the invariant
• the signed sum over permutations is the determinant
• the solved board has parity zero
• the squares are drawn for three, four or five places
• the squares of the dimensions add to the size of the group
• the strings cross exactly as often as the pairs are out of order
• the table is between 3 and 8 columns wide
• the table is drawn for 2, 3 or 4 places
• the table is drawn for three, four or five places
• the tour starts from the blank's home square
• the two components account for every arrangement
• the view is one the family draws
• the whole graph is drawn only for the two-by-two board
• there are exactly as many irreducible characters as conjugacy classes
• two decompositions of one permutation have the same parity
• two pieces exchanged breaks its own invariant
• two pieces exchanged breaks only its own invariant
• with nothing odd among them

• round 1 still has a matching covering every vertex ×3
• and every neighbour named is a vertex on the right
• and no vertex is matched twice
• and so does every vertex on the right
• and the reason is a set with too few neighbours between them
• and the rounds use up every edge exactly once
• between two and four graphs the family knows are compared
• every edge gets exactly one colour
• every matched pair is an edge of the graph
• every vertex on the left has a neighbour list
• every vertex on the left has the same degree
• the graph is one the family knows
• the largest matching is exactly the left side less the worst deficiency
• the table has both a graph that succeeds and a graph that fails
• the view is one the family draws
• this view is for a graph that has no complete matching
• this view is for a graph whose matching covers the whole left side

• the walk reaches m × n / gcd cells on a 3 by 5 grid ×3
• a dial needs at least three positions
• a row is a permutation exactly when its multiplier is coprime to the modulus
• every non-zero row is a permutation exactly when the modulus is prime
• every residue has a place on the dial
• every row of the addition table is a permutation
• it fills the whole grid exactly when the moduli are coprime
• multiplying `order` times returns to 1
• the arithmetic and the walk agree
• the drawn entry is the arithmetic
• the drawn walk passes the top as often as the division says
• the first modulus is between 2 and 12
• the multiplier is not zero
• the orbit visits each residue once before closing
• the order divides m − 1, which is Fermat's little theorem
• the period is the least common multiple
• the second modulus is between 2 and 12
• the walk is between one and 144 steps
• the walk runs at least one full period

• on 3 points, avoiding a complete graph on 3 allows Turán's count ×7
• there is some n at which the expected count of 4-sets is below one ×5
• and the codes are all 3 sequences of length 1 ×4
• on 3 points no two trees share a code ×4
• K4: the counting bound and the best drawing found agree about whether it lies flat ×3
• and 6 is below the known R(4,4) of 18 ×2
• K5: the best drawing found has one crossing ×2
• the most triangle-free edges on 6 points is ⌊n²/4⌋ ×2
• there are exactly n to the n minus two trees on 4 labelled points ×2
• a tree is given as a list of pairs of points
• a tree with more than one edge always has a leaf to strip
• a vertex meets five others
• and every pair gets one colour either way round
• and in fact at least two, which is Goodman's bound
• and it is the edge count of the balanced two-part graph
• and no tree is listed twice
• and the bound for graphs with no triangle rules out K3,3
• and the layout kept is one a reader can follow
• and the next size up is the first with an expectation of at least one
• and the two colours take half the pairs each
• at least one colouring is tried, and at most twenty thousand
• each clique size counted is between 3 and 8
• each clique size is a whole number between 3 and 8
• EVERY colouring of six people contains a monochromatic triangle
• every edge the extremal graph is missing would complete a triangle
• every pair is coloured exactly once
• every tree is drawn for between 3 and 5 points
• five edges in the pentagon
• K5 has ten edges
• K6 has fifteen edges
• minus one is itself a square, so the rule does not depend on which way round the pair is taken
• no triangle has all three edges the same colour
• stretching the drawing to fill its panel changes no crossing
• the bijection is checked up to between 3 and 6 points
• the bound is of the order of 2 to the k over 2
• the code is two shorter than the number of points
• the colouring is drawn on 5, 13 or 17 points, each a prime one more than a multiple of four
• the colouring treats its two colours alike, so the largest sets match
• the dial has a radius between 40 and 400
• the expectation is swept to between 20 and 200 points
• the extremal graph splits the points in two with every edge crossing
• the first complete graph the bound rules out is K5
• the forbidden complete graphs have between three and five points
• the labelled points number between 3 and 7
• the majority colour is one of the two
• the Paley colouring is drawn on 5, 13 or 17 points
• the search found a layout that is not degenerate
• the search runs on between three and six points
• the seed is a whole number the colouring can be drawn from
• the sweep runs to between 20 and 200 points
• the table or the check runs to a size the enumeration can reach
• the table runs to between 5 and 12 points
• the table runs to between four and six points
• the three ends make three pairs, and any one of them closes a trio
• the tree drawn has one fewer edge than it has points
• the tree encoded has between 4 and 7 points
• the view is one the family draws
• this colouring has a monochromatic triangle
• two colours over five edges forces three of one
• two of the three cannot be drawn flat
• while it allows the two that can be drawn flat

• the number of cycles for words of 2 letters matches 2^(2^(n−1) − n) ×3
• the strings of length 4 found come in whole rotations ×3
• and 2 in ×2
• every vertex has 2 edges out ×2
• a closed circuit needs every degree even
• a closed circuit over every edge visits one more vertex than it has edges
• a closed walk over every edge visits one more vertex than it has edges
• all four landmasses have odd degree
• an even degree pairs every arrival with a departure
• an odd degree leaves exactly one edge unpaired
• an open walk needs exactly two odd degrees
• and none of them appears twice
• and one edge per word
• and the greedy rule gives the greatest of those that begin with n zeros
• and the position found is the one the patch was taken from
• each word is smaller than all its rotations
• every vertex drawn is a vertex used
• every window of two consecutive symbols is a different pair
• every word of the right length appears in the cycle
• in the other direction too
• Königsberg has seven bridges
• landmass E has degree 3
• landmass I has degree 5
• landmass N has degree 3
• landmass S has degree 3
• no cyclic sequence of 6 symbols shows every pair from 4 exactly once
• no Euler walk exists
• the alphabet has two or three letters
• the circuit uses every pair once
• the concatenation is a de Bruijn sequence
• the count grows with the word length
• the degrees add to twice the number of edges
• the even panel really shows an even degree
• the example graph is a path or a circuit
• the graph drawn has at most nine vertices
• the graph has one vertex per word one letter shorter
• the greedy rule also produces a de Bruijn sequence
• the ground set has between four and seven elements
• the lengths of the necklaces add to the number of words, which is Witt's identity
• the necklace concatenation is the least de Bruijn sequence
• the odd panel really shows an odd degree
• the patch appears at exactly one position of the array
• the patch is taken from inside the sheet drawn
• the rows are not all de Bruijn sequences in their own right
• the search runs to words of three or four letters
• the sequence is as long as the number of words
• the sixteen positions give sixteen different blocks
• the torus drawn is the four-by-four one
• the view is one the family draws
• the walk uses every edge
• the windows are all different
• the words are between two and five letters long
• the words are between two and four letters long
• the words are listed in lexicographic order

• the road a to d is carrying its full 1 ×2
• and it is also what reaches the sink
• and removing that many roads disconnects the two ends
• each route continues until it reaches the sink
• every road of this network has capacity one
• every way of splitting the middle places is a cut and is listed
• everything arriving at a leaves again
• everything arriving at b leaves again
• everything arriving at c leaves again
• everything arriving at d leaves again
• everything arriving at e leaves again
• everything arriving at f leaves again
• no cut is smaller than the flow, over every cut there is
• no road carries more than its capacity
• no two routes share a road
• the cut is either drawn or it is not
• the network is one this family draws
• the road a to c is carrying its full 2
• the road a to t is carrying its full 1
• the road b to d is carrying its full 2
• the road b to t is carrying its full 4
• the road s to b is carrying its full 3
• the road s to c is carrying its full 3
• the smallest cut and the largest flow are the same number
• the smallest of them equals the largest flow
• the value is what leaves the source
• the view is one the family draws
• there are as many routes as the flow's value

• the coefficient of x^0 is the number of combinations totalling 0 ×10
• the coefficient of x^0 is the number of objects of size 0 ×9
• the coefficient of x^0 is the sum along its diagonal ×7
• and the marked coefficient counts the combinations written out beside it
• between two and four boxes, each offering between two and five whole-number values
• both series are short lists of small whole numbers
• every combination of choices is listed
• the marked coefficient is a small whole power
• the marked coefficient is inside the product
• the marked total is one the boxes can reach
• the sequence is growing, so the check has content
• the sequence is one the family knows
• the series inverted has a non-zero constant term
• the series is drawn to between 5 and 10 terms
• the view is one the family draws

• region 0 touches region 1 ×21
• deleting and contracting account for the count at 0 colours ×6
• a colour has been freed for the middle
• after: no edge joins two regions of one colour
• an odd rim needs four and an even rim three
• before: no edge joins two regions of one colour
• between 3 and 6 rows are drawn
• colours are counted up to between 3 and 7
• every one of the seven regions touches every other
• four colours are enough
• no graph with an edge can be coloured with no colours
• no two neighbouring regions share a colour
• nor with one, when it has an edge
• the case is the one where the chain reaches or the one where it does not
• the chain reaches the far neighbour exactly in the blocked case
• the chain that is swapped does not reach the other neighbour of its pair
• the chromatic number is where the count first becomes positive
• the construction shown is the seven-region one
• the five neighbours use five different colours, which is the only hard case
• the graph is one the family draws
• the map can be coloured with at most five
• the map has between 4 and 24 regions
• the vertical step is 2, 3 or 4
• the view is one the family draws
• the wheel has between 3 and 23 rim regions
• with the middle coloured: no edge joins two regions of one colour

• a marked square holds a prime, and an unmarked one does not
• and 41² is where it fails
• consecutive integers are neighbours on the spiral
• Euler's polynomial is prime for forty values in a row and then is not
• no two integers land on the same square
• the first polynomial really is the prime-richer one over the range drawn
• the sieve starts where the primes do
• there are primes to compare against

• the exponent of 2 in row 0, entry 0 is the number of carries ×756
• row 0, entry 0 of Pascal's triangle ×91
• row 0, entry 0 is the binomial coefficient ×45
• 2 divides the coefficient at least 1 times ×8
• the routes to the cut at height 0 are counted both ways ×4
• and no more than 3 times ×3
• between 6 and 40 rows ×2
• the exponent of 2 in the coefficient is the number of carries ×2
• the run down diagonal 2 adds to the entry at row 7, place 3 ×2
• at most twenty routes are drawn
• between 1 and 6 steps up
• between 1 and 8 steps across
• classifying routes by where they cross the diagonal accounts for all of them
• every entry of the drawn triangle was checked both ways
• k is strictly between nought and n
• n is between 2 and 200
• row 6 adds to two to the power 6
• row 6 with alternating signs adds to nought
• shallow diagonal 8 adds to a Fibonacci number
• the base is a prime between 2 and 11
• the cut is a diagonal strictly inside the grid
• the diagonal is one the triangle holds
• the identity is one the mode draws
• the mode of the pascal family is one of numbers, parity, mod, sierpinski, sums, paths, carries
• the routes counted by the additive rule agree with the coefficient
• the row count is a whole number between 1 and 64
• the row is one the triangle holds
• the row's double is inside the triangle, or the total cannot be pointed at
• the run starts inside the triangle and leaves room for its total
• the shallow diagonal is one the triangle holds
• the squares of row 4 add to the middle entry of row 8
• the subdivision depth is a whole number between 1 and 8

• each case really does overflow its boxes
• every item is in a box
• more items than boxes forces a box with two
• the fullest box holds ⌈items/holes⌉
• there are at least as many things as boxes
• there is at least one box

• nothing survives at 9 either ×7
• a point set in general position was found for every seed
• and all three sizes of hull turned up in the test
• and four of them in convex position
• and its longest fall is n
• and neither colour can be given to the next number without making one
• and the extra term's label leaves the n by n square it would have to fit in
• and the surviving lengths run right up to it with no gap
• between one and four seeded point sets are drawn
• every five points in general position hold four in convex position
• every term carries a different pair of counters
• its longest climb is n
• one more term and a climb or a fall of n+1 appears
• the claim is tested on between 200 and 20,000 point sets
• the colouring drawn contains no such pattern
• the exhaustive search runs to between 5 and 14 numbers
• the extremal sequence has n² terms
• the panels drawn show different hull sizes
• the pattern is a progression or a sum
• the search reaches a length at which no colouring survives
• the search runs to between 5 and 14 numbers
• the sequence is built from between 2 and 5 blocks
• the view is one the family draws

• and it agrees with the closed form at n = 0 ×13
• the coefficient of x^1 is the convolution the equation demands ×12
• C(0) two ways ×9
• a 6-gon has C(4) triangulations ×4
• every triangulation has 3 flips available ×3
• the paths that stay above number the 5-th Catalan number ×3
• the non-crossing pairings of 8 points number C(4) ×2
• a hexagon has fourteen triangulations
• a polygon to be cut into triangles has between 3 and 10 sides
• a triangle that is not rainbow has exactly one other door
• a triangulation of the polygon has three fewer chords than the polygon has sides
• and every path is one or the other
• and one step below it
• and six are pentagons
• and so is the number of doors along the bottom edge
• and stays under four
• and the edge count is the handshake count
• and the one drawn ends in a triangle carrying all three colours
• and the scaled sequence climbs towards one from below
• and the third the third
• and twenty-one flips between them
• at least one corridor from the bottom edge ends in a rainbow triangle
• by the largest term drawn the scaled value is as close to one as Stirling predicts
• each cut uses n−3 diagonals
• Euler's relation holds on the solid these faces make
• every diagram pairs off all the points
• every pairing is drawn, so the point count is small
• every path is drawn, so the grid is small
• every reflected path ends one step right of the corner
• every triangulation of a hexagon has three flips
• every vertex carries a colour its position allows
• no two chords cross
• no two chords of the triangulation cross
• one flip at a time reaches every triangulation
• so the good ones are the difference of two binomial coefficients
• the bad paths number the paths to the shifted corner
• the first corner takes the first colour
• the flip graph is drawn for a polygon of 4 to 6 sides
• the good paths number the Catalan number
• the grid is between 3 and 8 on a side
• the grid the triangle is cut into has side between 3 and 12
• the number of small triangles carrying all three colours is odd
• the reflection is one-to-one on the bad paths
• the second corner the second
• the successive ratio climbs at every step
• the triangle is cut into a grid of side between 3 and 12
• the view is one the family draws
• the walk never enters the same triangle twice
• the word is a balanced one that never goes negative
• the words of this length number the Catalan number
• three of the faces are squares

• V − E + F is 0 for this solid ×3
• a spanning tree has one fewer edge than the graph has vertices
• and 720° is 360° times V − E + F
• and its twelve edges
• between one and three holes are punched
• each unfolded face closes back on the corner
• every edge lies between exactly two faces
• every edge of the cube is shared by two faces
• every edge of the dodecahedron is shared by two faces
• every edge of the icosahedron is shared by two faces
• every edge of the octahedron is shared by two faces
• every edge of the surface is shared by exactly two faces
• every edge of the tetrahedron is shared by two faces
• no leftover edge closes a loop among the faces
• the alternating sum is not 2, on a solid with flat faces and straight edges
• the defects on the cube come to 720°
• the defects on the dodecahedron come to 720°
• the defects on the icosahedron come to 720°
• the defects on the octahedron come to 720°
• the defects on the tetrahedron come to 720°
• the faces at a corner of the cube do not close up flat
• the faces at a corner of the dodecahedron do not close up flat
• the faces at a corner of the icosahedron do not close up flat
• the faces at a corner of the octahedron do not close up flat
• the faces at a corner of the tetrahedron do not close up flat
• the flattened cube keeps its eight corners
• the flattened cube still gives 2
• the leftover edges are what the tree did not use
• the leftover edges join every face to every other
• the leftover edges number one fewer than there are faces
• the mode is one the family draws
• the solid is one of the five regular ones
• the solid is one the family builds
• the tree is grown from one of the graph's own vertices
• the tree reaches every vertex
• the two trees together use every edge exactly once
• V − E + F is -2 for this solid
• V − E + F is 2 for the cube
• V − E + F is 2 for the dodecahedron
• V − E + F is 2 for the icosahedron
• V − E + F is 2 for the octahedron
• V − E + F is 2 for the tetrahedron
• which is V − E + F = 2

• and has no direction to choose there
• and it is inside the disc
• and it lies inside the disc
• and it sits exactly on the rim
• and second
• and sits on it
• and the crossing found is a genuine zero
• and the second
• and the sweep finds points that barely move, near it
• at the left end the map moves the point right or not at all
• at the right end it moves it left or not at all
• between 8 and 120 arrows are drawn per panel
• dipole has the index it claims
• each set is swept at between 200 and 20,000 points
• every arrow is tangent to the sphere
• every orbit runs into the fixed point
• no point of a ring, turned stays where it is, and none comes close
• no point of the whole plane, shifted stays where it is, and none comes close
• saddle has the index it claims
• so every point of the open disc moves
• so it has opposite signs at the two ends of a half turn
• so the sweep crosses zero at least twice
• source has the index it claims
• the annulus is turned by between 5 and 175 degrees
• the control's fixed point is fixed, first coordinate
• the difference reverses sign between opposite points
• the field is one the figure knows
• the field vanishes at the pole
• the fixed point is unique
• the halving map does have a fixed point
• the imbalance changes sign somewhere in half a turn
• the line halves the first shape
• the map is one the figure knows
• the map keeps the interval inside itself
• the map sends the disc into itself
• the pair found really does take the same value
• the point found really is fixed
• the ring is turned by between 5 and 175 degrees
• the solved point is fixed, first coordinate
• the solved point is fixed, second
• the sweep takes between 36 and 2000 samples
• the two shapes cut have between 3 and 12 sides each
• the view is one the family draws
• turning the line through half a turn reverses the imbalance
• which the open disc does not contain

• a sphere with 3 cross-caps has characteristic -1 ×3
• a surface with 2 handles has characteristic -2 ×2
• the classification is drawn up to 3 of them in this view ×2
• a non-orientable surface and an orientable one share the characteristic and differ
• a sphere has characteristic 2
• a sphere with 1 cross-cap has characteristic 1
• a sphere with 2 cross-caps has characteristic 0
• a torus has characteristic 0
• aba⁻¹b⁻¹ gives the characteristic of a torus
• aba⁻¹b⁻¹ has the Euler characteristic a torus has
• aba⁻¹b⁻¹ is orientable
• abab gives the characteristic of a projective plane
• abab has the Euler characteristic a projective plane has
• abab is not orientable
• abab⁻¹ gives the characteristic of a Klein bottle
• abab⁻¹ has the Euler characteristic a Klein bottle has
• abab⁻¹ is not orientable
• abb⁻¹a⁻¹ has the Euler characteristic a sphere has
• abb⁻¹a⁻¹ is orientable
• and every face to a face
• and no edge to itself
• and no face to itself
• and no vertex is its own opposite, so the map moves every point
• and ten faces
• and the quotient's is a projective plane's
• and thirty edges
• and twenty faces
• between one and five cross-caps are drawn
• between one and five gluings are drawn side by side
• edge a is glued to exactly one other edge
• edge b is glued to exactly one other edge
• edge c is glued to exactly one other edge
• edge d is glued to exactly one other edge
• every vertex has its opposite among the vertices
• fifteen edges
• no two of the surfaces share a characteristic
• the antipodal map carries every edge to an edge
• the camera angles are numbers
• the cover's characteristic is twice the surface's, because every cell is doubled
• the drawing scale is sane
• the drawn range reaches at least one characteristic that both sides hold
• the icosahedron has twelve vertices
• the icosahedron's characteristic is a sphere's
• the name follows from the number
• the name follows from the two numbers
• the orientable row is on or off
• the polygon has an even number of edges
• the quotient has six vertices
• the two numbers name the surface without anything else being consulted
• the two surfaces with the same characteristic are told apart by orientability
• the view is one the family draws
• the word glues every edge to exactly one other

