The ledger

What the figures prove — page 2

Page 2 of 6, continuing through the fields in the same order. Every claim here was checked as the drawing was made rather than typed beside it.

Algebra

8 families

`algebra-tiles`

57 kinds of claim · 21 placements

the diagonal point (0, 0) lands on the parabola ×13
the diagonal point (-3, -3) lands on the parabola ×12
and the line for root 0 never rises above it ×7
the line for 0 touches the cusped curve ×7
and the line for root -3 never rises above it ×6
the line for -2.4 touches the cusped curve ×6
the line for root -3 touches the parabola at p = 6 ×6
the line for root 0.5 touches the parabola at p = -1 ×6
the area under e^(−1x² + 0x) is √(π/a)·e^(b²/4a) ×5
the area under e^(−x²)cos(0x) is √π·e^(−0²/4) ×3
the line for root 2 passes through the point ×3
the line for root -1.8100 passes through the point ×2
the sign of the discriminant says how many roots x² − 2x + 1 has ×2
x³ − 3x + 0.5: the discriminant's sign predicts the root count bisection finds ×2
a pair and its swap give the same p
and its height is b²/4a
and the reassembled rectangle has the same area
and the rings out to 1.3 hold π(1 − e^(−1.69))
and the same q, so the map is two-to-one
each exponent opens downward
each marked root is a root
every marked point is inside the plane drawn
the area under the parabola is 13/24 of the square
the bell's area squared is π
the completed form is the same function
the drawn sample's share agrees with 13/24
the exponent opens downward, or there is no bell
the four tiles are the square
the highlighted point is below the curve and has two roots
the larger length is larger
the length really solves the equation
the line for root −1 passes through the point
the line for root 0 touches the parabola at p = 0
the marked pair has two different roots
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 product is a constant times one bell
the product is narrower than either factor
the rings add to π
the sampled general share agrees with (41 + 6 ln 2)/72
the sampled top of the exponent is at b/2a
the second point is above the curve and has none
the sign of the discriminant says how many roots x² − 2x − 3 has
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
two to four members of the family
which is completing the square, as arithmetic
which is the identity
x² + 3x + 1 lies below the curve and has that many real roots
x² + x − 2 lies below the curve and has that many real roots
x² − 2x + 1 lies on the curve and has that many real roots
x² − 2x + 3 lies above the curve and has that many real roots
x² − 2x − 3 lies below the curve and has that many real roots

`complex-turn`

228 kinds of claim · 50 placements

the composite 4 does not divide P(4) ×17737
the prime 2 divides P(2) ×2262
at n = 0, 3 times the class-0 total less 2ⁿ is what the other roots contribute ×98
(3 + 4i)^1 is not a real number ×60
and it has length 5^1 ×60
the coefficient of x^0 at power 1 ×40
P(0) − ρ^0 is twice the real part of σ^0 ×31
the derivative of z^3 − 1 is 3z^2: coefficient 0 vanishes ×27
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
and the average over the 1th roots of ∏(1 + ζᵏ) gives the same ×12
the count is the necklace count exactly when 1 is odd ×12
the odd-divisor sum for 1 is a multiple of 1 ×12
the primitive roots of order 1 add to a real number ×12
the subsets of 1…1 with a total divisible by 1, listed, match the formula ×12
the product 1,4,6,4,1 was found by the search ×10
row 1: the terms written out add to the entry ×8
the factor belonging to the roots of order 1 has degree φ(1) ×8
the gaps at n = 6 repeat those at n = 0 ×8
the product 1,-1,-1,1 was found by the search ×8
the real part of (3 + 4i)^1 is not a multiple of 5, so the fraction is in lowest terms ×8
1 + ζ^0 has length 2|cos(0π/3)| ×7
and it is the coefficients of index 0 mod 3 added one by one ×7
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 product 1,2,1 was found by the search ×7
the walk along the image of the circle of radius 0.4 is fine enough to see which way it turned ×7
the share is exactly 1/2 precisely when the die's polynomial vanishes at the other roots ×6
the throws with a total divisible by 2, counted through the roots ×6
the roots of the 1-th derivative lie in the hull of the ones before ×5
at degree 4 nothing sits between measure 1 and the smallest found ×4
exactly one root of p′ between the 1-th and 2-th roots ×4
the circle of radius 0.4 has 0 roots inside and turns 0 times ×4
the class-0 total found through the roots is real ×4
the curve the image of radius 0.4 never passes exactly through the origin ×4
the degree-1 cyclotomic products, multiplied out, are exactly the polynomials found ×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
x-polynomial x⁴ + 2x³ + 2x² + 2x + 1 has every root in the disc and is a product of cyclotomic polynomials ×4
x-polynomial x⁴ − 4x³ + 6x² − 4x + 1 has every root in the disc and is a product of cyclotomic polynomials ×4
side 1 stays outside the ellipse and touches it at its midpoint (equilateral) ×3
side 1 stays outside the ellipse and touches it at its midpoint (obtuse) ×3
side 1 stays outside the ellipse and touches it at its midpoint (scalene) ×3
the product -1,-1,1,1 was found by the search ×3
the product -1,3,-3,1 was found by the search ×3
far out, the image turns once for each power of z, which is 3 times ×2
the product -1,0,1 was found by the search ×2
the product -1,2,0,-2,1 was found by the search ×2
the product 1,-2,1 was found by the search ×2
the product 1,0,-2,0,1 was found by the search ×2
the roots agree on p₁ at c = -3 ×2
the roots agree on p₁ at c = 1 ×2
the roots agree on p₂ at c = -3 ×2
the roots agree on p₂ at c = 1 ×2
the roots agree on p₃ at c = -3 ×2
the roots agree on p₃ at c = 1 ×2
x-polynomial x³ + 2x² + 2x + 1 has every root in the disc and is a product of cyclotomic polynomials ×2
x-polynomial x³ − 3x² + 3x − 1 has every root in the disc and is a product of cyclotomic polynomials ×2
271441 is 521 squared
a polynomial of degree 3 has 3 roots
all 12 roots add to zero
and |σ|²ρ = 1, the constant term
and close in it does not go round the origin at all
and has no imaginary part left
and it crosses between them
and it divides its own Perrin number
and its coefficients are whole numbers
and its measure is 1.17628
and mod 4 it does outgrow anything mod 3 allows
and multiply to the constant term, with a sign that follows the degree
and one inside, its reciprocal
and so is the necklace sum
and the angles add
and the degrees of the factors add to n, because every root has exactly one order
and the derivative's roots have the same average as the roots
and the middle coefficient lies strictly between −2 and 2
and the two off the circle multiply to 1
and their imaginary parts cancel
and their imaginary parts cancel, because the coefficients are real
and whole ones
and Σz³ is −3c
at a place where the discriminant is passing through zero
at most eight powers are labelled
between 1 and 8 dice are thrown
between 2 and 8 moduli from 2 to 12 are compared
between 20 and 300 polynomials a degree
between 8 and 400 powers are drawn
between two and five whole-number constant terms
between two and six powers are chained
each hull is no larger than the one before
each root found really is a root
each step of the division gives a whole number
eight of its ten roots are on the unit circle
every root found is a root of unity
every root found really is a root
every root is on the unit circle
every root of p′ is real
every root of p′ lies in the hull of the roots (cluster)
every root of p′ lies in the hull of the roots (line)
every root of p′ lies in the hull of the roots (scatter)
in both coordinates
Lehmer's polynomial has four roots on the upper half of the circle
mod 3 no class is ever more than 2/3 away from a third of 2ⁿ
mod 4 the gap is no larger than what 1 + i and 1 − i can contribute
no power of (3 + 4i)/5 up to the 60th returns to 1
no power of a root of Lehmer's polynomial up to the 80th returns to 1
no sampled root is as far as 1 from every root of the derivative
one fewer root of p′ than of p
one is outside
one real root and a complex pair
one to three of the named root sets
one to three of the named triangles
outside the hull the pushes never cancel
row 1: the identity and the roots agree on p₁
row 2: the identity and the roots agree on p₂
row 3: the identity and the roots agree on p₃
row 4: the identity and the roots agree on p₄
row 5: the identity and the roots agree on p₅
row 6: the identity and the roots agree on p₆
row 7: the identity and the roots agree on p₇
row 8: the identity and the roots agree on p₈
six roots, then five, four, three, two and one
subsets are listed for sets of at most 14 numbers
the 1-th powers of the roots add to p₁, found from the coefficients alone
the 2-th powers of the roots add to p₂, found from the coefficients alone
the 3-th powers of the roots add to p₃, found from the coefficients alone
the 4-th powers of the roots add to p₄, found from the coefficients alone
the bar chart shows a set of 3 to 9 numbers inside the table
the classes are remainders mod 3 or mod 4
the classes are remainders on division by 2 to 6
the classes between them hold all 2ⁿ of the coefficients' total
the complex pair lies inside the unit circle
the derivative of a degree-5 polynomial has 4 roots
the distances to the two roots of p′ add to the same total at every midpoint (equilateral)
the distances to the two roots of p′ add to the same total at every midpoint (obtuse)
the distances to the two roots of p′ add to the same total at every midpoint (scalene)
the division leaves no remainder
the ellipse is centred on the centroid
the equation is zⁿ = 1 for n between 4 and 20
the factors multiply out to a polynomial of the right degree
the family passes from three real roots to one
the fifth derivative's one root is the average of the six
the fifth-root polynomial has real coefficients
the grid runs to between 21 and 81
the lengths multiply
the loops sit inside their panel, below its header
the modulus is small enough for exact products in a double
the nearest polynomial outside the disc is well clear of the tolerance
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 one outside is real
the point drawn is a root of Lehmer's polynomial
the point is on the unit circle
the point whose powers are drawn is one the family names
the polygon drawn has between 3 and 16 corners
the polynomial has a degree
the polynomial has a degree between 1 and 6
the polynomial is monic, so its power sums are whole numbers
the polynomials searched have degree 2 to 4
the power drawn is between 2 and 16
the power sum is still an exact whole number
the product -1,0,0,0,1 was found by the search
the product -1,1 was found by the search
the product 1,1 was found by the search
the pushes cancel at each root of p′
the radii drawn are between 2 and 5 positive numbers no larger than 4
the real root is the plastic number, 1.3247…
the remainders are taken on division by 2 to 12
the ring outside the hull was checked
the root set is one the family names
the roots add to minus the next coefficient over the leading one
the roots are distinct
the roots sum to zero
the sequence is drawn to between 12 and 40 terms
the sequence settles on the nearest whole number to ρⁿ well inside the range
the seventh root returns to 1 at the seventh power
the sign of the discriminant says how many roots are real
the smallest measure at degree 10 is Lehmer's polynomial
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 at most 12 powers
the table runs to between 3 and 10 powers
the table runs to between 6 and 16
the third derivative's three roots still span a triangle
the totals account for every throw
the two chains drawn are for moduli in the table
the view is one the family draws
three to seven real roots
with nothing imaginary left over
x-polynomial x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x² + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x² + 2x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x² − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x² − 2x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x³ + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x³ + x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x³ + x² − x − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x³ − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x³ − x² + x − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x³ − x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + 2x² + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + 2x³ − 2x − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + x² + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + x³ + 2x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + x³ + x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + x³ + x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ + x³ − x − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − 2x² + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − 2x³ + 2x − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − x² + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − x³ + 2x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − x³ + x − 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − x³ + x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials
x-polynomial x⁴ − x³ − x + 1 has every root in the disc and is a product of cyclotomic polynomials
Σz is 0 for every member
Σz² is 6 for every member