• the Borromean rings: the diagram's count and Gauss's integral agree for components 1 and 2 ×3
• the figure-eight knot: 3 colours are available exactly when 3 divides its determinant ×3
• the figure-eight knot: the count with 3 colours ×3
• the trefoil: 3 colours are available exactly when 3 divides its determinant ×3
• the trefoil: the count with 3 colours ×3
• the unknot: 3 colours are available exactly when 3 divides its determinant ×3
• the unknot: the count with 3 colours ×3
• three rings in a chain: the diagram's count and Gauss's integral agree for components 1 and 2 ×3
• a colouring is drawn on a diagram that has crossings
• a colouring uses three, five or seven colours
• a diagram with k crossings is cut into k arcs
• a loop that dips through and back: punctures and crossings agree
• a loop that dips through and back: the diagram's count and Gauss's integral agree for components 1 and 2
• and Gauss's integral, which sees no disc at all, gives it too
• and one whose linking number is bigger than one
• and the one found uses all three
• and their signs add to an even number
• and they reach zero in two different ways — one with no punctures, one with two that cancel
• at every crossing the three arcs are all alike or all different
• at least two of the links have linking number zero
• between three and five deformation sizes, none of them large
• between two and four links the family knows
• each crossing has exactly one of its two strands going underneath
• no single arc passes over every crossing
• the (2, 4) torus link: the diagram's count and Gauss's integral agree for components 1 and 2
• the constant colourings are always allowed, so there are at least p
• the figure-eight knot admits a colouring using more than one of the 5 colours
• the figure-eight knot has the crossings it claims
• the first component of this link is a flat circle, so its disc is the obvious one
• the Hopf link: punctures and crossings agree
• the Hopf link: the diagram's count and Gauss's integral agree for components 1 and 2
• the knot is one the figure knows
• the knots in the table have different determinants
• the link is one the family knows
• the linking number is the same at every deformation
• the punctures added with signs give the linking number
• the strand is broken where it goes underneath, not somewhere else
• the table contains a link whose components cross and whose linking numbers are all zero
• the table is ordered by crossing number
• the table's columns are small primes
• the trefoil admits a colouring in more than one colour
• the trefoil admits a colouring using more than one of the 3 colours
• the trefoil has the crossings it claims
• the trefoil is three arcs
• the two computations agree at this deformation
• the two loops stay clear of each other at every amplitude
• the unknot is a single arc
• the view is one the family draws
• there are nine three-colourings of the trefoil in all
• there are three moves, and Reidemeister proved there are no others
• twice the over-strand equals the two under-strands added, at every crossing
• two circles side by side: punctures and crossings agree
• two circles side by side: the diagram's count and Gauss's integral agree for components 1 and 2
• two closed curves cross an even number of times
• two or three links whose first component is a flat circle
• while the number of crossings is not

• on the 1-sheeted cover, the lift of a loop winding 0 times closes exactly when it should ×24
• generator 1 acts as a permutation of the 3 sheets ×14
• the rank comes out at 1 + 3(2 − 1), which is the index times one less than the rank below ×9
• a loop winding 0 times lifts to a closed path exactly when 0 is a multiple of 3 ×7
• vertex 1 has at most one a leaving it ×4
• vertex 1 has at most one b leaving it ×4
• the loop drawn for class 0 is a whole number ×3
• the loop drawn for class -2 is a whole number ×2
• a class on the torus is a pair of whole numbers, not both zero
• a free group of rank two has four words of length one
• a spanning tree of a connected graph has one edge fewer than it has vertices
• and are not under the other one
• and at most one arriving
• and at the same height
• and by the end the loop has been dragged through the hole
• and consecutive ones differ by exactly one turn
• and exactly one arriving
• and in the second, so it closes up on the torus
• and inside the outer boundary
• and it goes round each hole a net zero times
• and its degree is even
• and its degree is odd
• and the same word cancels to nothing on the wedge, where it stays four letters long
• and the top and bottom as many times as its second
• and their number divides the number of sheets
• and they are those multiples in order
• and twelve of length two
• between one and five words to be refused
• between one and four generator words in a, b and their inverses
• both loops start at the same point of the ring
• carrying the loop along either path leaves its class alone
• each loop on the torus is a pair of whole numbers, not both zero
• each sheet has exactly one edge of each label leaving it
• every cover's counted rank matches the formula
• every point of the fibre projects onto the marked point of the circle
• every sheet is reached by some word, so the cover is connected
• in the disc every stage of the shrink stays inside
• no sample of the loop lands on the point it is wound about
• no step of the walk turns more than a quarter of a turn
• no two deck transformations agree on the base vertex
• over the line, the only loop that lifts to a loop is the one that goes nowhere
• running one loop then the other adds the counts
• so the two maps are not the same map
• so the whole fibre sits over one point
• so there are at most as many symmetries as sheets
• the a's in the word add up to the winding number about that hole
• the b's in the word add up to the winding number about that hole
• the base circle itself does not lift to a loop
• the bouquet below has two or three circles
• the bouquet has between one and four circles
• the bouquet has two to four circles
• the classes drawn are whole numbers no bigger than four
• the classes that lift to loops are the multiples of the sheet count
• the commutator does not cancel down any further
• the commutator has exponent sum zero in the first generator
• the composition of two deck transformations is one
• the cover has between two and five sheets
• the cover has between two and four sheets
• the cover is connected, so its deck group acts on one object
• the cover is connected, so the index argument applies
• the cover is connected, so the sheets are one orbit
• the curve about the left hole is a whole number
• the curve about the right hole is a whole number
• the degree of the map that is not odd is a whole number
• the degree of the odd map is a whole number
• the drawing shows between two and five turns
• the edges outside the tree are exactly the free generators the graph carries
• the first loop is a whole number
• the first map really does send opposite points to opposite points
• the folded graph has at least one independent loop
• the free group outgrows the abelian group of the same rank
• the generator aa is accepted by the folded graph
• the generator ab is accepted by the folded graph
• the generator abab is accepted by the folded graph
• the generator abaB is accepted by the folded graph
• the generator ba is accepted by the folded graph
• the generator bb is accepted by the folded graph
• the hole is between a fifth and three fifths of the ring
• the identity is a deck transformation
• the index of the subgroup is the number of sheets
• the lift ends a whole number of turns above where it started
• the loop at p is a whole number
• the loop being lifted is a whole number
• the loop drawn at p has the class claimed
• the loop drawn for a class really has that class
• the loop lifted goes round between one and four times
• the loop lifted is the one asked for
• the loop lifted winds between zero and six times
• the loop stays clear of the hole
• the loops composed have counts no bigger than three
• the loops tabulated wind between zero and eight times
• the marked point is on the base circle
• the monodromy table is drawn over two generators
• the number of sheets is what one lap advances by
• the path crosses the side edges as many times as its first count says
• the probe's two images are exactly opposite under the odd map
• the second loop is a whole number
• the second map does not, which is what makes it a control
• the shrink is legal to begin with and stops being legal
• the slide is drawn in three to five stages
• the symmetries reach every sheet exactly when there are as many of them as sheets
• the table runs to between three and six sheets
• the table runs to between two and six sheets
• the two run one after the other is a whole number
• the two squares never overlap during the slide
• the view is one the family draws
• the winding number is a whole number
• the word a is refused
• the word aa is refused
• the word ab is refused
• the word b is refused
• the word is one this family realises
• the word read off the curve is the word the curve was built from
• the words that come back are closed under composition
• two or three circles below
• two words land on the same sheet exactly when one undoes the other into the subgroup
• while a word that really is trivial reduces away, so the test can fail
• words of length up to two to four are listed

• and between 24 and 140 the long way
• between none and sixteen cross-sections are outlined
• between three and nine frames are drawn along the band
• carrying the surface round once brings it back on the other face
• going once round the long way returns to the point with the cross-section reversed
• going once round the tube returns to the same point
• the band is drawn at a whole number of half-twists
• the frame comes back the same size — only its sense can change
• the frame returns with a reversed sign exactly when the band is one-sided
• the gluing arrow matches the twist the figure is about
• the mesh is between 16 and 96 round the tube
• the self-intersection circle is on or off
• the tube's distance from the axis leaves the figure-eight room to close
• the view is one the family draws
• u = 0 and u = π are two different places on the surface and one place in space
• which is not the same point unless it is one the reversal fixes

• and one lap leaves one loop while two laps leave two
• the cut is inside the band
• the scissors take one lap down the middle and two off-centre

• none of the 594 pairs of non-neighbouring edges meet ×5
• a pair of horns at stage 1 is clasped, of linking number ±1 ×4
• stage 3 has 2^3 − 1 clasped pairs ×4
• ear clipping: a polygon of 20 corners is cut into 18 triangles ×2
• the image triangulation: a polygon of 20 corners is cut into 18 triangles ×2
• the source triangulation: a polygon of 20 corners is cut into 18 triangles ×2
• a line with a point removed falls into two pieces
• a plane with a line removed falls into two pieces
• a plane with a point removed stays in one piece
• a triangulation of a polygon has one diagonal fewer than it has triangles
• all 16 rays agree on the parity
• and a ray from the centre crosses it an even number of times
• and both insides keep a substantial part of the frame
• and is not linked with that horn's partner
• and the clasp shows as at least two crossings of the projection
• and the steepest chord anywhere grows geometrically as the scale shrinks
• and they disagree on the count, which is the point
• at every angle tried the chords steepen by more than fifteenfold over the scales measured
• between 2 and 36 rays are drawn
• consecutive squares in the visiting order are neighbours
• each square's side is at most 2⁻ⁿ, so the diameters go to zero
• ear clipping: no two triangles overlap
• ear clipping: the triangles account for exactly the polygon's own area
• every cell of the grid is visited exactly once
• every direction gives the same parity
• every direction of ray gives the same verdict at this point
• every simple polygon of four or more corners has an ear to clip
• halving the pitch more than halves the largest disc that fits
• horns from different clasps are not linked with each other
• no two of the kept squares overlap
• some sampled point is inside the curve
• space with a line removed stays in one piece
• the arms of the spiral are between 22 and 70 units apart
• the clipping terminates
• the counts themselves differ, which is why the parity is the claim
• the curve goes twice round its own centre
• the disc drawn touches no edge of the curve
• the drawn ray crosses the surface more than once
• the five-pointed loop crosses itself five times
• the horns shrink geometrically, so their tips converge
• the image triangulation: no two triangles overlap
• the image triangulation: the triangles account for exactly the polygon's own area
• the infinite product is bounded away from zero
• the kept area is the product of the stages' own factors
• the loop drawn goes once round one horn
• the magnification of the second panel is between 3 and 27
• the marked point is inside the frame, given as fractions of it
• the number of stages drawn is between 1 and 4
• the number of stages of horns is between 1 and 4
• the number of terms summed is between 2 and 6
• the outside point is clear of the tube
• the plane comes apart into exactly two pieces, no more and no fewer
• the point is put inside or outside
• the radius stays positive, so no two angles share a point
• the resolution of the flood fill is between 24 and 90
• the sample grid is between 8 and 40 columns
• the sampled inside agrees with the polygon's own area to within six per cent
• the second corridor is the narrower one
• the source triangulation: no two triangles overlap
• the source triangulation: the triangles account for exactly the polygon's own area
• the spiral turns between 1 and 6 times
• the terms roughen faster than they shrink
• the tube is closed up out of two triangles per patch
• the two triangles either side of a diagonal send a point on it to the same place
• the view is one the family draws
• two pitches, both inside the range the family draws

• the image of 0 lies on the circle ×14
• the ray to 0 passes through the pole ×14
• the image of -2 lies on the circle ×13
• the ray to -2 passes through the pole ×13
• a circle missing the pole projects to a circle, to within a fitted residual
• a circle through the pole projects to a straight line
• and no two of them meet
• and the south chart does have one there
• at least two finite points are carried across the projection
• between 2 and 24 rays are drawn — one ray is not a correspondence
• between a hundred and four thousand points checked
• between three and seven fibres, each at a latitude strictly inside the sphere
• dim and rays are read by the ray views alone
• each fibre lies on the unit three-sphere
• each transformation is normalised to determinant one
• every patch has the same area on the sphere
• every two fibres are linked exactly once, counted from the drawn curves
• nearly every sample lay in the overlap
• none of the circles in this list passes through the pole
• only the parabolic map has a repeated fixed point
• the crossing angle is the same on the sphere and in the plane
• the fitted transformation is invertible
• the image is on the ray from the pole through the point
• the image lands on the plane
• the north chart has no value at its own pole
• the projected patches are wildly unequal even though the originals are equal
• the projection is drawn from a circle or from a sphere
• the rotation, read in the plane, is exactly the fitted Möbius transformation
• the three chosen points determine the transformation
• the trace classifies the transformation as the label claims
• the two charts differ by the reciprocal at every point of the overlap
• the two charts miss different points
• the view is one the family draws
• there are pairs to link

• and at three doors it is two thirds
• everybody has it, so everybody positive has it
• nobody has it, so nobody who tests positive has it
• one of the two strategies wins, and only one
• prior is a probability, between 0 and 1
• specificity is a probability, between 0 and 1
• staying wins as often as the first pick was right
• switching gets better with more doors
• switching wins two thirds against a host who knows, and half against one who does not
• the answer rises with the base rate
• the four cells fill the square
• the game needs at least three doors
• the ignorant host's story throws two of the six worlds away
• the ignorant-host variant is only drawn at three doors, where switching has one meaning
• the shaded fraction is Bayes' theorem
• the two strategies exhaust the possibilities

• 22 is still under a half
• 23 is over it
• one person shares with nobody
• the half-way point is 23 people
• the pair count is k choose 2

• at least one needle is dropped
• every needle claimed was drawn
• so a needle crosses with probability 2/π
• the area under half a sine wave is 1
• the box has area π/2
• the estimate is in a plausible range for pi
• the printed estimate is 2Ln / dc

• the wait for a new one at 0 seen is 6/6 ×21
• the 6 waits add to 6 times the harmonic sum ×3
• a run of the game reproduces the expected length
• a walk of 40000 steps spends about the solved share of its time at A
• a walk of 40000 steps spends about the solved share of its time at B
• a walk of 40000 steps spends about the solved share of its time at C
• and comes back to A about as often as the solve says
• and comes back to B about as often as the solve says
• and comes back to C about as often as the solve says
• and it is still climbing at the same rate
• and it really is stationary
• and that sum is bounded, so the shares can be normalised
• and the chance of ending at the top satisfies B = R + Q B
• and the chance of ever coming back is less than one
• and the chance of winning is the share of the stake held
• and the share of wins reproduces the computed chance
• and the traffic round the cycle is not equal in the two directions
• and they are the shares the rule leaves alone
• as much traffic goes from A to B as comes back
• as much traffic goes from A to C as comes back
• as much traffic goes from A to D as comes back
• as much traffic goes from A to E as comes back
• as much traffic goes from B to C as comes back
• as much traffic goes from B to D as comes back
• as much traffic goes from B to E as comes back
• as much traffic goes from C to D as comes back
• as much traffic goes from C to E as comes back
• as much traffic goes from D to E as comes back
• at even odds the expected length is k times N minus k
• at least two chains, so the comparison says something
• away from even odds the chance of winning is the ratio the odds give
• between six and sixty steps are drawn
• between two and four step-up chances are drawn
• every chain named is one the family carries
• every start can go either way
• every state has somewhere to go
• every state holds a share
• iterating from A reaches the same share for A
• iterating from A reaches the same share for B
• iterating from A reaches the same share for C
• no two edge weights are drawn on top of one another
• row A holds probabilities
• row B holds probabilities
• row C holds probabilities
• row D holds probabilities
• row E holds probabilities
• row F holds probabilities
• row G holds probabilities
• row H holds probabilities
• the chain is one of mixing, cycle, reducible, lazy
• the chain is one the family carries
• the chain with the smaller gap really does take longer
• the chance of a win is strictly between nothing and everything
• the chances of leaving A add to one
• the chances of leaving B add to one
• the chances of leaving C add to one
• the chances of leaving D add to one
• the chances of leaving E add to one
• the chances of leaving F add to one
• the chances of leaving G add to one
• the chances of leaving H add to one
• the corroborating run is between 500 and 40,000 games
• the cycle's forward chance is between a half and one
• the cycle's shares are stationary even so
• the cycle's stationary shares are equal
• the distance to stationarity never rises
• the drawing runs out to between 10 and 60 states
• the expected number of steps satisfies t = 1 + Q t
• the expected return to A is one over its share
• the expected return to B is one over its share
• the expected return to C is one over its share
• the game is played over between four and nine totals
• the long-run shares are drawn only for a chain that has them
• the measured decay rate is the second eigenvalue
• the measured mixing time follows the eigenvalue's prediction
• the number of kinds to collect is between 2 and 16
• the panels drawn are genuinely different cases
• the return chances satisfy the rule that defines them
• the rows never move further apart as the power rises
• the run is between 2,000 and 400,000 steps
• the run spends its time in A in the computed share
• the run spends its time in B in the computed share
• the run spends its time in C in the computed share
• the same current flows across every edge of the cycle
• the series is added over between 60 and 4,000 terms
• the share read off the weights is the share the equations give
• the shares add to one
• the solved share for A survives a step
• the solved share for B survives a step
• the solved share for C survives a step
• the stationary distribution adds to one
• the system is not singular
• the traffic between neighbours balances
• the view is one the family draws
• the walk is between a thousand and four hundred thousand steps
• the weights add to the geometric sum they are
• the weights are a square table of between three and six states
• the weights are all one, so the sum grows without bound
• the weights are symmetric and never negative
• the weights grow, so there is nothing to normalise

• the count for threshold 0 agrees with the formula ×7
• every one of the 5040 orders was tried ×4
• the counted best for 5 matches the formula ×4
• between three and twelve sizes, each between 2 and 20000 ×2
• the alternating sum lands on the 9 the search found ×2
• 9 of them leave nothing where it started
• all 24 arrangements are drawn
• an exhaustive point is at most 8 objects
• and comes within a few per cent of it
• and it does better than taking one at random
• and its height there is the same number
• and sends no two objects to the same place
• and settles above the 0.5802 the limit is, rather than falling to 1/e
• and so is the chance of success
• and stays below the limit it is climbing towards
• and the best single threshold secures at least half of it
• and the half is never breached
• and with three, 0.6842
• between three and ten sizes
• each correction crosses the answer rather than approaching it from one side
• each proportion is within one over the next factorial of 1/e
• no online rule beats the oracle
• skipping the certain one is worth exactly one, whatever the setting
• so does the threshold set at the median of the maximum
• the alternating sum reaches the same number
• the arrangement is a permutation of its own places
• the arrangement names every object exactly once
• the best achievable rank rises with the size of the field
• the best rule looks at some of them and not all of them
• the best threshold is heading for one over e
• the best threshold rises with the field
• the case is uniform or tight
• the chance falls as the field grows
• the exhaustive count is drawn for between 3 and 8 objects
• the grid of every arrangement is drawn for at most 5 objects
• the largest field drawn is already close to that limit
• the limit curve peaks at one over e
• the number of objects is in the range this view can enumerate
• the oracle is never beaten
• the rank the rule will accept gets stricter as more remain
• the ratio is drawn up to between 2 and 10 objects
• the settings are between nought and one
• the shortfall approaches the half as the setting shrinks
• the sizes drawn are between 3 and 400
• the standard demanded rises with the number still to come
• the value grid has between 200 and 2000 steps
• the view is one the family draws
• which is far above what the relative-ranking rule achieves
• with one object there is nothing to find, and the search finds nothing
• with three it is 3/2
• with two candidates the best expected rank is 5/4
• with two it wins three times in four