`eliminate`

16 kinds of claim · 5 placements

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

`group`

221 kinds of claim · 59 placements

a subgroup of 1 divides the group of 8 ×8
the ball of radius 0 in F₂ has the size the formula gives ×8
the ball of radius 0 in Z has the size the formula gives ×8
the ball of radius 0 in Z² has the size the formula gives ×8
D3 has an element of order 2 ×7
D3: every solution but e has order exactly 2 ×7
D3: the solutions of g^2 = e are a multiple of 2 ×7
the ball of radius 0 in Z/2 ∗ Z/2 has the size the formula gives ×7
the ball of radius 0 in Z³ has the size the formula gives ×7
the solutions of x^1 = e are a multiple of 1 ×6
a subgroup of order 8 exists ×5
A4 has an element of order 2 ×5
A4: every solution but e has order exactly 2 ×5
A4: the solutions of g^2 = e are a multiple of 2 ×5
every motion sending corner 1 to 0 is one of 2 ×4
S4 has the number of elements it should ×3
the elements reachable in 1 steps are exactly those at distance 1 or less ×3
A4 has the number of elements it should ×2
colouring 1100: orbit times stabiliser is the group ×2
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
S4 has an element of order 2 ×2
S4: every solution but e has order exactly 2 ×2
S4: the solutions of g^2 = e are a multiple of 2 ×2
the classes hold all 16 colourings between them and share none ×2
the radius runs to between 3 and 8 ×2
a class's size times the relabellings fixing one graph is n!
a class's size times what commutes with one member is the group
a colouring gives one colour per corner
a pair names two different corners
a subgroup whose blocks differ has an element that shows it
a tuple no turn changes repeats one element
a turned tuple still has product e
all four pieces are non-empty
all graphs are drawn for at most four points
and divides the index
and does not fall away
and it is already below three fifths
and no element is in two of them
and none of them is the identity, which would generate nothing
and that does not change as the ball grows
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
and there is a multiple of p of them
at most 20,000 tuples are enumerated
between one and four groups this family knows
between one and six things are tabled
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
corner 1: orbit times stabiliser is the group
corners times the rotations holding one still is every rotation
diagonal 1–3: orbit times stabiliser is the group
e preserves every distance between corners
each step's size divides the last
edge 1–2: orbit times stabiliser is the group
edges times the rotations holding one still is every rotation
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 corner of the cube can be carried to every other
every corner of the dodecahedron can be carried to every other
every corner of the icosahedron can be carried to every other
every corner of the octahedron can be carried to every other
every corner of the tetrahedron can be carried to every other
every edge of the cube can be carried to every other
every edge of the dodecahedron can be carried to every other
every edge of the icosahedron can be carried to every other
every edge of the octahedron can be carried to every other
every edge of the tetrahedron can be carried to every other
every element found is accounted for in the shells
every element sits in exactly one shell
every face of the cube can be carried to every other
every face of the dodecahedron can be carried to every other
every face of the icosahedron can be carried to every other
every face of the octahedron can be carried to every other
every face of the tetrahedron can be carried to every other
every generator names an element of the group
every named element is in the group
every reduced word up to the stated length is listed
every ring's size divides the length
every set has a non-empty edge inside it
every shorter word is in exactly one piece of the a split
every shorter word is in exactly one piece of the b split
every tuple built has product e
F₂ multiplies its ball by more than two at each step
faces times the rotations holding one still is every rotation
graphs on three to five points
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
most of a ball in F₂ is its outermost shell
multiplying on the left carries every edge to an edge
no two of the motions do the same thing to the corners
no word is in both pieces of the a split
no word is in both pieces of the b split
one block for each place in the orbit
one rotation for each way of putting down a directed edge
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 ball is drawn for the groups with a two-dimensional picture
the ball keeps growing
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 classes hold every element once
the classes hold every graph exactly once
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 drawing has extent in both directions
the dual's corners point at the faces in one of its orientations
the edge of a ball in F₂ stays a large share of it
the edge of a ball in Z² is a falling share of it
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 four pieces and the identity account for every word
the generators are distinct
the group is a dihedral group D3 to D8 or a named permutation group
the group is one of Z, Z2, Z3, F2, D, T
the group is one this family knows
the group is the dihedral one or the cyclic one
the groups are among Z, Z2, Z3, F2, D, T
the groups drawn do not all grow alike
the identity is a class of its own
the labelled trees number n to the power n − 2, as Cayley's formula says
the m₁ and m₂ generate the whole group
the motions sending the thing to one place are one coset of the stabiliser
the named elements are closed under composition
the outermost shell of Z is a falling share of the ball
the outermost shell of Z² is a falling share of the ball
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 radius is between 2 and 6
the rings hold every tuple once
the second is too
the series never grows
the sets run to size parameter 4 to 12
the singletons are exactly the elements whose p-th power is e
the solid is one of the five
the subgroup is named by between one and eight element labels
the Sylow subgroups are exactly the conjugates of one of them
the trivial subgroup and the whole group are both among them
the tuples are drawn when there are at most 64 of them
the tuples with product e number |G| to the power n − 1
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
the words run to between 3 and 8 letters
the words strictly inside the bound are counted
their number leaves remainder 1 on division by p
they differ in something a reader can see — the number of steps across, or the number of arrows
two generating sets are compared
two motions with the same target differ by a motion holding the thing still
under a prime turn every ring has 1 or p tuples
Z does not
Z/2 ∗ Z/2 does not
Z² does not
Z³ does not