• a path bounces once per row
• and are centred under the funnel
• and ends in a bin the board has
• the exact bin shares add to one
• the path ends over the bin its right-turns name
• the row count is a whole number between 1 and 24

• the connected graphs on 4 points, counted twice by different means ×4
• above the threshold the measured share matches the equation's root
• and below it the largest piece is a vanishing share
• and below the threshold the equation has only the zero root
• and it is nonetheless not connected
• and it stays put to within half
• and the chance of at least one never exceeds the expected count, as Markov requires
• and the triangle's stays wider than it
• and well above it a triangle is nearly certain
• being connected is at most as likely as having no isolated point
• between four and forty samples at each size
• between one and four sizes, each at most six points
• between three and five sizes, each between fifty and twelve hundred
• between twenty and two hundred samples per point
• between two and four edge chances, each strictly between zero and one
• between two and four sizes, each between twenty and a hundred and forty
• connectivity's transition narrows relative to its threshold as the graph grows
• dividing by the two-thirds power is the division that stays put
• each sweep runs from mostly-absent to mostly-present
• every edge present is connected with probability one
• every point is in exactly one piece
• no edges is connected with probability zero
• the chance of being connected rises with p
• the comparison runs on between three and six points
• the component grows with the size, as it must
• the count of isolated points follows n·e⁻ᶜ
• the drawn graphs have between six and twenty points
• the enumeration runs on between two and six points
• the measured triangle count matches the expectation it is supposed to
• the number of points is between three and twenty
• the predicted share solves its own equation
• the range drawn reaches the connectivity threshold
• the sampled graphs have between sixty and three thousand points
• the sampled graphs have between two hundred and three thousand points
• the seed is a whole number the graphs can be drawn from
• the view is one the family draws
• well below the threshold a triangle is rare
• well past average degree one, most of the graph is in one piece

• row 0 of the transition matrix adds to one ×20
• the grid's error in 4 dimensions falls like N^(−2/4) ×5
• in 2 dimensions the grid is the faster ×2
• in 8 dimensions the grid is the slower ×2
• and a run of the chain reproduces it to within a twentieth
• and at four dimensions the two rates meet
• and holds no negative probability
• and is between 5 and 400 sharp
• and is positive everywhere on the interval, so no weight is unbounded
• and is repeated between 40 and 800 times
• and is run for between 2,000 and 200,000 steps
• and so is the weighted one — neither is biased
• and the Halton points' falls faster than that
• and the random one falls like N^(−1/2), whatever the dimension
• and the weighted one is appreciably quieter
• as much probability flows from i to j as back again
• between 32 and 2,048 points are drawn
• each dimension is a whole number between 1 and 12
• each estimate uses between 50 and 5,000 draws
• in 1 dimension the grid is the faster
• so a step leaves the target distribution where it was
• the chain has between 5 and 24 states
• the discarded start is shorter than the run
• the discrepancy is probed on a grid of between 40 and 300 boxes a side
• the grid's error in 1 dimension falls like N^(−2/1)
• the Halton points are more evenly spread than the random ones on this measure
• the peak sits inside the interval
• the plain average is centred on the answer
• the points to fit a slope through are not all at one place
• the proposal is a probability density
• the random points' discrepancy falls like one over the square root
• the seed is a whole number the drawing can be made from
• the spread is measured over between 4 and 200 runs
• the sweep runs over between 3 and 8 sizes
• the sweep runs to between 2^8 and 2^18 points
• the view is one the family draws

• coin summed 2 times has probabilities adding to one ×7
• coin summed 2 times has some spread to speak of ×7
• skew summed 2 times has probabilities adding to one ×7
• skew summed 2 times has some spread to speak of ×7
• die summed 2 times has probabilities adding to one ×6
• die summed 2 times has some spread to speak of ×6
• the sum of 4 divided by 4 has probabilities adding to one ×6
• the sum of 4 divided by 4 has some spread to speak of ×6
• the sum of 4 divided by the root of 4 has probabilities adding to one ×6
• the sum of 4 divided by the root of 4 has some spread to speak of ×6
• the average of 1 draws still has the distribution of one draw ×5
• the sum of 4 has probabilities adding to one ×4
• the sum of 4 has some spread to speak of ×4
• and doubling the draws moves it toward the computed one at 0.3 ×3
• at 1.5 standard deviations the mass outside is within the bound ×3
• the measured rate at 0.3 is never below the computed one ×3
• there are exactly 91 possible histograms of 12 draws over three faces ×3
• the largest sum is a power of two between 16 and 128 ×2
• 0 nine times in ten, 10 otherwise has probabilities adding to one
• 0 nine times in ten, 10 otherwise has some spread to speak of
• 0 nine times in ten, 10 otherwise stays under the bound at every width
• a die has probabilities adding to one
• a die has some spread to speak of
• a fair coin, ±1 has probabilities adding to one
• a fair coin, ±1 has some spread to speak of
• a fair coin, ±1 stays under the bound at every width
• a fair die has probabilities adding to one
• a fair die has some spread to speak of
• a fair die stays under the bound at every width
• and dividing by the root of n leaves it exactly where it started
• and increasing away from the mean
• and it comes within half the bound somewhere
• and its mean is exactly the level asked about
• and none is smaller than that divided by a polynomial in n
• and scaled so that one standard deviation is one unit
• and the more lopsided summand is further from the bell at every n
• and the rate at the mean itself is zero
• and the rate is the information distance from the original to the tilted law
• and the runs are further apart early than late
• and the second panel's settle on the height of the bell curve
• and the variance adds too
• between 2 and 12 runs are drawn
• between 2 and 3 summands are compared
• between 2 and 4 averages, each of at most 64 draws
• between 2,000 and 60,000 averages are formed
• between 3 and 40 draws
• between 3 and 6 sums, each of at most 200 copies
• between two and four levels are marked
• dividing by n shrinks the spread like 1/√n
• each distribution is one the family knows
• each n is between 2 and 200
• each run is between 50 and 5000 draws
• each sum is closer to the bell curve than the last
• each sum is of between 1 and 400 copies
• each window is between half a standard deviation and six
• every distribution swept is one the family knows
• every level is between the mean and the largest value
• every run finishes inside four standard errors of the mean
• five values, unevenly weighted has probabilities adding to one
• five values, unevenly weighted has some spread to speak of
• five values, unevenly weighted stays under the bound at every width
• it is centred at zero
• no histogram is more likely than exp(−nD)
• sixteen dice has probabilities adding to one
• sixteen dice has some spread to speak of
• so one scaling is collapsing
• the average of two Cauchys has the density of one Cauchy, by the convolution integral
• the bell curve convolved with itself is the bell curve again, widened by √2
• the bell curve's exponent differs from the rate by enough to show over the range drawn
• the bound is met at between 1.2 and 5 standard deviations
• the constrained region is not empty
• the constrained set has a probability strictly between nothing and everything
• the constraint asks for a mean above the true one and below the largest face
• the distribution has probabilities adding to one
• the distribution has some spread to speak of
• the distribution is one the family knows
• the event has positive probability at every n drawn
• the exact probability stays under the exponential the rate predicts
• the extremal distribution has probabilities adding to one
• the extremal distribution has some spread to speak of
• the first panel's curves grow taller as they narrow
• the gap for 0 nine times in ten, 10 otherwise falls like one over the square root of n
• the gap for a fair coin, ±1 falls like one over the square root of n
• the gap for a fair die falls like one over the square root of n
• the histograms' chances add to one
• the level asked about is above the mean and below the largest value the summand can take
• the level is between the mean and the largest value the summand takes
• the mass at exactly k standard deviations is the whole of the bound
• the mean of a sum is the sum of the means
• the measured decay rate agrees with the rate function computed from the summand alone
• the measured rate moves towards the smallest relative entropy in the region
• the measured rates come from between 16 and 96 draws
• the rate at a level away from the mean is positive
• the rate function is convex all the way along
• the scaled values are increasing
• the sweep includes a lopsided distribution
• the sweep runs to between 24 and 200 draws
• the tilted distribution has probabilities adding to one
• the tilted distribution has some spread to speak of
• the tilted weights are a distribution
• the view is one the family draws
• three face values
• three positive chances
• three positive chances adding to one
• while a light-tailed average narrows by the square root of the count

• the formula agrees with the enumeration at 0 steps above ×7
• the sum of 4 steps has probabilities adding to one ×4
• the sum of 4 steps has some spread to speak of ×4
• a walk in space escapes, and the simulation sees it
• a walk on a line comes back, and the horizon barely hides it
• a walk on a plane comes back too, but more slowly
• a walk takes a whole positive number of steps
• an endpoint has the parity of the step count
• an even number of steps between 8 and 200
• and impossible with everything
• and moves the start to its mirror image
• and so are the reflected ones
• and the distribution is symmetric
• and the gap to the bell curve closes at every step
• and the variance of its position is exactly the number of steps
• and their spread is the square root of the number of steps
• between 4 and 16 steps are drawn, since every path of them is enumerated
• each exact distribution is of between 2 and 400 steps
• each step moves one place along one axis
• every game drawn ends at a barrier
• every path from below the level has to cross it
• every path is counted once
• every step is drawn
• every step is one place, either way
• every walk is counted once
• folding leaves the endpoint alone
• folding matches the touching paths with the paths from the mirrored start one for one
• ruin is certain with nothing left
• some walk goes outside the envelope, because √n is typical and not maximal
• the distribution sums to one
• the drawn path really does touch the level
• the drawn walk's mean square displacement per step is near one in every window
• the durations satisfy their own recurrence
• the enumerated check runs at an even length between 4 and 16
• the extremes are more likely than the middle, which is the whole surprise
• the least likely outcome is an even split
• the paths ending where they end are counted by a binomial coefficient
• the player starts with something and not everything
• the position has the parity of the step count
• the quoted constant is Pólya's
• the ruin probabilities satisfy their own recurrence
• the spread of the endpoints is of the order of the square root of the steps
• the table holds between 4 and 40 units
• the three windows are whole numbers of steps
• the three windows are whole numbers of steps inside the walk
• the time spent above is always even
• the view is one the family draws
• the walk can reach its endpoint in the steps allowed
• the walk comes back to where it started at least once
• the walk has no drift
• the walk is between 400 and 60,000 steps long
• the walk starts above the level it may touch
• the walks are centred on where they started
• the win probability is a probability

• |qα − p| comes out below 1/N
• and every convergent from the first onward breaks a record
• every record-breaking fraction is a convergent
• neighbouring convergents differ by a determinant of one
• so p/q is within one over qN of the number itself
• the constant is one this figure knows
• the largest denominator is a whole number between 8 and 200
• the number of boxes is a whole number between 3 and 14
• there are record-breaking denominators to mark
• two of the N + 1 points share a box, as they must

• the quotients rebuild 34/13 ×3
• the number of terms is a whole number between 2 and 12 ×2
• √2 has a convergent above √5, as Hurwitz says every irrational must
• and the first that does is the fundamental solution found by searching
• and the last of them has closed on √5
• between four and twelve terms are expanded
• between two and five constants
• D is a whole number between 2 and 40 and not a perfect square
• D is between 2 and 40
• D is not a perfect square
• D is not a perfect square, or the equation has only the trivial solution
• D names the Pell equation and is read only by the pell view; the other views take `of`
• dilations is between 2 and 6 and is read only by the dilate and ehrhart views
• e has a convergent above √5, as Hurwitz says every irrational must
• each convergent is closer than the one before
• every constant expanded is one this family knows
• every convergent stays close to the hyperbola
• neighbouring convergents differ by a determinant of one
• rs is a short list of whole heights and is read only by the reeve and ehrhart views
• some convergent solves the equation exactly
• the constant is one this figure knows
• the convergents alternate above and below the value
• the denominator is a whole number between 1 and 10000000
• the error view compares a list of constants and every other view expands one
• the expansion repeats
• the golden ratio's convergents never leave the neighbourhood of √5
• the numerator is a whole number between 1 and 10000000
• the recurrence stays in whole numbers
• the solution found satisfies the equation
• the tower's last convergent is a number
• the view is one the family draws
• there are enough convergents to draw
• π has a convergent above √5, as Hurwitz says every irrational must
• π has a convergent that beats √5 by two orders of magnitude
• φ has a convergent above √5, as Hurwitz says every irrational must

• the side of the big square is a whole number between 2 and 200 ×2
• the side of the small squares is a whole number between 1 and 200 ×2
• and the discrepancy never changes size
• and the new pair is strictly smaller
• every step is strictly smaller than the one before
• the descent carries the discrepancy to minus itself
• the descent runs out of room before the discrepancy runs out
• the descent takes at least three steps
• the pair is one the descent can start from
• the two small squares overlap and still reach the corners
• the two small squares, less their overlap and plus the corners, are the big one

• 6 is called what it is ×7
• the bar for 6 is its divisors laid end to end ×7
• the lattice draws every divisor of 30 exactly once ×5
• 496 is perfect ×2
• the number is a whole number between 4 and 100000 ×2
• and so do the norms of two and three
• between one and eight whole numbers above one
• both first splits divide the number
• both trees end in the same multiset of primes
• every divisor including the number itself adds to twice the number
• no divisor is drawn twice
• no element of this ring has norm two or three, so neither factor can split further
• the cells add up to the divisor sum
• the divisor count is the product of one more than each exponent
• the divisor sum factorises as the two row totals multiplied
• the exponent is a whole number between 2 and 7
• the first tree's leaves multiply back to the number
• the lattice half-width is a whole number between 2 and 5
• the Mersenne number for this exponent is prime
• the number has a factor tree to draw
• the number has at least three prime factors, so its trees can differ
• the number has at most three distinct primes
• the number is a product of exactly two prime powers
• the powers of two below 2^k add up to the Mersenne number
• the rectangle holds every divisor
• the second tree's leaves multiply back to the number
• the two conjugate factors have norms multiplying to 36
• the two trees start differently

• the coefficient at 0 is what the theorem says ×27
• the product's coefficient at 0 is p(0) ×19
• the two products agree at 0 ×19
• p(4) is divisible by 5 ×11
• log p(10) is below π√(2n/3), as it is for every n ×10
• the estimate at 10 is within a factor of two of the count ×10
• the ranks modulo 5 split the 30 partitions of 9 into 5 equal classes ×4
• the partitions of 8 into odd parts and into distinct parts come out equal ×3
• the number of coefficients shown is a whole number between 6 and 20 ×2
• the number sorted by rank leaves 4 on division by 5 ×2
• the number whose partitions are sorted by rank is a whole number between 4 and 14 ×2
• and changes the number of parts by exactly one
• and it is the largest one that fits
• and the estimate is closer at the far end than at the near one
• and the partitions the move cannot touch are exactly what survives
• and the ratio climbs towards that ceiling rather than away
• at least one of the divisible values is in the table
• doing it twice gives back what it started as
• every listed partition adds to the number it partitions
• the congruence drawn is the one modulo 5 or modulo 7
• the conjugate is a partition of the same number
• the largest number counted is a whole number between 20 and 120
• the largest number in the table is a whole number between 12 and 40
• the marked values are whole numbers inside the range drawn
• the move gives another partition of the same number into distinct parts
• the number being partitioned is a whole number between 3 and 12
• the number whose distinct partitions are paired off is a whole number between 3 and 14
• the numbers outside the congruence class are not all divisible either
• the partition view is one of durfee, glaisher, series, euler, pentagonal, growth, congruence
• the partitions listed are all of them
• the parts are a descending list of whole numbers
• the restricted counts are not the unrestricted one
• the series drawn is the partition product or Euler's identity
• the signed count of distinct-part partitions is the product's coefficient
• the square, the arm and the leg account for every dot
• there is a Durfee square to draw
• turning the diagram over twice gives it back

• the search over divisors of 2 agrees with whether 2 is a perfect square power ×14
• the derivative of order 7 at zero is a whole number ×12
• and below 1/2 ×11
• at q = 2 the tail is positive ×11
• q! times the first 3 terms is a whole number ×11
• the degree-1 integral is positive, because the integrand is ×9
• the fraction 1/1 is no closer to √2 than 1/(3q²) allows ×9
• the largest value of the degree-1 polynomial is (π²/4)ᵏ/k! ×9
• and 4 has one such root, not several ×7
• the search over divisors of 2 agrees with whether 2 is a perfect cube power ×7
• the 1th truncation beats the exponent 2 ×5
• and it does so with room to spare: the error is below q^(−2) ×4
• some truncation beats the barrier of degree 2 ×4
• the search over divisors of 2 agrees with whether 2 is a perfect fourth power ×4
• a fraction's denominator is not zero
• and is below twice the peak, since sin integrates to two over the interval
• and it is closer than 1/q², which is what makes the exponent exactly two
• at least one of the numbers drawn has an irrational root, or the figure argues nothing
• between one and three increasing barrier degrees, none above the number of truncations drawn
• between two and five increasing degrees, each between 1 and 12
• between two and ten whole numbers, each between 2 and 64
• its denominator is a whole number between 1 and 200
• so it is strictly between 0 and 1, where no whole number is
• the fraction is at least close enough to π for the picture to be about π
• the largest denominator tried is a whole number between 3 and 14
• the number of convergents drawn is a whole number between 4 and 9
• the number of tail terms summed exactly is a whole number between 12 and 60
• the number of truncations of the constructed number is a whole number between 3 and 5
• the numerator of the fraction π is supposed to be is a whole number between 2 and 400
• the range drawn reaches a degree whose bound is below one, which is the whole argument
• the root taken is a whole number between 2 and 4
• the tail falls as the denominator grows
• the view is one the family draws
• there are derivatives to show