`linear-map`

145 kinds of claim · 100 placements

and the 5th power agrees with 5 multiplications ×2
a graph with a cycle has at least as many edges as vertices
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 is at most the smaller of the two counts
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 the two add to the number of edges
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 its nullity is the number of independent cycles, counted from the graph itself
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 array is one the family knows
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 incidence matrix's rank is the number of vertices less the number of components
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 number of independent rows and the number of independent columns are one number
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 the source's dimension
what survives and what is lost add to two

`projection`

137 kinds of claim · 42 placements

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 plane and an arrow need the same number of coordinates only in three dimensions
a plane in seven dimensions needs twenty-one numbers
a rational is two whole numbers
a squared length is never negative
a, b and a × b, in that order, form a right-handed frame
a·b is |a||b| cos θ
a·b is the shadow's length times |b|
an ordinary fit is compared only against a weighted one
and holds in three
and in the second
and is not perpendicular in the ordinary sense unless the weights agree
and it is linear in its first slot
and its squared length is what Cauchy–Schwarz leaves over
and only there is the perpendicular to a plane a single line
and so has the second
and so it lies along the adjoint's kernel
and the weighted fit does worse under the ordinary one
and to the second
b is reachable exactly when it casts no shadow on the kernel
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
each of the three pairs multiplies to the fourth unit
every neighbouring point of the plane is further from the target
every pair of distinct units lies on exactly one line
every point is a finite pair
every point of the drawn curve has length one under the product
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
everything the map reaches is perpendicular, under the product, to what the adjoint kills
its length is the parallelogram's area, measured in the parallelogram's own plane
its square and the square of the dot product add to |a|²|b|²
Lagrange's identity holds for every pair tried
moving the map across the product leaves the number where it was
multiplying the reflections gives minus the reflection of the product
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
so the two arrows point into opposite half-spaces
some pair of vectors tells the transpose apart from the adjoint under this product
the adjoint of the adjoint is the map that started
the adjoint sends the drawn direction to nought
the camera angles are within a half turn
the classical order is never the better of the two
the cofactor matrix carries the old normal to the new one exactly
the count runs to between 4 and 10 dimensions
the degree is 1, 2 or 3
the drawn direction is an eigenvector, in the first coordinate
the dropped line meets b at a right angle
the eigen-directions are named by two short strings, or not at all
the eigenvectors are perpendicular under the product exactly when the map is its own adjoint under it
the first basis vector has been scaled to length one
the first edge is three finite coordinates no larger than six
the first vector is not zero
the first vector is seven whole numbers under ten
the first vector is three finite coordinates no larger than six
the first vector is two finite coordinates no larger than twelve
the identity holds for every pair tried, not only for the drawn one
the inequality holds for the weighted product too
the inner product is positive definite, so every non-zero vector has a positive length
the inner product is three finite numbers [p, q, r] no larger than twenty-five
the Jacobi identity fails in seven dimensions
the least value is zero exactly when the two vectors are parallel
the map applied to the old normal is visibly not the new normal
the map has rank one, so what it reaches is a line
the map has two real eigenvalues
the map is four finite entries no larger than twelve
the map is invertible, so the patch stays a patch
the map is three rows of three finite entries no larger than four
the matrix being inverted is invertible
the mirror is a coordinate plane, named by the axis it reverses
the mode of the projection family is one of shadow, signs, schwarz, gram, qr, least, columns, weight, orthloss, legendre, adjoint, selfadjoint, ranges, cross, shadows, mirror, normal, planes, seven
the modified order stays near the arithmetic's own precision
the new normal is perpendicular to the first new edge
the normal equations have a solution
the nudged fit has one coefficient per column
the nudged line is worse than the fitted one
the ordinary fit does worse under the weighted product
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 product is perpendicular to the first factor
the product is perpendicular to the first vector
the quadratic cannot have two distinct real roots
the remainder is perpendicular to the line the map reaches
the residual is perpendicular to the first column
the second drawn vector is perpendicular in the weighted sense
the second edge is three finite coordinates no larger than six
the second vector is seven whole numbers under ten
the second vector is three finite coordinates no larger than six
the second vector is two finite coordinates no larger than twelve
the shadow on the x–y plane has the signed area of the product's third component
the shadow on the y–z plane has the signed area of the product's first component
the shadow on the z–x plane has the signed area of the product's second component
the sign of a·b follows the angle
the squares of the three shadows add to the square of the area itself
the tangent at each eigenvector's tip runs parallel to the other eigenvector
the target is three finite coordinates
the target is two finite coordinates no larger than twelve
the target's squared length splits into the projection's and the residual's
the three pairs span three planes with no direction in common
the two columns are independent, or they span a line rather than a plane
the two eigenvalues are different, so the directions are determined
the two vectors are not parallel
the two vectors are not parallel, or there is nothing left after the subtraction
the two vectors are not parallel, so there is a plane to write down
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 vectors span a plane
the weighted product does not care about the order
the weights are one positive number per point, no larger than a hundred
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
under the ordinary product the adjoint is the transpose
what is left after the subtraction is perpendicular to the first vector