• the count on the circle of squared radius 1 is 4(d₁ − d₃) ×40
• 3, 4, 5 is a Pythagorean triple ×10
• the points on the circle of radius √25 are 4(d₁ − d₃) ×8
• the tetrahedron of height 1 has no lattice point inside it ×7
• 5 = 2² + 1² ×5
• the area is 20 interior points plus half of 7 on the edge, less one ×5
• the count at dilation 1 is the Ehrhart polynomial's value ×4
• the count at k = 1 matches the cubic fitted from the first value ×4
• the number of columns in the grid is a whole number between 4 and 8 ×2
• a group with entries only up to two fails to identify two of the classes
• a polygon's boundary count and its dual's add to twelve
• a prime is a sum of two squares exactly when it is one more than a multiple of four
• and a self-dual one has six boundary points, which is half of twelve
• and dividing by the radius itself tames it further still
• and exactly four on it, which are its own corners
• and from entries up to three onward the count stops moving
• and Scott's inequality caps the boundary count at nine
• and some attain the bound
• and stays inside three times the two-thirds power throughout
• and stays inside twice the cube root over the range drawn
• and the disc of radius two catches thirteen
• and the flat ones are excluded rather than ignored
• and the point it came from is on the unit circle
• and the rest pair off
• and they have between three and six corners
• and twelve has six
• at least two heights are drawn
• between one and five powers are drawn
• D is a whole number between 2 and 40 and not a perfect square
• D is between 2 and 40
• D is not a perfect square, or the equation has only the trivial solution
• every edge lies at lattice distance one from the interior point
• every lattice triangle in the grid satisfies the identity
• every marked point really is on the circle
• every one of the 2300 triples of grid points was reached
• every power is again a solution
• every solution up to the largest drawn is a power of the smallest
• for some height the linear coefficient is negative, so it counts nothing
• four of the sixteen are their own dual
• n names one circle and is read only by the default view; the other views take their own parameters
• no polygon with an interior point has more boundary points than Scott allows
• no two slopes give the same triple
• one extra for the hole repairs it, which is the Euler characteristic in disguise
• one has one divisor
• one of them attains Scott's bound
• Pick's formula with one interior point makes twice the area the boundary count
• Pick's theorem holds for every polygon swept
• some circles miss the lattice entirely, which is the whole question
• the bound is attained only at one interior point
• the box the polygons are drawn from is a whole number between 3 and 4
• the circle of squared radius one carries four points
• the disc of radius one catches five points
• the divisors of everything up to six add to fourteen
• the dual of one of the sixteen is another of the sixteen
• the error divided by the cube root is the tamer ratio
• the error divided by the two-thirds power is the tamer ratio
• the error is not bounded by twice the square root over this range
• the identity as stated gives the wrong area for a region with a hole
• the largest bound counted is a whole number between 200 and 4000
• the largest prime to test is a whole number between 10 and 200
• the largest radius counted is a whole number between 40 and 400
• the largest squared radius drawn is a whole number between 12 and 64
• the lattice half-width is a whole number between 1 and 16
• the leading coefficient is the volume for every height drawn
• the number is a whole number between 1 and 200
• the polygon has between 3 and 16 whole-numbered corners and some area
• the polygon is one of blob, triangle, comb, thin
• the second differences are twice the area
• the slopes are proper positive fractions
• the solution found satisfies the equation
• the sweep found a substantial family
• the sweep grid is between 3 and 6 points across
• the view is one the family draws
• the volumes differ while the counts do not
• there are sixteen classes to pair up
• there are sixteen lattice polygons with a single interior point
• twice the area is a whole number, which is why halves are the only fractions in the formula
• upTo names a range and is read only by the gaussian and count views
• while a polygon with no interior point can exceed the bound, which is why it needs one

• 7/2 is in lowest terms ×108
• the circles on 0/1 and 1/7 touch ×56
• s(1) and s(2) are coprime ×48
• the 1th ratio's numerator agrees with the Calkin–Wilf walk ×48
• the ratio 1/1 appears once ×48
• 0/1 and 1/7 are Farey neighbours ×24
• 1/1 arrives already in lowest terms ×15
• and its mediant 1/1 is already in lowest terms ×13
• s(2) counts the hyperbinary representations of 1 ×12
• the sequence has one entry per coprime pair up to 7 ×2
• and at most its right bound
• and is closer to the target
• and its denominator does too
• and the path closes on the target
• circles that are not neighbours stay clear of each other
• each path is a word of at most eight L's and R's
• each record has a larger denominator than the last
• how many terms are drawn is a whole number between 8 and 64
• how many turns are taken is a whole number between 6 and 34
• no fraction is produced twice
• no two words produce the same matrix, so the tree never rejoins
• reading the drawing left to right reads the fractions in increasing order
• the depth is a whole number between 2 and 5
• the descent passes at least three fractions within 1/q² of the target
• the descent produces several record approximations
• the drawing scale is a whole number between 200 and 400
• the matrix for "L" has determinant ±1
• the matrix for "LR" has determinant ±1
• the matrix for "LRL" has determinant ±1
• the matrix for "LRLR" has determinant ±1
• the matrix for "LRLRL" has determinant ±1
• the matrix for "R" has determinant ±1
• the matrix for "RL" has determinant ±1
• the matrix for "RLR" has determinant ±1
• the matrix for "RLRR" has determinant ±1
• the matrix for "RR" has determinant ±1
• the matrix for "RRL" has determinant ±1
• the matrix for "RRLR" has determinant ±1
• the matrix for "the root" has determinant ±1
• the mediant is at least its left bound
• the mode of the mediant family is one of tree, farey, ford, matrices, diatomic, approx
• the order is a whole number between 2 and 12
• the target is one of phi, root2, pi, e
• the tree is full to its stated depth
• up to which index the hyperbinary count is checked is a whole number between 4 and 24

• the length is a whole number between 3 and 7 ×2
• the number of colours is a whole number between 2 and 4 ×2
• and exactly a of them are single
• and the number of non-constant classes is (a^p − a)/p
• every class is a single string or a full ring of p
• every other class holds exactly p strings
• every string of the given length is drawn
• p divides a^p − a, which is what the count just showed
• the classes account for every string
• the classes of size one are exactly the constant strings
• the figure draws every string, so there is a ceiling on how many
• the length is prime
• the length is prime — the whole argument needs it to be

• two primes agree modulo 3 and disagree about x² + 27y² ×158
• 3 is represented by x² + 1y² exactly when 3 mod 4 is one of the classes ×134
• every step is a whole unit, because 1 is not a multiple of 13 ×70
• the sign of multiplication by 3 is (3 | 11) ×22
• (−1 | 3) is decided by 3 mod 4 ×17
• (2 | 3) is decided by 3 mod 8 ×17
• Gauss's lemma counts 1 folds for −1, and its parity is the symbol ×17
• Gauss's lemma counts 1 folds for 2, and its parity is the symbol ×14
• the modulus deciding x² + 1y² is at most 4n ×3
• the modulus is a whole number between 5 and 31 ×3
• 3 folds gives the same answer as Euler's criterion ×2
• (p | q) comes out of the count above it
• (q | p) comes out of the count below the line
• a prime with both symbols positive is 1 modulo 8, which is the two conditions met at once
• and every one of them lands in the bottom half
• and exactly half the multipliers are squares
• and it has no imaginary part
• and points up, which is the part Gauss took four years over
• and positive, which is the part Gauss took four years over
• and their product is minus one to the power of the whole rectangle
• between two and five forms are drawn
• both are odd primes and they are different
• each form's coefficient is a whole number between 1 and 27
• every moved cycle has the same length, which is the order of the multiplier
• every point is counted exactly once, on one side or the other
• for a prime 1 modulo 4 the sum is real
• for a prime 3 modulo 4 the sum is purely imaginary
• no lattice point lies on the diagonal, because the primes are coprime
• the bound the congruence check runs to is a whole number between 2000 and 40000
• the cycle count and the inversion count give the same sign
• the drawing holds both kinds of form, which is what it is for
• the first prime is a whole number between 3 and 31
• the folded values are all different
• the folds for 2 are the k above p/4
• the largest modulus tried is a whole number between 20 and 400
• the largest prime drawn is a whole number between 60 and 400
• the largest prime in the table is a whole number between 11 and 97
• the length of the sum, squared, is the modulus
• the modulus is an odd prime
• the multiplier is a whole number between 2 and 30
• the multiplier is not a multiple of the modulus
• the points below the line are the sum of the floors of kq/p
• the second prime is a whole number between 3 and 31
• the square of the sum is plus or minus the modulus
• the view is one the family draws

• the class 0 shares a factor with 4, so it can hold at most the factor itself ×20
• 211 leaves remainder 1 on division by 2 ×10
• the limit is a whole number between 10 and 400 ×5
• 2 classes share no factor with 4 ×4
• the staircase ends at π(3000) ×4
• every listed prime is already in the class 3 mod 4 ×2
• so some prime factor of it is 3 mod 4 — the classes multiply, and a product of ones is one ×2
• the class 3 leads at almost every bound ×2
• the modulus is a whole number between 3 and 14 ×2
• the number built is 3 mod 4 ×2
• a Mersenne prime has a prime exponent
• a square left unstruck holds a prime, and a struck one does not
• an even truncation is an upper bound
• and 168 below a thousand
• and an odd one a lower bound
• and every combination of exponents appears once
• and it tracks the logarithm of a logarithm to within a twelfth throughout
• and modulo three it never falls behind over this range
• and over the last decade of the range the diverging sum gains more than twice as much
• and some of the lower bounds are negative, which is no bound at all
• and the first bound at which it does not is 26,861
• and the number of them in the interval is never far below n / ln n
• and the one prime it holds divides the modulus
• and the ratio to x / ln x comes down as x grows
• and the twin pairs track theirs
• and there are almost none of them in this range
• and they hold them in near-equal numbers, which is more than Dirichlet's theorem claims
• between two and five odd primes are listed
• both classes hold primes below the bound
• each is near the share an even split would give
• every class coprime to the modulus holds primes
• every interval from n to 2n holds a prime
• every prime below the bound lands in exactly one class
• every prime factor of one more than a square of an even number is 1 mod 4
• every prime factor of the constructed number is outside the list
• multiplying the factors out gives exactly the sum over the numbers they build
• no listed prime divides it
• one gap between each consecutive pair
• the bound the postulate is checked to is a whole number between 100 and 50000
• the column count is a whole number between 4 and 25
• the construction is the one that subtracts or the one that squares
• the count runs above x / ln x over this range
• the factorisation multiplies back to the number built
• the factors are between 2 and 4 primes
• the full inclusion-exclusion agrees with the count made by sifting
• the gaps add up to the distance from 2 to the last prime
• the grid of whole numbers drawn is a whole number between 24 and 144
• the largest bound counted is a whole number between 5000 and 200000
• the largest number sieved is a whole number between 100000 and 2000000
• the largest prime sifted by is a whole number between 11 and 43
• the list is between two and six primes
• the logarithmic integral runs above the count at every checkpoint drawn
• the marked checkpoints are whole numbers inside the range
• the number built stays inside exact arithmetic
• the number of intervals drawn is a whole number between 6 and 40
• the numbers marked are exactly those the product builds
• the partial sums swing far past the answer before settling
• the pass number is between 0 and 8
• the power each factor is truncated at is a whole number between 2 and 5
• the primes 1 mod 4 are about half of all of them
• the primes one more than a square track the conjectured count
• the race is drawn modulo three or four
• the range sifted is a whole number between 10000 and 1000000
• the range the sum over primes is drawn to is a whole number between 1000 and 1000000
• the sieve found every prime below the bound
• the squares left standing below 100 are the primes
• the sum of the reciprocals of the primes keeps growing
• the sum over all primes has passed 2.8 and is still climbing
• the view is one the family draws
• there are 25 primes below 100
• while the ratio to the logarithmic integral is nearer one by more than a factor of four
• while the sum over the twins is still under 1.8
• π(x) stays above x / ln x throughout this range

• above the critical parameter the trajectory visits both lobes
• and each is narrower by the stretch factor
• and even at the cusp, where a straight line cannot fit, the departure stays small
• and it is the Lorenz system's own exponent, near nine tenths
• and settles on neither fixed point
• and the strips are laid out in order without overlapping
• areas shrink at every step
• below the critical parameter it spirals into a fixed point
• between two and four starting gaps, each a whole power of ten from −2 to −12
• enough returns to see a shape
• every surviving strip lies inside the square
• every window still holds some of the orbit
• so the total width falls, and the limit has none
• the closed-form fixed points really are fixed
• the contraction is between 0.1 and 0.4
• the geometric parameter is between 0.1 and 8
• the growth rate does not depend on how close the two starts were
• the Hénon parameter is between 1 and 1.45
• the Jacobian determinant is the contraction
• the map's maximum is in its interior, so it folds
• the number of magnifications is a whole number between 1 and 4
• the number of stages is a whole number between 2 and 5
• the number of steps is a whole number between 500 and 40000
• the orbit stays bounded, which is what makes it an attractor
• the Prandtl parameter is between 0.1 and 40
• the Rayleigh parameter is between 0.1 and 100
• the return map is a curve, not a cloud — its typical thickness is under one per cent of its range
• the step size is between 0.0005 and 0.05
• the stretching factor is between 2.2 and 5
• the strips double at every stage
• the surviving set has a dimension between nought and one
• the trajectory stays in a bounded region

• rule 90 row 0 matches Pascal's triangle mod 2 ×40
• every cell is on or off and nothing else
• the drawn table reads back as the rule number
• the number of rows is a whole number between 4 and 160
• the rule number is a whole number between 0 and 255
• the seed is single or random

• a point beside root 1 runs to root 1 ×3
• each root cubes to one
• the iteration cap is a whole number between 8 and 200
• the sample count is a whole number between 60 and 700
• the span is between 0.02 and 6
• with no imaginary part left over

• the orbit at r = 2.8 has period 1 ×4
• the column count is a whole number between 40 and 1400
• the diagram is drawn as marks rather than as a bitmap of elements
• the doublings come in order
• the last ratio measured is Feigenbaum's constant to within a twentieth
• the lower parameter is between 0 and 4
• the number of doublings is a whole number between 3 and 5
• the points kept per column is a whole number between 10 and 400
• the range runs upward and is wide enough to draw
• the upper parameter is between 0 and 4

• a trajectory in this table takes exactly 4 directions and no more ×3
• the path bounces correctly off side 1 ×3
• a closed path is one that returns to within a tolerance
• a path closing on itself was found within 1e-9 — the search at this depth found none otherwise
• and arrives and leaves at the same angle to the radius
• and does so after twice the sum of the two whole numbers
• and its angles add to a straight angle
• and its invariant is the same at every bounce
• and never returns exactly to where it started
• and no nearby triangle inscribed in the same three sides is shorter
• and that genus is a whole number
• and the direction along the wall is untouched
• and the path does not close before it is supposed to
• and with the obstacle the gap grows by far more than the distance does
• at the value it was aimed with
• between 10 and 200 bounces
• between 20 and 200 bounces are followed
• between 3 and 24 bounces are drawn
• between 3 and 40 chords are drawn
• each altitude meets its side between the corners
• every chord passes the same distance from the centre
• in both coordinates
• on the empty table the gap grows no faster than the distance travelled
• on the empty table the gap stays proportional to the distance travelled
• so no part of it is below the axis
• the angle of arrival equals the angle of departure at every wall
• the angles of the table add to what a polygon's angles must
• the ball is moving and has a wall to reach
• the ball starts inside the table
• the bouncing path and the folded straight line agree everywhere along their length
• the drawn triangle has the angles it was asked for
• the escaping trajectory's invariant is below it
• the first hit is somewhere on the rim
• the irrational path enters nearly every cell of the grid
• the irrational path is followed for between 60 and 600 bounces
• the obstacle fits inside the table with room to pass
• the path is followed for a drawable number of bounces
• the rational slope is a ratio of small whole numbers
• the rational slope is in lowest terms
• the rational slope's path returns to its exact starting state
• the reflected copies drawn fit inside the panel they are drawn in
• the search runs to between three and sixteen bounces
• the slope is drawable
• the star closes exactly
• the star polygon is drawn in a single closed loop
• the star wraps between 1 and 6 times
• the stem is between a fifth and a whole radius deep
• the stem is between a twelfth and half the cap wide
• the surface the table unfolds into has a whole genus
• the sweep is a grid of a workable size
• the sweep produced something to compare
• the table is a triangle
• the table is one the family knows
• the trajectory is followed for between 40 and 600 bounces
• the trajectory stayed inside the table
• the trapped trajectory never touches a stem wall
• the trapped trajectory's invariant is above the stem's half-width and inside the cap
• the triangle is acute, which is when this path exists
• the two invariants sit on opposite sides of the stem's half-width
• the two paths start very nearly together
• the view is one the family draws
• two angles of a triangle, in degrees, leaving a third
• while the trajectory aimed under the half-width does go down the stem
• with the obstacle the same two paths end up hundreds of times further apart

• at depth 2 the cover's total agrees with the closed form at exponent 1.037 ×36
• after 0 steps there are two to the power 0 pieces ×7
• at the dimension itself the total is exactly one, at depth 2 ×6
• after 6 steps there are as many pieces as the rule makes ×2
• a smaller box never covers the set in fewer boxes
• and above it the total shrinks to nought
• below the dimension the total grows without bound as the cover is refined
• between 20 and 400 thousand steps
• between one and four sets this family draws
• between two and six depths, each between 1 and 14
• each count is exactly the number of pieces at that depth
• how many box sizes are counted is a whole number between 3 and 6
• how many times the box size is divided is a whole number between 1 and 5
• how many times the rule is applied is a whole number between 2 and 6
• the chosen cells are distinct positions of the grid
• the contraction beats the stretch, so the formula's first case applies
• the counted box dimension closes on the closed form
• the counted dimension closes on the one the rule forces
• the counted dimension lies strictly between a curve's and a region's
• the curve is drawn finer than the smallest box counted
• the drawn boxes are the counted boxes
• the exponent is swept between 0.05 and 1 either side of the dimension
• the first exponent is positive, which is what makes the orbit sensitive to its start
• the formula returns a value between a curve's dimension and a region's
• the Hausdorff dimension never exceeds the box dimension
• the horizontal contraction is the stronger one, which is what makes the carpet self-affine
• the map contracts area but does not collapse it
• the map's parameter is in the range that has an attractor
• the number of columns is a whole number between 2 and 6
• the number of pieces and the ratio give the dimension the rule forces
• the number of rows is a whole number between 2 and 5
• the number of times the rule is applied is a whole number between 1 and 7
• the orbit stays on the attractor rather than escaping
• the rows carry different numbers of cells, so the two dimensions differ
• the rows carry equal numbers, so the two dimensions agree
• the sample is finer than the smallest box counted
• the second is negative, which is what collapses the attractor onto a set of no area
• the set is one built by a rule, or there is no exponent to pivot about
• the set is one this family draws
• the two exponents add to the logarithm of the area factor, which the map fixes
• the view is one the family draws

• the nonlinearity is between 0.4 and 1.2 ×2
• and both boundary circles still stay where they are
• and the reason is that this map does not preserve area
• and the second really is outside it
• and they turn in opposite directions
• both boundary circles stay where they are
• each crossing is a fixed point of the map
• each orbit is followed for between 60 and 600 steps
• inside a plateau the rotation number does not depend on where the orbit started
• no orbit leaves the annulus
• no point of the annulus comes near to staying where it is
• one fixed point is a centre and the other a saddle
• the angle is a number or one of the named constants
• the angular shift changes sign across the annulus at every angle
• the boundaries still turn in opposite directions
• the conserved quantity stays put along every orbit drawn
• the first parameter really is inside the plateau
• the gaps take at most three distinct values, as the three-distance theorem says
• the largest nonlinearity is between 0.4 and 1.2
• the locked share of the parameter axis grows with the nonlinearity
• the loop and its image cross exactly twice
• the loop and its image enclose the same area
• the lower parameter is between 0 and 1
• the map locks onto one step in two over an interval of parameters
• the map preserves area everywhere on the annulus
• the measured curve is flat over a stretch at more than one rational
• the number of nonlinearity levels is a whole number between 6 and 40
• the number of parameters per level is a whole number between 40 and 320
• the number of parameters sampled is a whole number between 60 and 900
• the number of steps drawn is a whole number between 10 and 60
• the number of steps is a whole number between 3 and 120
• the orbit repeats exactly when the angle is rational
• the outward push is a sensible size
• the perturbation is a sensible size
• the perturbation is between a fiftieth and two fifths
• the rotation number never falls as the parameter rises
• the upper parameter is between 0 and 1
• the view is one the family draws
• the window runs upward and is wide enough to draw
• with no nonlinearity the curve is a straight line and has no plateaus to speak of

• bin 1 holds the share the density says it should ×60
• every point over 1 comes back after 1 doublings ×6
• 0.00000 returns to itself after 2 steps ×5
• a full two-branch fold has 2^1 points fixed by its 1th power ×5
• both maps have the same number of points of period dividing 1 ×5
• 0.6154 is fixed at r = 2.6 ×4
• the number of steps is a whole number between 2 and 400 ×4
• the numerator of the starting point is a whole number between 1 and 10000000 ×3
• at r = 2.4 the error shrinks by the slope at the fixed point each step ×2
• the number of steps drawn is a whole number between 3 and 20 ×2
• the orbit at r = 2.4 converges far enough to measure a rate ×2
• the parameter is between 0 and 2 ×2
• √2 is fixed by the Newton step
• a new crossing has a period above one that divides 2
• after the computed orbit is dead the exact one is still moving
• and carries the right end to the right end
• and it never reaches zero, because an odd denominator cannot be halved away
• and never grows
• and shrinks every distance by a factor under one
• and stays there, because zero is fixed
• and the constant the ratio approaches is 1/(2√2)
• and the repeat found really is a repeat
• and then separate visibly, at a step the figure marks
• between 6 and 60 steps are taken
• between one and three logistic parameters, each with an attracting fixed point
• each letter of the itinerary is the next binary place of the starting point
• each Newton error is the square of the last over twice the current guess
• every bin is visited
• every corner of the staircase lies on the curve or on the diagonal
• every crossing of the map is also a crossing of the composed map
• every exact iterate is a whole numerator over the same denominator
• every start reaches the same point
• five Newton steps from 1.9 reach twelve decimal places
• how many steps of the orbit are drawn is a whole number between 3 and 12
• its denominator is a whole number between 3 and 10000000
• one point drawn per step, and the start
• the 2 new crossings of period 2 are whole orbits of 2 points
• the average of one long orbit is the average over the whole interval
• the change of coordinate fixes the left end
• the change of coordinate is increasing, so it is reversible
• the computed orbit reaches exactly zero inside the drawn window, having run out of binary places
• the contraction is one the family knows
• the coordinate change carries the tent map to the logistic map
• the coordinate is one whose invariant density this figure knows
• the denominator is odd, or the exact orbit reaches zero as surely as the float one does
• the error after n steps is inside k to the n times the error at the start
• the exact orbit closes up inside the drawn window
• the exact shares add to one
• the fixed point attracts, so there is a rate to measure
• the longest cycle counted is a whole number between 2 and 8
• the longest word counted is a whole number between 2 and 8
• the longest word counted is between two and eight
• the map is one the family knows
• the map is one this figure knows
• the map itself has a fixed point
• the map sends the interval into itself
• the number of binary places the starting point has is a whole number between 3 and 8
• the number of bins is a whole number between 8 and 80
• the number of drawn steps is a whole number between 2 and 400
• the number of times the map is composed is a whole number between 2 and 4
• the orbit approaches the fixed point exactly when the slope is shallow
• the orbit stays inside the unit interval
• the point a repeating word names has that word as its itinerary
• the point found by bisection is fixed
• the quadratic curve is on or off
• the slope at the fixed point is 2 − r
• the starting point has between three and eight binary places
• the starting point is between 0 and 1
• the starting point is inside the interval
• the transported orbit is the orbit of the transported point
• the transported orbit obeys the tent map's own recurrence
• the two agree exactly for at least eight steps
• the view is one the family draws
• the words of each length and the points of that period are equally many
• two logistic parameters above one
• two points are at most k times as far apart after the map as before

• step 1 follows the rule ×178
• every one of the 2 patterns of length 1 occurs exactly once ×12
• the orbit of 27 reaches one ×7
• a power of two is never a power of three
• a starting number is a whole number between 3 and 1000000
• across a large sample of starts, half the steps are odd ones
• and 27 is the famous long one
• and by the largest run drawn it is above nine tenths
• and it is the one at one
• and no two residues give the same parity string
• and reaches a few thousandths within the range drawn
• and the average fall per step is the value the pairing predicts
• at least one pattern gives a whole number
• between two and four odd offsets, each at most eleven
• each convergent beats every fraction with a smaller denominator
• every number the reverse tree reaches does come back to one
• every orbit trends downwards in the logarithm
• every pattern of the length was solved
• most parity strings shrink the number by the end of the run
• no pattern with a negative denominator gives a positive whole solution
• one is already at one
• one length per start
• so the residues and the parity strings match one for one
• the continued fraction of log₂3 starts 1, 1, 1, 2
• the denominator is always odd, so the solution stays where the map is defined
• the depth of the tree is a whole number between 3 and 12
• the largest number of steps is a whole number between 4 and 15
• the largest start is a whole number between 20 and 4000
• the longest pattern length checked is a whole number between 6 and 14
• the longest pattern searched is a whole number between 8 and 20
• the map is a bijection at every length checked
• the multiplier condition predicts the drop, on a large member of the class
• the number of convergents listed is a whole number between 5 and 10
• the number of steps is a whole number between 3 and 7
• the ordinary rule has exactly one cycle at these lengths
• the ordinary rule is among those searched
• the parity of the first steps depends only on the start modulo two to the k
• the pattern length is a whole number between 4 and 10
• the relative gap shrinks along the list
• the rule with minus one has at least three, which no drift argument distinguishes
• the share provably driven down grows with the number of steps allowed
• the starting number is a whole number between 1 and 100000
• the tree is still growing at the depth drawn
• the view is one the family draws

• and neither is four tenths
• and so is minus one
• one is not
• the critical point is in this Julia set exactly when its parameter is in the Mandelbrot set
• the escape time separates into bands
• the iteration cap is a whole number between 20 and 400
• the Julia parameter is a complex number
• the sample count is a whole number between 60 and 700
• the set is the Mandelbrot set or a Julia set
• the span is between 0.0005 and 4
• zero is in the Mandelbrot set

• and below the onset of chaos it is negative
• at r = 4 the exponent is log 2, which is exactly computable
• the column count is a whole number between 40 and 1200
• the lower parameter is between 0 and 4
• the number of steps is a whole number between 10 and 200
• the orbits separate exactly when the parameter is past the onset of chaos
• the parameter is between 0 and 4
• the range runs upward and is wide enough to draw
• the starting point is between 0 and 1
• the two orbits begin the stated distance apart
• the two starts are close together
• the upper parameter is between 0 and 4

• a choice from each of the pairs is one of two to the power of their number
• between two and five pairs, each of two named things
• each element belongs to exactly one chain
• every moved point lands strictly inside the open interval
• every point the map has to reach is reached
• no two moved points land on the same place
• the assembled map sends no two elements to the same place
• the left-to-right map sends different elements to different places
• the left-to-right shift is a whole number between 1 and 4
• the members of a pair are named apart exactly when an order is claimed
• the number of elements drawn from each side is a whole number between 5 and 14
• the number of sequence terms drawn is a whole number between 4 and 12
• the pairs either carry an order or they do not
• the right-to-left map sends different elements to different places
• the right-to-left shift is a whole number between 1 and 4
• the rule fits two to the power of the pairs it cannot tell apart
• the rule names exactly one of the rows exactly when the pairs carry an order
• the two maps are different, or there are no chains to see
• the view is one the family draws
• the window shows at least three chains

• between one and five bases are compared
• between two and eight connectives are tabulated
• no two of the sixteen have the same table
• Post's criterion and the closure agree about whether a connective is complete on its own
• the closure and Post's criterion agree about this basis
• the connective is one of the sixteen
• the number of columns is a whole number between 4 and 8
• there are sixteen binary connectives and no more

• a line drawn as parallel was checked and does not meet the base line inside the disc
• a triangle has three vertices
• each crossing point was computed and lies on both lines
• every angle of the triangle is positive
• every geodesic arc meets the boundary circle at a right angle
• every vertex is inside the disc
• exactly one of the lines through the point never meets the base line
• more than one geodesic through the point misses the base line
• the angles of a triangle in this model add to less than a straight angle
• the defect is a positive area
• the number of lines tried through the point is a whole number between 3 and 9
• the number of parallels drawn is a whole number between 1 and 9
• the outside point is inside the disc
• the point is off the base geodesic

• at 1 rounds the search agrees with the rule that says same length or both at least 1 ×3
• at depth 1 the words must be 1 letters before Duplicator survives ×3
• by the largest size drawn, Duplicator survives 2 rounds on every pair sampled ×2
• Duplicator survives 3 rounds on a cycle of 22 against two of 11 ×2
• the search agrees with the rule that says Duplicator survives 3 rounds from 5 points up ×2
• a star splits at the first round
• an even number of edges is a coin toss at every size, as the flip bijection says it must be
• and does not at the smallest
• and does not survive the three-pebble one
• and Duplicator survives at every length past it
• and it does grow
• and once Duplicator survives at one length it survives at every longer one
• and so do two triangles, which is why refinement cannot separate them
• and that object is a path
• and the group has order two
• and the share rises with the size rather than wandering
• and the two graphs have the same number of edges
• at every depth there is a length past which the two words cannot be separated
• between fifty and four hundred graphs at each size
• between four and twelve pairs of graphs at each size
• between ten and eighty graphs are sampled at each size
• between three and six graph sizes, each of four to sixty points
• between three and six rounds shown
• between three and six sizes, each of four to forty-four points
• between three and six sizes, each of three to forty points
• between two and four depths
• between two and three round counts are tabulated
• Duplicator survives the two-pebble bijective game
• every point of both graphs has two neighbours
• exactly half of the 64 graphs on 4 points have an even number of edges
• large ones almost never do
• one graph is connected and the other is not, so a sentence separating them would express connectedness
• one or two round counts, each of at most three
• one-dimensional refinement gives both graphs the same colours and cannot tell them apart
• small graphs usually fail the property
• so the count grows like a logarithm rather than like the length
• the cycle stays one colour at every round, since every point looks alike
• the deeper game needs a larger graph before Duplicator always survives it
• the depth is between one and three
• the first-order property "every pair has a common neighbour" has settled on 1 by the largest size drawn
• the first-order property "some point joined to nothing" has settled on 0 by the largest size drawn
• the first-order property "three points all joined" has settled on 1 by the largest size drawn
• the game agrees with the neighbourhood argument about who wins
• the game is played to at most three rounds, which is as far as a search finishes
• the game runs between one and four rounds
• the measured share agrees with the count of expected failures
• the number of colours never falls
• the number of rounds Spoiler needs is computed
• the other two monoids are aperiodic, so both languages are first-order definable
• the parity language's monoid contains a non-trivial group, so no first-order sentence says it
• the rounds needed against a chain one longer follow the logarithm of its length
• the sizes are listed in increasing order
• the sweep runs over between three and eight cycle lengths
• the sweep runs to between five and nine
• the table runs to between four and nine
• the transcript ends the way the exhaustive search says the game ends
• the two cycles are of different lengths
• the two cycles have three points each
• the two graphs have the same number of points and the same number of edges
• the two neighbourhoods are the same object, computed from the cycles rather than named
• the two small cycles run to between three and thirteen points
• the view is one the family draws
• two chains of between two and nine elements
• two cycles of between five and thirty points
• two-dimensional refinement does tell them apart
• words up to between four and six letters
• words up to between six and twelve letters

• a set of 0 has more subsets than members ×13
• height 2 holds finitely many polynomials, and the enumeration produced them ×5
• −1 is among the roots the enumeration found
• √2 is among the roots the enumeration found
• 1 is among the roots the enumeration found
• a half is among the roots the enumeration found
• a larger height admits more polynomials
• all 512 listings of 3 subsets of a set of 3 were tried
• and the second
• and yet the two images are far apart, so the weave is not continuous
• by a factor of more than a thousand at this many places
• every fraction the grid reaches is on the list
• every kept cell got exactly one place in the list
• no fraction is listed twice
• the complemented diagonal is on no row of the listing
• the digit strings are as long as the drawing says
• the first number's digits are between four and ten decimal places
• the golden ratio is among the roots the enumeration found
• the largest height enumerated is a whole number between 3 and 6
• the largest sum of numerator and denominator drawn is a whole number between 3 and 9
• the number of places drawn is a whole number between 4 and 10
• the number of rungs drawn is a whole number between 3 and 5
• the places run from one with no gaps
• the second number has the same number of places
• the size at which every listing is tried is a whole number between 2 and 3
• the two expansions differ by less than one unit in the last place drawn
• the view is one the family draws
• the walk's count and the sum of totients agree, having been computed independently
• the window holds enough of them to draw
• the window on the line is between one and twelve wide
• the woven number has one place from each, alternately
• unweaving recovers the first number

• consecutive entries of the Gray code differ in exactly one bit
• every cell of a prime implicant is one the function makes true
• every leaf of the formula is a variable
• every opening bracket in the formula is closed
• squares next to each other on the map differ in one variable
• the covering rectangles and the formula agree on every assignment
• the formula and its disjunctive normal form agree on every row
• the formula has between one and five variables
• the formula is a non-empty string
• the map is drawn for three or four variables
• the table has one row per assignment
• the whole formula is consumed by the parser

• □□p → □p is valid exactly on the dense frames
• □p → □□p is valid exactly on the transitive frames
• □p → ◇p is valid exactly on the serial frames
• □p → p is valid exactly on the reflexive frames
• □p at a world is p at every world it can reach
• ◇□p → □◇p is valid exactly on the confluent frames
• ◇p → □◇p is valid exactly on the euclidean frames
• ◇p at a world is p at some world it can reach
• 4 is valid on a frame exactly when the frame is transitive
• 5 is valid on a frame exactly when the frame is euclidean
• a formula separating the other pair exists among those searched
• a transitive model filters to a transitive quotient here, checked rather than assumed
• a world and a valuation were found at which the axiom fails
• a world that reaches nowhere makes every necessity true and every possibility false
• and fewer worlds than the model it came from
• and it is deeper than anything the closure could ask
• and no frame with any worlds in it validates both Löb's axiom and □p → p
• and on every one of them ¬□⊥ is a fixed point of ¬□p
• and some class of the quotient can — the collapse invents a way back that the model has none of
• and the other two starting worlds are not
• and two frames agreeing on all of them disagree about the axiom
• and validates what is valid
• B is valid on a frame exactly when the frame is symmetric
• between one and five axioms are tabulated
• between two and six frames are compared
• D is valid on a frame exactly when the frame is serial
• excluded middle at a stage is exactly p there or ¬p there
• formulas up to depth two to four are compared
• Löb's axiom is valid exactly on the transitive frames with no cycles
• no condition in the catalogue matches McKinsey's axiom
• no stage forces both p and its negation
• no world of the model can be returned to
• on a frame with the property the axiom holds everywhere, under every valuation
• on that frame every deeper modality agrees with the first, not only the second
• one of the two frames collapses the tower and the other does not
• p → □◇p is valid exactly on the symmetric frames
• so every frame of the logic is transitive, which Löb's axiom implies
• some depth of looking ahead separates the chain from its quotient
• some frame this family knows has the property
• some frames validate it
• some stage forces neither p nor its negation, so p ∨ ¬p fails there
• T is valid on a frame exactly when the frame is reflexive
• the axiom is one of the five
• the bisimilar worlds agree on every formula up to the drawn depth
• the chain runs between eight and twenty-four worlds
• the deeper closure is asked for or not
• the frame is one this figure knows
• the gallery's verdict for a branch, both ends dead is the sweep's
• the gallery's verdict for a single dead end is the sweep's
• the gallery's verdict for a transitive chain is the sweep's
• the gallery's verdict for a two-cycle is the sweep's
• the gallery's verdict for a world seeing itself is the sweep's
• the gallery's verdict for one step and stop is the sweep's
• the model is the chain or the strict order on its worlds
• the model refutes □p → p, which is not valid in this logic
• the quotient has at most one world for each way of answering the closure
• the sweep runs over frames on three or four worlds
• the tower is followed to between two and four boxes
• the truth lemma holds for □□p: it is true at a world exactly when it is true at that world's class
• the truth lemma holds for □p: it is true at a world exactly when it is true at that world's class
• the truth lemma holds for p: it is true at a world exactly when it is true at that world's class
• the truth lemma holds: □□p is true exactly where it is accepted
• the truth lemma holds: □p is true exactly where it is accepted
• the truth lemma holds: p is true exactly where it is accepted
• the two starting worlds are related by the bisimulation
• the view is one the family draws
• this frame is not reflexive, so the axiom has a chance of failing
• this frame is not symmetric, so the axiom has a chance of failing
• this frame is not transitive, so the axiom has a chance of failing
• two frames this family knows
• what is established at a stage stays established later
• worlds share a class exactly when no formula of the closure tells them apart

• the size of the underlying set is a whole number between 2 and 4 ×3
• a cyclic order has exactly n arcs of length k
• an element's share of the maximal chains is one over its layer's size
• and a star is a family of that size
• and every chain meets the middle layer exactly once
• and every one of them is a whole layer
• and every two of its members really do meet
• and everything sits below something maximal
• and some antichain's shares add to exactly one
• and takes one step of size at each rank between its ends
• at most k of the n arcs pairwise meet, and k of them do
• each chain is symmetric about the middle of the cube
• each chain runs upward in the order
• each named target is a subset index is a whole number between 0 and 7
• each step adds exactly one element
• each subset is joined to the ones one element larger
• every chain has an upper bound inside the order
• every element of the order is on some chain
• every relation of the order is respected by every extension
• every subset is on exactly one chain
• exactly n + 1 antichains use the chains up exactly
• no antichain's shares add to more than one
• no chain meets the antichain twice
• no element is on two chains
• no element is sent to the built set
• no subset is on two chains
• no two elements of one layer are comparable
• so there is something maximal
• some incomparable pair splits the extensions between a third and two thirds
• the built set disagrees with f(k) about whether k belongs
• the chain count is the elements less the matched pairs
• the extensions enumerated and the extensions counted by peeling agree
• the fewest chains covering the order is the size of the largest antichain
• the largest antichain is at least as large as the widest layer
• the largest family in which every two sets meet is as large as a star
• the map names one subset per element
• the number has between four and sixteen divisors
• the number has between four and sixteen proper divisors
• the number of chains is the size of the middle layer
• the number of subsets is two to the power of the set's size
• the number whose divisors are ordered is a whole number between 4 and 210
• the number whose divisors are ordered is between 4 and 210
• the number whose proper divisors are ordered is a whole number between 6 and 210
• the order does not already decide everything
• the order has an incomparable pair to ask about
• the order is one this family draws
• the order is one this mode draws
• the order is small enough to search every collection of its elements
• the order is small enough to search every subset of
• the sets of the right size were all generated
• the size of the sets in the family is a whole number between 2 and 4
• the theorem needs the sets to be small enough that two can miss each other
• the two ends of an edge differ by exactly one element
• the view is one the family draws
• there are more subsets than elements

• a set is contained in its double negation
• and here it is strictly contained — the punched-out points come back
• and so does double-negation elimination
• between one and four points are punched out
• every element sits below its double negation
• every punched-out point is inside the set
• excluded middle fails somewhere in this algebra
• no open set larger than the implication satisfies the condition
• the implication really does meet the antecedent inside the consequent
• the negation of an open set is open
• the set sits strictly inside the ambient line
• the set together with its negation does not fill the line
• the union has as many pieces as the two sets between them
• three negations are the same as one
• three negations collapse to one