`quaternion`

109 kinds of claim · 33 placements

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

`wiring`

84 kinds of claim · 28 placements

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

Discrete

12 families

`bipartite`

54 kinds of claim · 24 placements

round 1 still has a matching covering every vertex ×3
a largest matching of the contracted graph, plus two edges inside the cycle, is a largest matching of the original — which is why contracting is allowed
a largest matching of this graph holds three edges
and every neighbour named is a vertex on the right
and it is a smallest one, against a scan of every subset of the vertices
and it leaves the stem's end unmatched, which is where a search would start
and no vertex is matched twice
and one whose obstruction is a set that has to be deleted before the odd pieces appear
and one whose obstruction needs no vertices deleted at all — an odd graph is already short by one
and so does every vertex on the right
and the empty set already gives an excess of at least nothing
and the reason is a set with too few neighbours between them
and the rounds use up every edge exactly once
between three and seven graphs the family knows are compared
between two and five graphs with sides are compared
between two and four graphs the family knows are compared
blocked: and it is the size of the largest matching
blocked: the cover the search leaves is a smallest one
cycle5: the matching's shortfall is the worst odd excess
each of which is reached trivially
every edge gets exactly one colour
every matched pair is an edge of the graph
every right vertex reached is matched — an unmatched one would be an augmenting path and the matching would not be largest
every vertex on the left has a neighbour list
every vertex on the left has the same degree
jobs: and it is the size of the largest matching
jobs: the cover the search leaves is a smallest one
narrow: and it is the size of the largest matching
narrow: the cover the search leaves is a smallest one
net: the matching's shortfall is the worst odd excess
regular: and it is the size of the largest matching
regular: the cover the search leaves is a smallest one
so exactly one vertex is left out
star: the matching's shortfall is the worst odd excess
tail: the matching's shortfall is the worst odd excess
the cover is strictly larger than the matching, which is where König's equality fails
the cycle is odd, which is what makes it a blossom
the graph is one the family knows
the largest matching is exactly the left side less the worst deficiency
the matching drawn is made of edges of the graph
the search starts at every left vertex the matching missed
the set built from the failed search touches every edge of the graph
the table has both a graph that succeeds and a graph that fails
the table holds a graph that succeeds and a graph that fails
the table holds both a graph with a perfect matching and one without
the vertices a largest matching misses are exactly the worst excess of odd components over the set removed
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
this view is for a graph with an odd cycle in it
triangle: the matching's shortfall is the worst odd excess
trident: the matching's shortfall is the worst odd excess
two of the matching's edges lie inside the cycle, and the fifth vertex is matched outside it
which is König's theorem: the smallest cover and the largest matching are one number