• the number of ticks per block is a whole number between 4 and 14 ×2
• a fundamental sequence belongs to a limit ordinal
• a larger ordinal grows at least as fast at every argument
• a limit mark is where the marks before it pile up
• a successor is one step past the ordinal below it
• an ordinal expression is built from whole numbers, w, +, * and ^
• and so do the two multiplications on the other side
• and the terms strictly increase
• at least one entry needs an exponent that is itself a power
• at least one of the ordinals drawn is a limit
• between one and five order types are drawn
• between one and four pairs of expressions are compared
• between three and six rates are drawn
• between two and six ordinals are drawn
• each number's ordinal is strictly larger than the one before it
• each part of the expression is a number, w, or a bracket
• each rate is increasing in its argument
• every limit mark has marks before it
• every limit mark has marks piling up before it
• every term of the sequence is below the ordinal it approaches
• F₀ adds one
• F₁ doubles its argument
• F₂ multiplies by two to the power of its argument
• nothing sits between the pile-up and the limit it accumulates at
• ordinal addition is associative on the drawn values
• the arrow clears both the last term and the supremum it points at
• the base the numbers are written in is a whole number between 2 and 4
• the brackets in the expression close
• the drawn pair is an ordinal of the form ω·k + m with k between 1 and 3
• the exponents fall and the coefficients are positive
• the hereditary form reads back as the number it came from
• the index into a fundamental sequence is a positive whole number
• the largest argument in the table is a whole number between 3 and 5
• the largest number in the table is a whole number between 6 and 34
• the marks along each line strictly increase
• the number of steps shown is a whole number between 3 and 12
• the number of terms shown is a whole number between 3 and 7
• the ordinal beside each term is strictly smaller than the one before it
• the ordinal view is one of goodstein, arith, cnf, fund, growth
• the pair drawn on the lines is one of the pairs in the table, or null for none
• the starting value is a whole number between 2 and 6
• the two additions agree
• the two multiplications by a whole number agree
• the whole expression was read
• zero has nothing below it to approach

• a full column forces every row to carry a mark, so one reading implies the other
• no drawn row agrees with the built row everywhere it has been checked
• no row of the table is the Russell row
• the built row differs from row n in place n
• the drawn window is at least as wide as it is tall
• the labelling is one this figure knows
• the number of columns is a whole number between 2 and 14
• the number of drawn columns is a whole number between 3 and 16
• the number of drawn rows is a whole number between 3 and 12
• the number of rows is a whole number between 2 and 14
• the number of sets in the table is a whole number between 3 and 10
• the relation is one this figure knows how to draw
• the Russell row disagrees with row k at column k

• a form valid without existential import is valid with it
• every letter in the set expression names a drawn curve
• fifteen forms are valid with no assumption that anything exists
• nine more become valid once every term is assumed non-empty
• the form is three mood letters and a figure number, like AAA-1
• there are 256 syllogistic forms — four figures and sixty-four moods
• twenty-four forms are valid in the traditional list
• two circles realise all four patterns

• at 1 instances there is still a model, so the search has not finished ×5
• the assumption h2 is open where it is discharged ×3
• at 1 instances there is still a model ×2
• a conjunction is introduced from two derivations
• a conjunction is taken apart from one derivation
• a disjunction is introduced from one derivation
• a disjunction is used by considering both halves
• a negation is introduced from a derivation and its denial
• an assumption has nothing above it
• an implication is introduced from one derivation
• and both cases reach the same conclusion
• and every assumption it made has been discharged
• and it is the conjunction of what they derived
• and its antecedent is the discharged assumption
• and its consequent is what was derived
• and once it closes it stays closed
• and the count of models is positive at every stage drawn
• and the formula it proves is true in every row of its table
• and the goal follows from the lemma
• and the lemma the other proof uses is not a subformula of the goal
• and the named half is the conclusion
• and the named half is what was derived
• and the negated assumption is the conclusion
• and the second is its antecedent
• and the sweep contains both kinds
• between two and seven stages
• every drawn step is the resolvent of the two clauses above it
• every formula in the cut-free proof is a subformula of the goal, or its negation
• every leaf of the formula is a variable
• every level has a node that reaches the bottom
• every literal names a variable of the clause set
• every opening bracket in the formula is closed
• it closes exactly when the universal has been used 3 times
• modus ponens takes two derivations
• no node has more children than the branching factor
• resolution and the truth table agree on every one of the sets
• resolution reaches the empty clause exactly when no assignment satisfies the set
• so the conclusion is its consequent
• so the count of unsatisfiable sets and the count of refuted sets are one number
• the branching factor is a whole number between 2 and 3
• the case assumption h2 is open
• the clause set is one the family knows
• the conclusion of →I is an implication
• the conclusion of ¬I is a negation
• the conclusion of ∨I is a disjunction
• the cut-free proof closes, so the formula is a theorem
• the depth drawn is a whole number between 3 and 7
• the derivation ends at the stated goal
• the derivation is one of contraposition, transitivity, distribute
• the expansion does close, at some stage this figure reaches
• the first derivation gives a disjunction
• the first is an implication
• the formula and its disjunctive normal form agree on every row
• the formula has between one and five variables
• the formula is a non-empty string
• the lemma is itself a theorem, so it may be cut in
• the path passes through one node per level
• the pruning parameter is a whole number between 2 and 9
• the question is one of reaches, never
• the smallest unsatisfiable set of two-literal clauses needs four of them
• the sweep is over three variables
• the table has one row per assignment
• the tableau and the truth table reach the same verdict
• the tableau stays small enough to draw
• the tableau tests validity or satisfiability
• the tree drawn actually reaches the bottom row
• the two derivations are a formula and its denial
• the view is one the family draws
• the walk always has a surviving child to step to
• the whole formula is consumed by the parser
• there are twelve two-literal clauses on three variables
• this view draws a refutation, so the set has to be unsatisfiable
• which derived a conjunction

• every corner meets one edge per variable, each edge counted once
• every leaf of the formula is a variable
• every opening bracket in the formula is closed
• the cube is drawn for two, three or four variables
• the formula and its disjunctive normal form agree on every row
• the formula has between one and five variables
• the formula is a non-empty string
• the highlighted rows are the true ones, the false ones, or neither
• the table has one row per assignment
• the two ends of an edge differ in exactly one variable
• the whole formula is consumed by the parser

• 3 circles realise every one of the 8 patterns ×2
• 4 circles cannot realise all 16 patterns ×2
• and each of the sixteen is a single connected region
• every leaf of the formula is a variable
• every letter in the set expression names a drawn curve
• no pattern occupies more pieces than the arrangement has
• the formula is a non-empty string
• the four ellipses realise all sixteen patterns
• the number of sets is a whole number between 2 and 5
• the pieces the circles cut the plane into match Euler's count
• the whole formula is consumed by the parser

• a scheme that weights every position the same catches no transposition
• and every single-digit error too
• descending weights over a prime modulus catch every transposition
• every single-digit error is caught exactly when the modulus is at least ten and coprime to every weight
• the error kinds are single, transpose or twin
• the number is between four and fourteen digits
• the number is between six and fourteen digits
• the scheme is one of sum-9, sum-10, flat-11, weighted-10, isbn-11

• a rational is a whole numerator over a non-zero whole denominator
• a right angle can be trisected
• a root found by the theorem really is a root of the cubic
• a straight angle can be trisected
• a zero angle can be trisected
• and above it
• and exactly half way from the other end too
• and is not half way along after it
• and it fails when the given point is not the midpoint, so the midpoint is what is used
• and it is a different line from the segment
• and it is one radius beyond B
• and it lies on the segment
• and it sits over the midpoint
• and that crossing and the marked point are both on the circle
• and the diameter really is twice the radius
• and their images are not equally spaced
• between three and eight polygons, each with 3 to 24 sides
• both crossings exist
• cos(θ/3) is a root of the triple-angle cubic
• divisors are taken of a positive whole number
• each step lands short of the far end
• each step meets the circle
• each step of the compass round the circle meets it
• every image lands on the second line
• every point of the first round is where two drawn objects cross
• Gauss's criterion and the degree test agree about every polygon
• Gauss's criterion and the degree test agree about the tripled polygon
• nineteen of the polygons up to sixty can be drawn
• one round puts two points off the line
• six steps of the radius come back exactly to the start
• sixty degrees cannot be trisected
• some angle in the table the two instruments cannot cut, and the marked one can
• the angle comes from a polygon that can be drawn
• the angle is between 12 and 174 degrees
• the angle the placed straightedge makes is a third of the original
• the apex is near the segment
• the apex sees the diameter at a right angle
• the centre is equidistant from the ends of the diameter
• the circle has a sensible radius
• the constructed line is parallel to the diameter
• the cross ratio is the same on both lines
• the crossing is equidistant from the two ends of the remaining piece
• the denominator the cosines are taken over is a whole number between 2 and 16
• the diameter is tilted between 5 and 175 degrees
• the fixed opening is fixed
• the free point sits inside the segment PM
• the inverting circle meets the circle of radius AB
• the largest polygon tested is a whole number between 12 and 300
• the marked segment between the line and the circle is the radius
• the marked straightedge cuts every angle in the table
• the marks start equally spaced
• the middle mark is half way along before the projection
• the new height is √3/2
• the nine-gon is not — 3 is a Fermat prime but 9 repeats it
• the number being factorised is a whole number between 1 and 1000000
• the number of columns in the grid is a whole number between 6 and 20
• the number of compass steps is a whole number between 2 and 12
• the number of construction rounds is a whole number between 1 and 2
• the number of equally spaced marks is a whole number between 3 and 9
• the number of marks is odd so one of them is the midpoint
• the number whose square root is constructed is between 0 and 12
• the perpendicular at the join has height √n
• the point found is exactly half way along
• the second line is tilted between 0.15 and 0.75
• the second round is shown or not
• the segment cut off is longer than the radius far out and shorter close in
• the segment is between 0.4 and 6 units long
• the segment is between half a unit and two units long
• the segment really is longer than the opening can span in one go
• the seven-gon is not
• the seventeen-gon is constructible
• the sliding point and the centre are the same distance from the near crossing
• the two arcs about D and E meet
• the two arcs of the fixed opening meet
• the two crossings sit at the same height, so the line is parallel
• the view is one of root, trisect, polygons, compass, hexagon, straightedge, neusis, reach, parallel, poncelet, rusty
• the walk was built
• three steps of the radius reach the point twice as far away
• three steps reach the far side, which is the doubling
• twenty-four of the polygons up to a hundred can be drawn
• two points and one round of drawing give four more
• what is left is short enough for the fixed opening to span
• which makes it that piece's midpoint

• step 1 changes exactly one place ×7
• a code is more than one word and fewer than all of them
• and its worst single step changes all 4 places at once
• and the last corner is one step from the first, so the walk closes
• and visits none of them twice
• counting up in the ordinary way changes more places than that
• each cycle was met once in each direction
• every codeword has the length the cube has dimensions
• every corner meets one edge per coordinate
• every step of the code changes exactly one place
• every walk is enumerated for at most the 4-cube
• every word within 1 of a codeword decoded back to it
• the 3-cube carries exactly 6 such walks
• the 4-cube carries exactly 1,344
• the balls partition the cube exactly, so the code is perfect
• the code is a word list or one of repetition, parity, none, even2
• the dimension of the cube is a whole number between 2 and 4
• the drawn walks are the first of the ones counted
• the length of the words is a whole number between 2 and 4
• the radius of the balls drawn is a whole number between 0 and 2
• the view is one the family draws
• the walk visits all 8 corners

• ∛2 × ∛2 lands on a multiple of ∛2²
• ∛2 × ∛2² lands on a multiple of 1
• ∛2 × 1 lands on a multiple of ∛2
• ∛2² × ∛2 lands on a multiple of 1
• ∛2² × ∛2² lands on a multiple of ∛2
• ∛2² × 1 lands on a multiple of ∛2²
• 1 × ∛2 lands on a multiple of ∛2
• 1 × ∛2² lands on a multiple of ∛2²
• 1 × 1 lands on a multiple of 1
• a bigger subgroup names a smaller field, never the other way round
• a rational is a whole numerator over a non-zero whole denominator
• a subgroup of order two fixes exactly one root
• divisors are taken of a positive whole number
• each number under a root is a whole number between 2 and 40
• each number under a root is square-free, or the step is not a real step
• every coefficient is a whole number under 400
• every product of basis elements is a multiple of a basis element
• every rational root found by sweeping is one the theorem listed
• no integer polynomial in the searched range vanishes at π
• no power of two is a multiple of three
• no whole number the rational-root theorem allows is a root of x³ − m
• the closest miss is a genuine miss
• the coefficient bound is a whole number between 2 and 8
• the control polynomial really vanishes at its algebraic number
• the cube root really cubes to it
• the degree of a tower of square roots is a power of two
• the degree of the tower is the size of its basis
• the exact value and the decimal one agree
• the group of six permutations has exactly six subgroups
• the largest degree searched is a whole number between 1 and 4
• the leading coefficient is not zero
• the number being factorised is a whole number between 1 and 1000000
• the number being searched for is one of pi, sqrt2, cbrt2, phi
• the number whose cube root is taken is between 2 and 30
• the polynomial has degree between one and four
• the polynomial is named for the caption
• the powers of two are checked out to between 6 and 20 steps
• the product is the coefficient times the basis element it lands on
• the same search does find a polynomial for an algebraic number
• the searched number is a root of one of them
• the size of the subgroup times the degree of its field is the degree of the whole
• the tower has one basis element per subset of its roots
• the tower is built from one, two or three square roots
• the view is one the family draws

• the left piece keeps its 1th edge at 0° ×21
• the right piece keeps its 1th edge at 0° ×21
• the left piece still touches its pin at 0° ×7
• the right piece still touches its pin at 0° ×7
• the number of sides is a whole number between 3 and 8 ×2
• a kite: and vertically
• a kite: the outward normals weighted by edge length cancel horizontally
• a parallelogram: and vertically
• a parallelogram: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a parallelogram: the outward normals weighted by edge length cancel horizontally
• a polygon of 5 sides falls into 3 triangles
• a polygon of n sides falls into n − 2 triangles
• a rectangle: and vertically
• a rectangle: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a rectangle: the outward normals weighted by edge length cancel horizontally
• a regular hexagon: and vertically
• a regular hexagon: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a regular hexagon: the outward normals weighted by edge length cancel horizontally
• a trapezium: and vertically
• a trapezium: the outward normals weighted by edge length cancel horizontally
• a triangle: and vertically
• a triangle: the outward normals weighted by edge length cancel horizontally
• a triangle's quantity does not vanish, so no number of cuts slides it into a rectangle
• an L: and vertically
• an L: the outward normals weighted by edge length cancel horizontally
• and exactly one step up
• and the other side at its midpoint
• and the same two tile the square
• at half a turn the three pieces tile the rectangle
• between one and five swing angles, each between 0 and 180 degrees
• between two and five shapes
• each numerator over its power of three is the cosine of that multiple
• each shape is a named list of between three and ten finite points
• no edge has zero length
• no numerator in the sequence is a multiple of three
• the apex sits over the base, away from its two ends
• the apex sits over the base, between its two ends
• the cube's dihedral angle is a quarter turn exactly
• the directions are sampled between 90 and 2000 times
• the fan of triangles tiles the polygon
• the number of multiples tested is a whole number between 6 and 30
• the number of risers in the staircase is a whole number between 1 and 5
• the polygon is turned by at most a half turn
• the polygon's radius is between 0.5 and 1.5
• the polygon's radius is between a half and two
• the rectangle and the square have the same area
• the rectangle has the triangle's area
• the same three pieces tile the rectangle
• the side of the square is a whole number between 2 and 24
• the slice meets one side at its midpoint
• the slide is exactly one step across
• the square's side is divisible by n(n+1) so every corner lands on a whole number
• the three pieces tile the triangle
• the triangle is between 0.4 and 1.6 times as tall as its base is long
• the triangles' areas add up to the polygon's
• the two pieces tile the rectangle
• the view is one of steps, polygon, dehn, hinge, translate, count
• three pieces per triangle

• there is a field with 4 elements only if 4 is a prime power ×7
• the size of the field is a whole number between 2 and 9 ×3
• a ring of composite size has a pair of non-zero elements multiplying to zero
• addition and multiplication are commutative
• an irreducible polynomial of the right degree was found
• and it is exactly the composite ones that leave an element without a reciprocal
• every non-zero element has a reciprocal
• multiplication distributes over addition
• multiplication is associative
• no two non-zero elements multiply to zero
• the exponent of a product is the sum of the exponents
• the field is closed
• the number being factorised is a whole number between 1 and 1000000
• the number of primitive elements is φ(q−1)
• the powers of a primitive element close back onto one
• the size of the ring drawn beside it is a whole number between 0 and 12

• class 000 has as many words as the code has ×8
• class 000 has exactly one lightest word ×8
• a row called exact really divides to a power of two
• and always reaches what Gilbert and Varshamov guarantee
• and it weighs no more than one, which is what perfection means
• and none of them is zero
• and the climb is not monotone — some length is worse than the one before it
• at distance three the bound is only ever exact one short of a power of two
• each span is a distance between three and five and a longest length up to eight
• every codeword passes all three parity checks
• every rate is between zero and one
• every two words of the code found are at least the distance apart
• every word in the space is repaired to a codeword by its own syndrome
• four data bits give sixteen codewords
• nor Plotkin's, where it applies
• nor the Singleton bound
• some length meets the sphere-packing bound exactly
• the 128 words fall into eight classes
• the exact answer never beats the sphere-packing bound
• the highlighted codeword is a whole number between -1 and 15
• the lightest non-zero codeword weighs what the closest pair are apart
• the longest word length in the table is a whole number between 5 and 23
• the longest word length searched is a whole number between 5 and 8
• the minimum distance is a whole number between 3 and 5
• the minimum distance over all 120 pairs is three
• the minimum distance the bound is taken at is a whole number between 3 and 7
• the number of cosets shown across is a whole number between 4 and 8
• the rate at the longest length beats the rate at the shortest, at every distance drawn
• the received word is seven bits
• the repetition code and the Hamming code both meet the bound exactly
• the repetition code meets the bound at n = d
• the search finished rather than running out of budget
• the seven columns of the parity-check matrix are distinct
• the seven single errors give seven different syndromes
• the syndrome of a single error is the column of the position it hit
• the view is one of syndrome, cosets, bound, best, rate
• the word the syndrome repairs really is a codeword

• the pair 0,1 appears once ×105
• a construction is known here for this many points
• a design on this many points was actually found by search
• a projective plane over GF(q) has q² + q + 1 points
• any two lines meet in exactly one point
• each point lies in (v−1)/2 triples
• every line carries q + 1 points
• every pair of points appears
• every point lies on q + 1 lines
• exactly one line passes through any two points
• the design drawn is one that exists
• the design has v(v−1)/6 triples
• the divisibility test and the residue rule agree
• the drawing is the projective plane over GF(2), up to relabelling
• the largest number of points in the table is a whole number between 9 and 25
• the largest number of points tested is a whole number between 9 and 40
• the number of points in the design drawn is a whole number between 7 and 15
• the number of points in the design is a whole number between 3 and 21
• the plane is built over a prime between 2 and 5
• the plane is built over a prime field between 2 and 5
• the points are labelled or not
• the residue rule and the divisibility rule agree at every size
• the search agrees with the arithmetic
• the size of the permuted set is a whole number between 1 and 8
• the three midpoints are the same distance from the centre, so one circle holds them
• the two divisibility conditions are exactly v ≡ 1 or 3 mod 6