`clock`

19 kinds of claim · 19 placements

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

`complete-graph`

154 kinds of claim · 61 placements

the branch sets of 0 and 1 are joined by an edge ×13
the 2-point witness has no four-cycle ×8
on 3 points, avoiding a complete graph on 3 allows Turán's count ×7
K5: a subdivision of K5 is also a minor ×6
the plane over 2 gives a graph with no four-cycle ×5
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
K5, drawn flat as a solid's skeleton, has neither ×3
K5: an edge joins two different points of the graph ×3
K5: the minor test and the subdivision test agree about planarity ×3
the best weighting scores ½(1 − 1/ω) with ω = 2 ×3
and 6 is below the known R(4,4) of 18 ×2
cube: a subdivision of K5 is also a minor ×2
K5 with a point split: a subdivision of K5 is also a minor ×2
K5: the best drawing found has one crossing ×2
octahedron: a subdivision of K5 is also a minor ×2
Petersen graph: a subdivision of K5 is also a minor ×2
the grid side is between 6 and 20 ×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
Wagner graph: a subdivision of K5 is also a minor ×2
a named graph
a plane over the field with 2, 3 or 5 elements
a plane small enough to draw its pairs
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
after contracting the spokes, all ten pairs of the five merged points are joined
after deletion the random graph has no four-cycle
and every pair gets one colour either way round
and in fact at least two, which is Goodman's bound
and it contains a subdivision of K3,3, so it is not planar
and it has at most 2d + 1 points for a tree of depth d
and it is reached at a balanced split
and it is the edge count of the balanced two-part graph
and no move creates the forbidden clique
and no tree is listed twice
and on this many points it is the most any such graph has
and respects the cherry bound
and that is more than two thirds of the grid
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
at least one level is balanced
between 30 and 200 points
branch sets are disjoint
cube, drawn flat as a solid's skeleton, has neither
cube: an edge joins two different points of the graph
cube: the minor test and the subdivision test agree about planarity
each clique size counted is between 3 and 8
each clique size is a whole number between 3 and 8
each pair of neighbours of a point is a pair with that common neighbour, and none is counted twice
each path runs along edges of the graph
each piece holds at most half of its parent
enough well-spaced points were placed
EVERY colouring of six people contains a monochromatic triangle
every edge the extremal graph is missing would complete a triangle
every free row joins the free column's component, so that component holds at least k points per free row
every move strictly adds edges
every pair has at most one common neighbour
every pair is coloured exactly once
every point has three neighbours
every tree is drawn for between 3 and 5 points
fewer removed points than rows leave free rows and a free column
five edges in the pentagon
five to nine points
K5 has ten edges
K5 with a point split contains a subdivision of K3,3
K5 with a point split contains no subdivision of K5
K5 with a point split, drawn flat as a solid's skeleton, has neither
K5 with a point split: an edge joins two different points of the graph
K5 with a point split: the minor test and the subdivision test agree about planarity
K6 has fifteen edges
minus one is itself a square, so the rule does not depend on which way round the pair is taken
moving a point to a smaller part adds edges
no triangle has all three edges the same colour
no two edges of the triangulation cross
no two points share two neighbours, so there is no four-cycle
no weighting beats it
octahedron, drawn flat as a solid's skeleton, has neither
octahedron: an edge joins two different points of the graph
octahedron: the minor test and the subdivision test agree about planarity
Petersen graph contains a subdivision of K3,3
Petersen graph contains no subdivision of K5
Petersen graph, drawn flat as a solid's skeleton, has neither
Petersen graph: an edge joins two different points of the graph
Petersen graph: the minor test and the subdivision test agree about planarity
q + 1 points are orthogonal to themselves
six to fifteen points
so the cherries are at most the pairs
some fundamental cycle leaves at most two thirds on each side
some level leaves at most two thirds on each side
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 end is Turán's graph and Turán's count
the even weighting on a largest clique reaches the same score
the exhaustive search finds a K5 minor as well
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 clique has three to five points
the forbidden complete graphs have between three and five points
the graph has q(q + 1)²/2 edges
the graph is connected
the graph is small enough for bitmask search
the graph searched is the Petersen graph, the split K5 or the Wagner graph
the K3,3 columns never part
the K5 columns part on exactly the split K5 and the Petersen graph
the labelled points number between 3 and 7
the largest count is the balanced split's
the majority colour is one of the two
the Paley colouring is drawn on 5, 13 or 17 points
the paths share no interior point, and pass through no branch point
the Petersen graph has K5 as a minor and not as a subdivision
the Petersen graph's shortest cycle has five edges
the plane beats the random graph and stays under the bound at every size
the plane has q² + q + 1 points
the recursion is shallow enough to colour
the result is complete multipartite
the search found a layout that is not degenerate
the search runs on between three and six points
the search runs to between five and nine 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 thinnest balanced level grows like √n: its ratio to √n stays within a narrow band
the thinnest balanced level is within the square-root scale
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 triangle bound allows it and the girth bound does not
the view is one the family draws
the Wagner graph has no K5 minor
the Wagner graph has no K5 minor but a K3,3 minor
this colouring has a monochromatic triangle
two colours over five edges forces three of one
two of the three cannot be drawn flat
Wagner graph, drawn flat as a solid's skeleton, has neither
Wagner graph: an edge joins two different points of the graph
Wagner graph: the minor test and the subdivision test agree about planarity
while it allows the two that can be drawn flat
while the level itself grows several-fold
with at most one part fewer than the forbidden clique