• the squares for 1 and 2 produce every ordered pair once ×15
• the count at order 1 is a positive number ×8
• a symbol was found for column 1 ×6
• and once in each column of the square for 1 ×6
• column 1 repeats no symbol ×6
• column 1 was given a symbol it could take ×6
• every symbol appears once in each row of the square for 1 ×6
• relabelling left the pair 1, 2 orthogonal ×6
• symbol 0 is missing from exactly n−k of the columns ×6
• a field of order 4 has 3 non-zero multipliers ×5
• the relabelled square for 1 starts 0…3 along its first row ×5
• there is a field with 4 elements only if 4 is a prime power ×5
• row 1 uses every symbol once ×4
• every point lies on 4 lines, one from each class ×3
• the number of rows already placed is a whole number between 1 and 4 ×3
• the 3 available symbols are all used, so an extra square would repeat one ×2
• the order enumerated in full is a whole number between 1 and 4 ×2
• the order of the squares is a whole number between 3 and 8 ×2
• a square laid over itself produces only its own diagonal of pairs
• addition and multiplication are commutative
• an associative Latin square has an identity
• an irreducible polynomial of the right degree was found
• and every table whose elements are all self-inverse has one
• and fifty-six of order five
• and four have every element its own inverse
• and nine thousand four hundred and eight of order six
• and no pair of points is met by two lines
• and none of them carries the symbol already standing in that column
• and once in each column of the first square
• and once in each column of the second square
• and once in each column of the square for α
• and once in each column of the square for α+1
• and sixteen of them associate
• and three of them associate
• and twelve have an element of order four
• each class holds n lines
• each column may still take n−k symbols
• each square either associates or does not
• every line holds n of the points
• every non-zero element has a reciprocal
• every ordered pair of symbols occurs exactly once
• every pair of points is met by some line
• every set of columns can still take at least as many symbols as it has columns
• every symbol appears once in each row of the first square
• every symbol appears once in each row of the second square
• every symbol appears once in each row of the square for α
• every symbol appears once in each row of the square for α+1
• four of the sixteen have every element its own inverse
• multiplication distributes over addition
• multiplication is associative
• no table with an element of order four has a transversal
• no two non-zero elements multiply to zero
• no two standardised squares carry the same entry under the corner
• one shift per row already placed
• relabelling left the pair 1, α orthogonal
• relabelling left the pair 1, α+1 orthogonal
• relabelling left the pair α, α+1 orthogonal
• sixteen tables of order four associate
• the cyclic square has transversals exactly at odd orders
• the drawn order agrees with the table
• the field is closed
• the four self-inverse tables have the same number of transversals
• the largest order enumerated is a whole number between 3 and 6
• the Latin square view is one of transversal, mols, bound, net, count, extend, quasi, mates
• the multiplier of the second square is a whole number between 1 and 7
• the neighbouring orders are shown or not
• the new row uses every symbol once
• the number being factorised is a whole number between 1 and 1000000
• the number of columns is a whole number between 3 and 6
• the number of parallel classes drawn side by side is a whole number between 2 and 5
• the number of squares drawn side by side is a whole number between 1 and 4
• the order of the field the squares are built from is a whole number between 3 and 8
• the order of the plane is a whole number between 2 and 4
• the order of the square is a whole number between 3 and 7
• the pair is built from a prime order, where the construction works
• the parallel classes are the rows, the columns and one per square
• the quasigroup view is one of counts, tables
• the quoted orders are orders between 7 and 11 and larger than the ones enumerated
• the radius of the points is a whole number between 3 and 8
• the relabelled square for α starts 0…3 along its first row
• the relabelled square for α+1 starts 0…3 along its first row
• the second square uses a multiplier that makes it a different square
• the size of the field is a whole number between 2 and 16
• the size of the permuted set is a whole number between 1 and 8
• the squares for 1 and α produce every ordered pair once
• the squares for 1 and α+1 produce every ordered pair once
• the squares for α and α+1 produce every ordered pair once
• the total at order four agrees with every square of order four, listed
• there are 576 Latin squares of order four
• there are four reduced squares of order four
• there are twelve Latin squares of order three
• twelve of them have an element of order four
• two associative squares were found to draw

• and output 4, which was not ×11
• all 4 windows of 2 bits occur somewhere in the period ×7
• the recovered rule matches output 0, which was given ×4
• the modulus is a whole number between 16 and 65536 ×3
• the register of 4 bits visits every nonzero state before repeating ×3
• a multiplier consistent with the outputs exists
• a multiplier is a whole number between 2 and 1048576
• a short vector the modulus annihilates exists
• and it is a de Bruijn sequence: every window of that width, exactly once
• and so is the second
• and the all-zero window never appears
• and they occur equally often, but for the one the all-zero state would have supplied
• and visits none of them twice
• each of them exactly once
• each window is between one bit and the register's width
• every generator's points fall on a genuine family of parallel lines
• every point of the sequence satisfies the relation exactly
• how many high bits are predicted is a whole number between 1 and 8
• no short whole-number relation holds between consecutive outputs
• the all-zero state is the one the register cannot leave
• the all-zero window is the one that is short, by exactly one
• the criterion and the measured period agree
• the dimension is a whole number between 2 and 3
• the first prime is a whole number between 3 and 4000
• the first prime is three modulo four, which the construction requires
• the highest tap is the register's own width, or it is a shorter register
• the increment is a whole number between 0 and 1048576
• the linear solve, run on this generator, predicts the next output wrongly
• the modulus is small enough that the arithmetic stays exact
• the multiplier is a whole number
• the multiplier is a whole number between 2 and 1048576
• the multipliers really do differ, by more than a factor of two in spacing
• the number of outputs it then predicts is a whole number between 2 and 16
• the number of outputs the solver is given is a whole number between 3 and 8
• the number of points is a whole number between 16 and 4096
• the number of predictions is a whole number between 50 and 4000
• the number of triples is a whole number between 100 and 4000
• the occupied hyperplanes are consecutive, so the family has no gaps
• the output shows every nonzero window of the register's width
• the patched sequence is as long as there are windows
• the predictor gets essentially every output right
• the register does not repeat before its full period
• the register visits every non-zero state before repeating
• the register width is a whole number between 5 and 14
• the relation involves the middle coordinate, so an edge-on view exists
• the relation involves the second coordinate
• the relation's normal lies in the screen, so the planes are seen edge on
• the second prime is a whole number between 3 and 4000
• the solver is given only a handful of outputs
• the taps are positions inside the register
• the triples lie on a small number of planes
• the two multipliers give different numbers of lines
• the vector really does annihilate every point
• the view is one of compare, period, planes, lfsr, spectral, recover, nextbit, bbs, equi
• the width of the register is a whole number between 3 and 6
• three to eight multiplier-and-increment pairs
• two multipliers below the modulus
• two taps
• which is enormously better than guessing the leading bits

• the symbol lost at position 1 came back the same ×7
• the number of symbols transmitted is a whole number between 3 and 12 ×4
• the alphabet is a prime between 5 and 17 ×3
• the number of message symbols is a whole number between 2 and 4 ×2
• and inside the Johnson radius the list stays short
• and the count climbs steeply once the radius passes the Johnson bound
• and they differ in that many places
• and this far from the second
• any k of the n symbols recover the message exactly
• at least k symbols survive, or nothing can be recovered and the figure would be a lie
• at the full length every codeword is inside the ball
• every choice of surviving symbols was tried
• every lost position is one of the positions sent
• every non-zero remainder has a reciprocal mod a prime
• more symbols are sent than the message has
• no position is lost twice
• no received word has two codewords within half the minimum distance
• only the zero message gives the zero codeword
• so at least two codewords are within that radius
• the code has two words differing in all but one place
• the code is shorter than the field and longer than the message
• the code meets the Singleton bound exactly
• the field is a prime between 3 and 31
• the lightest non-zero codeword weighs n − k + 1
• the message is k symbols of the alphabet
• the number of received words examined is a whole number between 40 and 2000
• the received word sits this far from the first codeword
• the survivors recover the message
• the view is one of erase, distance, list, pair
• there are at most as many evaluation points as field elements
• which is further than unique decoding reaches
• with one symbol too few, exactly p messages fit the survivors

• district 1 gets exactly the seats it is due ×4
• party 1 gets exactly the seats it is due ×3
• the dial at δ = 0/20 is Adams's own answer ×2
• the dial at δ = 10/20 is Webster's own answer ×2
• the dial at δ = 20/20 is Jefferson's own answer ×2
• the house size is a whole number between 2 and 200 ×2
• a divisor above the interval awards fewer than the house size
• a divisor strictly inside the interval awards exactly the house size
• a quota violation is a whole number of seats away from the band
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• a region outside quota does not sit exactly on its quota
• a table meeting every district's target and every party's target exists
• a trial divisor awards no more seats than the search allows for
• Adams does the opposite
• Adams is found outside quota
• Adams is never found failing house or population monotonicity, which is a theorem about every divisor method
• Adams's awarded seats sum to the house size
• almost every instance drawn is usable
• and Hamilton is found losing a seat as the house grows
• and Hill's is the one no transfer improves for the relative difference
• and losing a seat to a region that grew more slowly
• and the smallest never gains one
• and Webster's bias for the largest region is smaller than either of theirs
• apportioning each district on its own gets at least one party's total wrong
• both ends of the interval land inside the drawn divisor axis
• each entry of the populations is a whole number between 1 and 10000000
• every house size swept has a seat for every region
• every method searched fails at least one of the three properties
• floor division is taken of a whole number by a positive whole number
• Hamilton gives every region the floor or the ceiling of its own quota
• Hamilton is never found outside quota, which is a theorem about it
• Hamilton's awarded seats sum to the house size
• Hamilton's seats sum to the house at every size swept
• Hill seats every region before any second seat
• Hill's awarded seats sum to the house size
• Jefferson gives the largest region more than its quota on average and the smallest less
• Jefferson is found outside quota
• Jefferson is never found failing house or population monotonicity, which is a theorem about every divisor method
• Jefferson's apportionment is improvable under every one of the three
• Jefferson's awarded seats sum to the house size
• no region's seat count falls anywhere in the swept range
• one seat target per district
• one seat target per party
• some allocation is stable for the district measure
• some allocation is stable for the relative measure
• some allocation is stable for the share measure
• the allocations stable for the district measure are all the same allocation
• the allocations stable for the relative measure are all the same allocation
• the allocations stable for the share measure are all the same allocation
• the awarded seats sum to exactly the house size
• the closed end of the interval awards exactly the house size
• the dial changes the answer at all
• the dial has a step exactly at one half
• the districts' seats and the parties' seats add to the same house
• the divisor and the priority ranking award the same seats
• the divisor method is one of jefferson, webster, adams
• the exact arithmetic stays inside the safe integer range
• the exact floor of a quota and its decimal floor agree
• the exact quotas sum to exactly the house size
• the exact quotas sum to the house size in both censuses
• the fall the sweep found is by a single seat
• the family of second censuses is small enough to walk in full
• the faster-growing region holds exactly one seat fewer after the census
• the floors leave between none and one seat per region over
• the growth step is a whole number between 1 and 100000
• the house has at least one seat for every region
• the house is large enough for every seat the method gives away for free
• the house is large enough for the seats the method gives away
• the interval of divisors is not empty
• the largest house swept is a whole number between 2 and 200
• the largest region never loses a seat as the dial turns toward Jefferson
• the marked region really holds fewer seats in the larger house
• the method is one of hamilton
• the method is one of hamilton, jefferson, webster, hill, adams
• the method is one of jefferson, webster, adams
• the mode is one of quota, alabama, divisor, population, family, bias, impossible, unfair, biproportional
• the named quota violation is a real comparison of the seats against the exact quota
• the number of growth steps is a whole number between 1 and 12
• the number of instances generated is a whole number between 50 and 1200
• the number of instances searched is a whole number between 20 and 400
• the number of regions is a whole number between 3 and 6
• the number of steps along the dial is a whole number between 8 and 40
• the open end of the interval awards more than the house size
• the populations is a list of 2 to 8 numbers
• the populations together stay inside the exact range
• the priority list reaches past the house size, so the interval has both ends
• the region that lost a seat grew by the larger exact ratio
• the seats awarded along the dial sum to the house size
• the second census fills exactly the same house
• the second census is larger than the first
• the second census totals what it says it does
• the signpost offset is a fraction between zero and one
• the slower-growing region holds exactly one seat more after the census
• the smallest house swept is a whole number between 2 and 200
• the sweep is short enough to draw one house size at a time
• the sweep runs over at least two house sizes
• Webster is found outside quota
• Webster is never found failing house or population monotonicity, which is a theorem about every divisor method
• Webster's apportionment is the one no transfer improves for the difference in seats per person
• Webster's awarded seats sum to the house size

• voter 1 ranks A and B the same way in both profiles ×5
• each ballot ranks all 3 of them exactly once ×3
• the 3-voter profiles fall into one class per pattern of the pair ×3
• the number of candidates is a whole number between 2 and 5 ×3
• the space of 3-voter profiles was built entire ×3
• B wins all 4 of its pairs ×2
• each candidate of the pair is a whole number between 0 and 2 ×2
• every one of the 64 profiles is examined ×2
• the manipulating voter's true ranking ranks all 3 of them exactly once ×2
• the number of judges is a whole number between 3 and 7 ×2
• A ≻ B ≻ D ≻ C is not an improvement and is not marked as one
• A ≻ C ≻ B ≻ D is not an improvement and is not marked as one
• A ≻ C ≻ B is not an improvement and is not marked as one
• A ≻ C ≻ D ≻ B is not an improvement and is not marked as one
• A ≻ D ≻ B ≻ C is not an improvement and is not marked as one
• A ≻ D ≻ C ≻ B is not an improvement and is not marked as one
• A does not beat everybody
• A has more Borda points than B
• A has more Borda points than C
• A has more Borda points than D
• A has more Borda points than E
• A has more first places than B
• A has more first places than C
• A has more first places than D
• A has more first places than E
• a profile is between one and eight voter groups
• a rational is a whole numerator over a non-zero whole denominator
• A really beats B
• A wins all 4 of its pairs
• an odd number of judges
• an odd number of judges, so majorities are decisive
• an odd number of judges, three to seven, each with a yes or no on three questions
• and it is silent on more profiles than the majority quota is
• and most do not
• and none is both, which is the impossibility this agenda exhibits
• and on some that are not aligned, so alignment is sufficient and not necessary
• and some are consistent
• and some do not, so neither procedure is always wrong
• and the aligned rule gives up universal domain
• and the drawn profile really is one of them
• B ≻ A ≻ C ≻ D elects B, which the voter ranks above D
• B ≻ A ≻ C elects B, which the voter ranks above C
• B ≻ A ≻ C is not an improvement and is not marked as one
• B ≻ A ≻ D ≻ C elects B, which the voter ranks above D
• B ≻ C ≻ A ≻ D elects B, which the voter ranks above D
• B ≻ C ≻ A elects B, which the voter ranks above C
• B ≻ C ≻ A is not an improvement and is not marked as one
• B ≻ C ≻ D ≻ A elects B, which the voter ranks above D
• B ≻ D ≻ A ≻ C elects B, which the voter ranks above D
• B ≻ D ≻ C ≻ A elects B, which the voter ranks above D
• B beats every other candidate head to head
• B does not beat everybody
• B has more Borda points than A
• B has more Borda points than C
• B has more Borda points than D
• B has more Borda points than E
• B really beats C
• Borda can be flipped by moving a third candidate somewhere in this range
• Borda puts A above B in the first profile
• Borda puts B above A in the second
• C ≻ A ≻ B ≻ D is not an improvement and is not marked as one
• C ≻ A ≻ B is not an improvement and is not marked as one
• C ≻ A ≻ D ≻ B is not an improvement and is not marked as one
• C ≻ B ≻ A ≻ D elects B, which the voter ranks above D
• C ≻ B ≻ A is not an improvement and is not marked as one
• C ≻ B ≻ D ≻ A elects B, which the voter ranks above D
• C ≻ D ≻ A ≻ B is not an improvement and is not marked as one
• C ≻ D ≻ B ≻ A is not an improvement and is not marked as one
• C does not beat everybody
• C really beats A
• C really beats D
• Coombs can be flipped by moving a third candidate somewhere in this range
• Coombs eliminated exactly one candidate in every round but the last
• Coombs puts A above B in the first profile
• Coombs puts B above A in the second
• Coombs reaches a winner without a level elimination
• counting the three-way cycles gives the same answer as looking for a winner
• D ≻ A ≻ B ≻ C is not an improvement and is not marked as one
• D ≻ A ≻ C ≻ B is not an improvement and is not marked as one
• D ≻ B ≻ A ≻ C is not an improvement and is not marked as one
• D ≻ B ≻ C ≻ A is not an improvement and is not marked as one
• D ≻ C ≻ A ≻ B is not an improvement and is not marked as one
• D ≻ C ≻ B ≻ A is not an improvement and is not marked as one
• D beats every other candidate head to head
• D does not beat everybody
• D really beats A
• D wins all 4 of its pairs
• dictatorship of judge 1 gives up at least one condition
• dictatorship of judge 1 is consistent wherever it is defined
• dictatorship of judge 1 is defined somewhere
• E does not beat everybody
• each judge is consistent
• each voter group carries a count and an order
• each voter group's size is a whole number between 1 and 500
• every assignment of positions to judges is considered
• every ballot the voter could submit produces a single winner
• every class holds the same number of profiles
• every consistent rule leaves some proposition undecided
• every inconsistent majority on this agenda has the same shape
• every judge's own three answers hang together
• every profile in the space was decided one way or the other
• every ranking the voter could submit was built
• every systematic anonymous rule on this agenda is considered
• instant runoff can be flipped by moving a third candidate somewhere in this range
• instant runoff eliminated exactly one candidate in every round but the last
• instant runoff puts A above B in the first profile
• instant runoff puts B above A in the second
• instant runoff reaches a winner without a level elimination
• majority is consistent on every aligned profile
• majority is defined somewhere
• majority, aligned profiles only gives up at least one condition
• majority, aligned profiles only is consistent wherever it is defined
• majority, aligned profiles only is defined somewhere
• no candidate beats every other, which is what having no Condorcet winner means
• no misreport helps the voter once only two candidates remain
• not every profile is aligned, which is what makes the restriction a restriction
• one candidate has strictly the most Borda points
• one candidate has strictly the most first places
• one disagreeing profile is drawn
• plain majority on every proposition is the rule the impossibility rules out
• plurality can be flipped by moving a third candidate somewhere in this range
• plurality puts A above B in the first profile
• plurality puts B above A in the second
• premise-based gives up at least one condition
• premise-based is consistent wherever it is defined
• premise-based is defined somewhere
• raising the quota never reduces how often the rule says nothing
• some profiles make the two procedures disagree
• some profiles produce an inconsistent majority
• some rules are complete
• the Coombs winner ends with a strict majority
• the dictatorship gives up anonymity
• the distinct winners counted by walking the list agree with the set
• the electorate is a whole number between 2 and 2000
• the electorate is odd, so the two-candidate control cannot end level
• the exact arithmetic stays inside the safe integer range
• the exhausted profile space is small enough to give every profile its own cell
• the exhausted profile space stays small enough to walk
• the honest ballot appears exactly once among the submissions
• the honest winner does not depend on the order the ballots are listed in
• the instant-runoff winner ends with a strict majority
• the largest electorate searched is a whole number between 3 and 6
• the majority between A and B is decided rather than level
• the majority between A and C is decided rather than level
• the majority between A and D is decided rather than level
• the majority between B and C is decided rather than level
• the majority between B and D is decided rather than level
• the majority between C and D is decided rather than level
• the majority quota is inconsistent on some profiles
• the majority tournament runs in a cycle through every candidate
• the majority's three answers do not hang together, which is the whole point
• the margin between A and B is antisymmetric
• the margin between A and C is antisymmetric
• the margin between A and D is antisymmetric
• the margin between A and E is antisymmetric
• the margin between B and C is antisymmetric
• the margin between B and D is antisymmetric
• the margin between B and E is antisymmetric
• the margin between C and D is antisymmetric
• the margin between C and E is antisymmetric
• the margin between D and E is antisymmetric
• the mode is one of profile, cycle, rules, iia, manipulate, judge, agenda, escape, doctrinal, impossible, escapes, quota, aligned
• the number of distinct winners lies between one and the number of rules
• the number of judges is odd, so no question ties
• the number of voters in the exhausted space is a whole number between 2 and 6
• the ordered pairs inside a class counted two ways agree
• the pair is two different candidates
• the premise-based answer is consistent
• the premise-based rule gives up systematicity
• the profile space has one entry for every assignment of ballots
• the rule being manipulated is one of plurality, borda, irv, coombs
• the rule to violate is one of plurality, borda, irv, coombs
• the two opposed counts for A against B sum to the electorate
• the two opposed counts for A against C sum to the electorate
• the two opposed counts for A against D sum to the electorate
• the two opposed counts for A against E sum to the electorate
• the two opposed counts for B against C sum to the electorate
• the two opposed counts for B against D sum to the electorate
• the two opposed counts for B against E sum to the electorate
• the two opposed counts for C against D sum to the electorate
• the two opposed counts for C against E sum to the electorate
• the two opposed counts for D against E sum to the electorate
• the two profiles really differ on some ballot
• the two rules disagree on this profile, which is what makes it a choice
• the two-candidate contest has a winner whichever ballot the voter submits
• the unanimity quota is inconsistent on none
• three judges with three yes-or-no answers each
• unanimity gives up at least one condition
• unanimity gives up completeness
• unanimity is consistent wherever it is defined
• unanimity is defined somewhere