`euler-path`

91 kinds of claim · 42 placements

the final circuit: step 1 walks along the edge it names ×28
Fleury's walk: step 1 walks along the edge it names ×6
the walk after the imaginary bridge is removed: step 1 walks along the edge it names ×5
the first-edge walk: step 1 walks along the edge it names ×3
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
a walk from a vertex of even degree can only get stuck where it started
after removing a cycle every degree is still even
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 each exactly once
and none of them appears twice
and one edge per word
and runs from one odd landmass to the other
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
Fleury's walk uses every edge
Fleury's walk: one edge between each pair of consecutive stops
Fleury's walk: the walk closes
in a graph of even degrees a walk on unused edges never meets a dead end
in the other direction too
it crosses every real bridge once
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
pairing the closest two first costs 6
the alphabet has two or three letters
the best pairing costs 4
the circuit uses every pair once
the concatenation is a de Bruijn sequence
the count grows with the word length
the cycles use every edge exactly once
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 final circuit: one edge between each pair of consecutive stops
the final circuit: the walk closes
the first walk gets stuck with edges left over
the first-edge walk stalls with edges unused
the first-edge walk: one edge between each pair of consecutive stops
the first-edge walk: the walk closes
the graph drawn has at most nine vertices
the graph has one vertex per word one letter shorter
the graph is one of the even-degree examples
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 network is one of the postman examples
the number of odd vertices is even, since the degrees add to twice the edges
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 route is every street once plus the cheapest pairing's paths
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 spliced tour uses every edge
the torus drawn is the four-by-four one
the view is one the family draws
the walk after the imaginary bridge is removed: one edge between each pair of consecutive stops
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
two landmasses have odd degree
vertex A has even degree
vertex B has even degree
vertex C has even degree
vertex D has even degree
vertex E has even degree
vertex F has even degree
vertex O has even degree
with the chosen paths doubled every degree is even

`flow`

74 kinds of claim · 16 placements

the tree gives the cheapest cut between places 0 and 1 ×15
with 3 spokes the pairs share k/2 ×5
the road a to d is carrying its full 1 ×2
among A, B and C the two smallest cheapest cuts are equal
among A, B and D the two smallest cheapest cuts are equal
among A, B and E the two smallest cheapest cuts are equal
among A, B and F the two smallest cheapest cuts are equal
among A, C and D the two smallest cheapest cuts are equal
among A, C and E the two smallest cheapest cuts are equal
among A, C and F the two smallest cheapest cuts are equal
among A, D and E the two smallest cheapest cuts are equal
among A, D and F the two smallest cheapest cuts are equal
among A, E and F the two smallest cheapest cuts are equal
among B, C and D the two smallest cheapest cuts are equal
among B, C and E the two smallest cheapest cuts are equal
among B, C and F the two smallest cheapest cuts are equal
among B, D and E the two smallest cheapest cuts are equal
among B, D and F the two smallest cheapest cuts are equal
among B, E and F the two smallest cheapest cuts are equal
among C, D and E the two smallest cheapest cuts are equal
among C, D and F the two smallest cheapest cuts are equal
among C, E and F the two smallest cheapest cuts are equal
among D, E and F the two smallest cheapest cuts are equal
and in whole units a matching of leaves gets through
and in whole units only one pair can be served
and it is also what reaches the sink
and removing that many roads disconnects the two ends
and separating every pair takes all but one spoke
between 50 and 400 networks
each road is priced at one half
each route continues until it reaches the sink
every place is placed in the tree drawing
every road of this network has capacity one
every road's price is non-negative
every route carries a whole or half unit, as Hu's theorem allows
every route costs at least 1 at the prices
every two-pair network in the family meets its cut, as Hu's theorem says it must
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
family sizes between 20 and 200
fifteen cheapest cuts take at most five values
for two pairs the largest shared flow equals the cheapest separating cut
no cut is smaller than the flow, over every cut there is
no road carries more than its capacity
no two places overlap in the tree drawing
no two routes share a road
nothing is drawn over the headings
stars with 3 to at most 8 spokes
the cheapest set of roads separating every pair has 2 roads
the cut is either drawn or it is not
the fifteen cheapest cuts take at most five values
the linear program is bounded
the network is one this family draws
the prices add up to exactly the flow's value, so both are optimal
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 three pairs can share 3/2 units at once, and no more
the two pairs compete: together they send less than apart
the value is what leaves the source
the view is one the family draws
there are as many routes as the flow's value
whole units fall short of the shared flow here
whole-number flow ≤ fractional flow ≤ cheapest separating cut

`genfun`

49 kinds of claim · 21 placements

at size 1 the two statistics give the same count at value 0 ×25
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 xᵐ/m! at m = 0 is the number of objects on 0 labels ×8
the coefficient of x^0 is the sum along its diagonal ×7
row 1 adds to the plain count at that size ×6
the row for k = 0 adds to the ordinary binomial coefficient ×6
the row for k = 0 has degree k(n − k) ×6
at size 1 the derivative at q = 1 is the total of the statistic over the objects ×5
the mean number of inversions at n = 1 is n(n − 1)/4 ×5
the bars at size 3 add to the plain count ×4
the series is drawn to between 3 and 10 terms ×3
a sequence of singletons on n labels is a permutation of them
an ordered pair on n labels is a two-way split of them
and the marked coefficient counts the combinations written out beside it
at least one coefficient exceeds one, so the polynomial is not merely a string of ones
between two and four boxes, each offering between two and five whole-number values
between two and four sizes, each one this statistic is enumerated at
both label-counts are short lists of small whole numbers
both series are short lists of small whole numbers
every combination of choices is listed
the checked size is between 3 and 6
the coefficient at size 4 counts a deal of the labels together with a structure on each side
the counts rise to one peak and fall away from it
the deal is a choice rather than a formality — the ordinary product would count one
the first piece takes between 1 and n − 1 of them
the four rows give four different counts, so the table is separating the constructions
the labelled count exceeds the unlabelled one, which is what the binomial buys
the labelled family is one this family knows
the labelled object has between 3 and 6 labels
the last row is spread over more than one value, so the statistic is doing something
the marked coefficient is a small whole power
the marked coefficient is inside the product
the marked size is inside both series
the marked total is one the boxes can reach
the mean grows with the size, so the reading is not a constant
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 sizes drawn are between 3 and 5
the sizes drawn are ones this statistic is enumerated at
the statistic is one this family tracks
the table is drawn to a size this statistic is enumerated at
the top of the coefficient is between 3 and 6
the two statistics disagree on individual objects, so the agreement is of distributions
the value at q = 1 is the plain count
the view is one the family draws
there are C(4, 2) ways to deal the labels between the two pieces
two different statistics this family tracks

`map-colour`

26 kinds of claim · 23 placements

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

`number-spiral`

8 kinds of claim · 8 placements

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

`pascal`

80 kinds of claim · 43 placements

Granville's rule gives row 0, entry 0 modulo 4 once its 2s are removed ×906
Lucas' theorem gives row 0, entry 0 modulo 2 ×906
the exponent of 2 in row 0, entry 0 is the number of carries ×756
the row adds to one at x = 0 ×401
row 0, entry 0 is 3 modulo 4 exactly when an odd number of pairs of ones are split ×243
row 0, entry 0 of Pascal's triangle ×91
the additive rule built C(32, 0) ×61
row 0, entry 0 is the binomial coefficient ×45
2 divides the coefficient at least 1 times ×10
and no more than 0 times ×6
degree 2 lies above the convex function ×6
degree 2: averaging down the triangle and the closed form agree ×6
column 0 of the sum is n's digit ×4
the polynomial for k = 0 peaks at k/n ×4
the routes to the cut at height 0 are counted both ways ×4
the weight on P0 is C(3, 0) t^0 (1 − t)^3 ×4
between 3 and 8 rows ×3
degree 2: the gap for x² is x(1 − x)/2 ×3
C(5n, 5k) − C(n, k) is a multiple of 5^3 for every n up to 24 ×2
Granville's product, with its sign, is C(13, 4) with its 2s removed, modulo 4 ×2
n is between 2 and 1000 ×2
the digit product is wrong modulo 4 somewhere in 32 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
a higher degree has a smaller largest gap
a p-free factorial is a unit modulo the prime power
and encloses 1/4
and ends at the last
and the two are not simply equal
and up
at least 8 rows
at most twenty routes are drawn
between 1 and 6 steps up
between 1 and 8 steps across
between three and six control points, each inside the unit square
classifying routes by where they cross the diagonal accounts for all of them
every control point's label sits inside the frame
every drawn entry was checked
every drawn pair agrees modulo 5^3, as the theorem promises
every drawn pair agrees modulo p, which is Lucas' theorem
every entry of the drawn triangle was checked both ways
every point of the curve lies inside the control points' hull
four to ten increasing degrees, up to 1024
k is strictly between nought and n
one to four functions from square, sine, kink, runge
one to three increasing degrees, up to 256
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
t is strictly between nought and one
t, if given, is strictly between nought and one
the addition has as many columns as n has digits
the averaged point is the weighted sum, across
the base is a prime between 2 and 11
the base is a prime up to 7
the curve starts at the first control point
the cut is a diagonal strictly inside the grid
the diagonal is one the triangle holds
the function is one of square, sine, kink, runge
the hull test accepts the hull's own centroid
the identity is one the mode draws
the kink converges like 1/√n
the mode of the pascal family is one of numbers, parity, mod, sierpinski, sums, paths, carries, power, granville, scaled, casteljau, basis, bernstein, rate
the modulus is p to a power from 1 to 4
the modulus is p, p² or p³
the picture is modulo a prime squared, whose remainders have two base-p digits to colour
the prime is 2, 3 or 5
the prime is one of 2, 3, 5 and 7
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 between 1 and 8
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 sine converges like 1/n, however smooth it is
the squares of row 4 add to the middle entry of row 8
the subdivision depth is a whole number between 1 and 8
the weights add to one
x² converges like 1/n

`pigeonhole`

6 kinds of claim · 11 placements

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

`progression`

23 kinds of claim · 11 placements

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

`triangulation`

51 kinds of claim · 36 placements

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

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