• the layer formula and the average over all 3! orders agree for A ×2
• the layer formula and the average over all 3! orders agree for B ×2
• the layer formula and the average over all 3! orders agree for C ×2
• the quota is a whole number between 1 and 120 ×2
• the weights is a list of 3 to 3 numbers ×2
• A and B are interchangeable
• a member who can never swing a vote has no power by either count
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• and at least one of the other rules fails one
• and D adds nothing
• and it falls at about the square-root rate over the whole sweep
• and satisfies the other three
• and the additivity column's rule gives a different answer here
• and the efficiency column's rule gives a different answer here
• and the symmetry column's rule gives a different answer here
• and the the null player column's rule gives a different answer here
• and their heights add to it
• at least one assembly drawn has power and weight ordered differently
• averaging what each adds satisfies all four on this game
• averaging what each player adds passes every condition on every game tested
• between one and four assemblies
• between three and eight increasing sample sizes
• C is not interchangeable with them, so symmetry is a real condition here
• each entry of the capacities the users need is a whole number between 1 and 200
• each entry of the weights is a whole number between 1 and 40
• each share lands on the grid the search used
• every coalition's value is a whole number
• every estimate is a share
• every ordering ends with everybody in
• no other split on the table has a quieter list of complaints
• on this game the nucleolus and the average split are different points
• some coalition objects to every split drawn here
• some split is individually rational
• somebody can swing something, or the assembly decides nothing
• the capacities the users need is a list of 2 to 5 numbers
• the closed-form shares pay the whole bill
• the coalitions together demand more than there is to give
• the core is drawn for three players
• the core, nucleolus and excess views are written for a game whose value is shared out, and this game's value is a bill
• the denominator the splits are searched over is a whole number between 12 and 180
• the empty coalition is worth nothing
• the error at the largest sample is a third of the error at the smallest
• the error is a distance
• the exact arithmetic stays inside the safe integer range
• the exact shares add to one
• the game is big enough that listing the orders is the point
• the game is one of partnership, majority, gloves, shops, capacity, dummy
• the game names a value for every coalition
• the grid divides the whole group's value evenly
• the group A pays no more together than it would alone
• the group AB pays no more together than it would alone
• the group ABC pays no more together than it would alone
• the group ABCD pays no more together than it would alone
• the group ABD pays no more together than it would alone
• the group AC pays no more together than it would alone
• the group ACD pays no more together than it would alone
• the group AD pays no more together than it would alone
• the group B pays no more together than it would alone
• the group BC pays no more together than it would alone
• the group BCD pays no more together than it would alone
• the group BD pays no more together than it would alone
• the group C pays no more together than it would alone
• the group CD pays no more together than it would alone
• the group D pays no more together than it would alone
• the layer formula and the average over all 4! orders agree for D
• the layers stack up to the largest requirement
• the marginal contributions along one ordering add up to the whole
• the nucleolus survives every objection exactly when some split does
• the power shares add to one
• the proportional rule needs somebody to be worth something alone
• the quota is more than half the weight and no more than all of it
• the requirements are given smallest first
• the resolution of the exhaustive sweep is a whole number between 12 and 120
• the rule in the additivity column fails additive
• the rule in the efficiency column fails efficient
• the rule in the symmetry column fails symmetric
• the rule in the the null player column fails null player
• the search is written for three players
• the second game names a whole value for each of the eight coalitions of three, starting at nothing
• the seed is a whole number between 0 and 10000
• the shares add to what the whole group is worth
• the shares add up to what the whole group is worth
• the splits achieving the minimum are one point to within a grid step
• the sweep and the balancedness condition agree about whether any split survives
• the users do not all need the same thing, or there is nothing to divide unevenly
• the view is one of axioms, core, nucleolus, excess, unique, power, sample, layers
• the whole group is worth a whole number
• there is more than one layer, or the figure has nothing to show
• two games on the same number of players
• two to four games this family knows
• what all four players are worth together is a whole number between 4 and 40

• person 1 values their own share at least as highly as person 1's ×9
• the value matrix, person 1's row is a list of 3 to 3 numbers ×7
• person 1's valuation is a list of 6 to 6 numbers ×6
• trimming piece 1 is worth exactly a third of the trimming to person 3 ×6
• the control matrix, person 1's row is a list of 3 to 3 numbers ×5
• each entry of person 1's valuation is a whole number between 0 and 100 ×4
• no allocation of these goods is envy-free — all 8 were tested and each failed ×4
• person 1 strictly prefers person 3's piece to their own ×4
• person 1's own piece is worth at least 25 to person 1 ×4
• person 1's valuation of the whole cake is 100 ×4
• person 1's values of the 4 pieces account for the whole cake ×4
• person 2's calling mark stays over the bar it belongs to ×4
• each entry of the control matrix, person 1's row is a whole number between 0 and 100 ×3
• each entry of the value matrix, person 1's row is a whole number between 0 and 100 ×3
• person 1's values of the three shares account for the whole cake ×3
• person 2 called the knife at exactly 25 ×3
• piece 1 is worth exactly a third of the cake to person 1 ×3
• the control matrix: person 1 values the whole set at 100 ×3
• the valuations are given for 2 to 2 people ×3
• the value matrix: person 1 values the whole set at 100 ×3
• a knife is asked to pass a non-negative amount of value
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• every envy-free allocation is also envy-free up to one item
• every person still waiting gets a calling mark
• every way of giving each item to one of the people was formed
• person 2 does not envy person 1's piece before the trimming is divided
• person 4, who never called, is left with at least 25
• the chooser envies nobody, having taken the piece it valued higher
• the chooser gains strictly more than half — the promise is not tight
• the chooser's piece is worth at least half the cake to the chooser
• the chooser's values of the two pieces sum to the whole cake
• the claim that somebody is left envious is either made or not made
• the claim that the chooser strictly gains is either made or not made
• the control matrix gives a row of values for two, three or four people
• the control was searched over every allocation too
• the control: a chooser with the cutter's own measure gains exactly nothing
• the control: the same search finds an envy-free allocation where one exists
• the cutter's envy is exactly zero: the piece left behind is worth what the other is
• the division being shown is one of cutchoose, trim, items, moving
• the division is proportional and not envy-free: somebody prefers another's piece
• the exact arithmetic stays inside the safe integer range
• the exact value of the left piece and the decimal one agree
• the number of items in the control matrix is a whole number between 2 and 6
• the number of items in the value matrix is a whole number between 2 and 6
• the number of segments the cake is cut into is a whole number between 2 and 12
• the round-robin allocation is envy-free up to one item
• the three pieces go to three different people
• the trimmed piece ties person 2's second largest exactly
• the value asked for is still ahead of the knife
• the value matrix gives a row of values for two, three or four people
• two calling marks never print on top of one another
• under the cutter's own measure the two pieces are exactly equal

• moving participant 1 from state 0 lowers the potential by exactly what they save ×12
• nobody improves by moving alone out of state 1 ×4
• the column chooser's options names all 2 of them in one to three characters ×2
• the row chooser's options names all 2 of them in one to three characters ×2
• a cell carries both marks exactly when neither chooser can gain by moving alone
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• all the traffic arrives at T
• all the traffic leaves S after the link is added
• all the traffic leaves S before the link is added
• and a move that does not improve raises it by exactly what it costs
• at least one column holds the row chooser exactly to the value
• at least one order of elimination was run
• at least one state has no improving move, so an equilibrium exists
• between two and four participants
• both of them sit on the diagonal, which is what makes the two tie-breakers comparable
• every cell of the matrix was put to both tests
• every distribution on the lattice was tested
• every order of elimination reaches the same surviving set
• every payoff is a whole number no larger than 20 in size
• every traveller is strictly slower after the zero-cost link is added
• no mixing weight on the sweep guarantees the row chooser more than the peak
• no mixture on the lattice holds the row chooser below the value
• no route is cheaper than the one everybody is on, so the flow after is an equilibrium
• no split of the traffic on the lattice costs less in total than the computed optimum
• some pair of columns achieves the column chooser's best mixture
• the best correlated equilibrium is at least as good as the best pure one
• the best use of the added link is somewhere between none of the traffic and all of it
• the bimatrix carries two payoffs in every cell
• the bimatrix drawn by the payoff mode is one of coordination, none, dominant, three
• the bimatrix has between 2 and 4 columns
• the bimatrix has between 2 and 4 rows
• the bimatrix is rectangular — every row offers the same columns
• the closed form for the least possible total agrees with pricing its own flow edge by edge
• the column chooser's mixture holds the row chooser to at most the value against every row
• the computed optimum is itself a point of the lattice
• the correlated mode draws a two-by-two game
• the cost of a fixed-cost link is a whole number between 1 and 400
• the cost per unit of flow on a congestible link is a whole number between 1 and 10
• the crossing sits strictly inside the unit interval
• the drawn distribution is a correlated equilibrium
• the drawn distribution is whole numerators
• the drawn distribution sums to one
• the equilibrium costs strictly more in total than the optimum
• the exact arithmetic stays inside the safe integer range
• the exact value and its decimal recomputation agree
• the first option is the better reply above the crossing and the worse below it
• the first resource never gets cheaper as more use it
• the first resource prices every load it can carry, as whole numbers no larger than forty
• the fixed-cost link is cheap enough that the old routes beat the new one
• the fixed-cost link is dear enough that the added link is taken by everybody
• the game drawn has exactly two pure equilibria
• the game the correlated mode draws is one of chicken, stag, battle
• the game the selection mode draws is one of chicken, stag, battle
• the lattice contains at least one correlated equilibrium
• the lattice of distributions has a denominator between 4 and 24
• the mode of the game family is one of payoff, mixed, dominance, network, correlated, potential, select
• the pure equilibrium AA is a correlated equilibrium too
• the pure equilibrium BB is a correlated equilibrium too
• the pure equilibrium CD is a correlated equilibrium too
• the pure equilibrium DC is a correlated equilibrium too
• the pure equilibrium HH is a correlated equilibrium too
• the pure equilibrium SS is a correlated equilibrium too
• the ratio of the equilibrium's total to the least possible total is exact
• the row chooser's maximin and the column chooser's minimax are the same number
• the row chooser's mixture is worth at least the value against every single column
• the second never gets cheaper as more use it
• the second prices every load it can carry, as whole numbers no larger than forty
• the selection mode draws a two-by-two game
• the state of least potential is one of them
• the surviving column is a best reply to the surviving row
• the surviving row is a best reply to the surviving column
• the traffic entering the network is a whole number between 3 and 40
• the two lines are not parallel, or there is no crossing to draw
• the two lines meet at the mixed equilibrium
• the two routes cost the same before the link is added, so nobody moves
• the zero-sum matrix has between 2 and 4 columns
• the zero-sum matrix has exactly two rows — the mixing probability is one number
• the zero-sum matrix is rectangular — every row offers the same columns
• what arrives at A leaves A
• what arrives at B leaves B
• with two columns

• 1's ranking of side one ranks all 4 of them exactly once ×14
• A's ranking of side two ranks all 4 of them exactly once ×4
• B's ranking of side two ranks all 4 of them exactly once ×4
• the matching drawn as unstable ranks all 4 of them exactly once ×4
• C's ranking of side two ranks all 4 of them exactly once ×3
• D's ranking of side two ranks all 4 of them exactly once ×2
• the member of side two whose lists are searched is a whole number between 0 and 3 ×2
• a profitable misreport is profitable under the ranking actually held
• a proposer still has a name left to propose to
• at least one of the matchings has no blocking pair at all
• at the top, every member of side two has its worst stable partner
• E's ranking of side two ranks all 5 of them exactly once
• each annotation fits inside its own cell with a gap to the cell beside it
• each member of side one settles on the last name it proposed to
• every matching of the two sides was formed
• every ordered pair of stable matchings was joined and met
• every pair of the two sides is put the blocking question
• every proposer ends the construction matched
• every ranking the participant could submit was formed
• every stable matching lies between the two the construction can reach
• no member of the proposing side has a profitable misreport
• nobody proposes to the same name twice
• nothing with a blocking pair is anything the construction returns
• side one has one ranking per member
• side two has one ranking per member
• submitting the true ranking returns the truthful matching
• the bottom of the order is what the construction returns with side two proposing
• the construction returns a one-to-one pairing
• the count of pairs put the blocking question
• the instance has a matching that stability excludes, or there is nothing to draw
• the instance has at least one stable matching
• the join of two stable matchings is stable
• the matching side one proposing returns is in the zero band of the census
• the matching side two proposing returns is in the zero band of the census
• the matching this mode draws is one stability actually excludes
• the meet of two stable matchings is stable
• the mode of the match family is one of run, blocking, lattice, strategy
• the pointwise better of two stable matchings is a matching
• the pointwise worse of two stable matchings is a matching
• the proposals counted and the proposals drawn are the same proposals
• the rounds cannot outnumber the proposals available
• the settled matching has no blocking pair
• the size of each side is a whole number between 2 and 6
• the tally accounts for every matching exactly once
• the top is at least as good as every stable matching for side one
• the top of the order is what the construction returns with side one proposing
• the true ranking is one of the rankings searched
• the whole strategy space of both sides was searched
• the zero column of the tally is the stable set

• the program at b₂ = 2 has at least one feasible vertex ×33
• the program at b₂ = 2 is bounded — an objective that increases without limit has no optimal vertex to draw ×33
• the program at b₂ = 5/2 has at least one feasible vertex ×32
• the program at b₂ = 5/2 is bounded — an objective that increases without limit has no optimal vertex to draw ×32
• the program at b₁ = 2 has at least one feasible vertex ×31
• the program at b₁ = 2 is bounded — an objective that increases without limit has no optimal vertex to draw ×31
• the program at b₁ = 5/2 has at least one feasible vertex ×30
• the program at b₁ = 5/2 is bounded — an objective that increases without limit has no optimal vertex to draw ×30
• the weights on person 1's task 1 add back to the share ×16
• round 1: what is left still has a whole assignment inside it ×5
• person 1's shares add to a whole task ×4
• task 1 is exactly covered ×4
• constraint row 1 constrains at least one variable ×3
• constraint row 1 has one coefficient per variable ×3
• the decomposition needs at most n² − 2n + 2 = 5 whole assignments ×2
• the right-hand side is a list of 2 to 2 numbers ×2
• a program here has between two and five constraint rows
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• a set of prices whose total equals the cheapest assignment exists
• an intersection is kept exactly when it satisfies every constraint
• and every corner is a whole assignment
• and every pair the assignment uses is tight
• and its weights add to one
• and the fractional corner is a half on every edge
• and the weight taken out is positive
• between three and six edges
• each decomposition rebuilds every entry exactly
• each entry of the right-hand side is a whole number between 0 and 400
• each piece's slope is the dual variable on that piece
• every complementary product is exactly zero
• every constraint coefficient is a whole number of size at most 40
• every dual certificate at this optimum gives the same price for the swept constraint
• every feasible primal value is at most every feasible dual value
• every objective coefficient is a whole number of size at most 40
• every pair of constraints was formed, not a selection of them
• every set of people is willing to take at least as many tasks as there are of them
• no dual variable and no reduced cost is negative
• no feasible lattice point beats the best vertex
• no pair's two prices exceed its cost
• no share is negative
• no slack and no variable is negative at the optimum
• so a mixture containing it would give a share to a pairing the corner refuses
• so its total is half the number of edges
• so the prices add to the cheapest assignment's cost, which certifies it
• some order gives a genuinely different decomposition of the same table
• the constraint whose right-hand side is swept is a whole number between 1 and 2
• the corners number the permutations, and no more
• the cost table is square
• the drawn dual region is convex
• the drawn primal region is convex
• the dual certificate is worth exactly what the primal optimum is worth
• the dual is drawn only for a program with two constraints — with more, the dual polytope has more than two variables and is not a polygon in the plane
• the dual optimum is unique, so each constraint has one price — a degenerate optimum, with more than two constraints through one point, has a whole set of them and no single table of products
• the dual program has at least one feasible vertex
• the dual program is bounded — an objective that increases without limit has no optimal vertex to draw
• the exact arithmetic stays inside the safe integer range
• the exact optimum and the decimal one agree
• the exact optimum at the top of the sweep and the decimal one agree
• the feasible vertices are in convex position
• the first order decomposes the table completely
• the fractional table needs at least two whole assignments
• the grid holds more matrices than there are permutations
• the grid the search runs on is a whole number between 3 and 8
• the high end of the sweep is a whole number between 1 and 400
• the low end of the sweep is a whole number between 0 and 400
• the multipliers reproduce the objective exactly
• the number of people is a whole number between 2 and 4
• the objective has one coefficient per variable
• the objective is not identically zero
• the optimal vertex has a dual certificate — a non-negative multiplier on each binding constraint
• the optimal vertex's coordinates are labelled clear of every dot, every value and the axis ticks
• the optimum has a dual certificate
• the optimum is linear across this interval, so the drawn segment is exact
• the polytope figure's mode is one of primal, dual, slack, shadow, birkhoff, extreme, support, assign, hungarian, lottery, fractional
• the primal maximum and the dual minimum are the same number
• the primal program has at least one feasible vertex
• the primal program is bounded — an objective that increases without limit has no optimal vertex to draw
• the reduced cost is the multiplier on the variable's own non-negativity
• the relaxation has a corner that is not whole
• the round empties at least one more cell than it found
• the share table is one of thirds, quarters, sparse
• the size of the assignment is a whole number between 2 and 4
• the slope of the optimum equals the dual variable for the swept constraint
• the support contains a whole assignment
• the sweep produced at least one linear piece
• the sweep runs over between two and sixty units of the right-hand side
• the sweep solved the program once at every whole right-hand side
• the table is one the family knows
• the table is square
• the table is used up exactly
• the two optima agree as decimals too
• the weights add to one
• there are n! = 6 whole assignments
• there is one breakpoint between consecutive pieces
• there is one product per constraint and one per variable
• two different assignments differ somewhere
• weak duality was checked on the whole cross product of the two vertex lists
• which beats every whole matching, so the relaxation is genuinely loose

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