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`

101 kinds of claim · 37 placements

negative coefficients of A − 0.008I against eigenvalues below it ×530
negative coefficients of A − -7.000I against eigenvalues below it ×371
step 1 keeps the eigenvalue in its interval ×14
the diagonal point (0, 0) lands on the parabola ×13
the diagonal point (-3, -3) lands on the parabola ×12
decile 1's retest average is ρ times its first average ×10
and the line for root 0 never rises above it ×7
the line for 0 touches the cusped curve ×7
the measured slice spread at ρ = 0 is √(1 − ρ²) ×7
and the line for root -3 never rises above it ×6
order 012 has the eigenvalues' signs ×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 slice at x = 0 has spread √(1 − ρ²) ×3
the slice at x = 0 is centred at ρx ×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
the slice at x = -2 has spread √(1 − ρ²) ×2
the slice at x = -2 is centred at ρx ×2
x³ − 3x + 0.5: the discriminant's sign predicts the root count bisection finds ×2
a coefficient reverses in 1/4 − 1/π² of uniformly drawn correlation matrices, and never both at once
a function on a torus has a pit, a peak and at least two passes
a pair and its swap give the same p
almost every sample completes without a zero pivot
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
both binned slopes are near ρ, well under 1
completions disagreeing with the eigenvalues' signs
each completion halves the interval
each exponent opens downward
each marked root is a root
each wedge holds arcsin ρ / π of the slice
each x first direction carries the sign of its own square
each y first direction carries the sign of its own square
every critical point is non-degenerate
every marked point is inside the plane drawn
in every band of ρ the share follows arcsin|ρ|/π
pits − passes + peaks on the torus
the area under the parabola is 13/24 of the square
the bell's area squared is π
the census completes enough forms to mean something
the completed form is the same function
the drawn sample's share agrees with 13/24
the eigenvalues have one sign each
the eigenvalues multiply to the determinant
the exponent opens downward, or there is no bell
the first pivot of the Hessian is not zero
the form takes both signs and both orders can start
the four tiles are the square
the highlighted point is below the curve and has two roots
the horizontal slice at y = 1.5 is centred at ρ·1.5
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 order x, y, z meets no zero pivot
the order x, z, y meets no zero pivot
the order y, x, z meets no zero pivot
the order y, z, x meets no zero pivot
the order z, x, y meets no zero pivot
the order z, y, x meets no zero pivot
the overall slope is positive and the slope inside every band of x₂ is negative
the picture is about positive lengths
the pivots multiply to the determinant
the population coefficient of x₁ holding x₂ fixed is −0.5
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 share reversed starts at 1/4 − 1/π² and climbs towards one half
the sign of the discriminant says how many roots x² − 2x − 3 has
the slope of the leftovers is exactly the fitted coefficient of x₁
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
the x first coefficients multiply to the determinant
the x first completion has one square of each sign
the y first coefficients multiply to the determinant
the y first completion has one square of each sign
two to four members of the family
which is completing the square, as arithmetic
which is the identity
with fifty observations the sign of a coefficient of −0.2 is usually right for independent predictors and wrong in about one sample in three at ρ = 0.95
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`

518 kinds of claim · 106 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
the rows for 1 and 2 are orthogonal over a common period ×66
(3 + 4i)^1 is not a real number ×60
and it has length 5^1 ×60
the search for 6 agrees with the prime-factor rule at 1 terms ×47
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
c₁(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₁(1) is a whole number ×25
c₁₀(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₁₀(1) is a whole number ×25
c₁₁(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₁₁(1) is a whole number ×25
c₁₂(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₁₂(1) is a whole number ×25
c₂(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₂(1) is a whole number ×25
c₃(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₃(1) is a whole number ×25
c₄(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₄(1) is a whole number ×25
c₅(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₅(1) is a whole number ×25
c₆(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₆(1) is a whole number ×25
c₇(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₇(1) is a whole number ×25
c₈(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₈(1) is a whole number ×25
c₉(1) has no imaginary part, because the primitive roots come in conjugate pairs ×25
c₉(1) is a whole number ×25
root 0 raised to the 5 is 1 ×24
the closed form matches the root sum at q = 1, n = 1 ×18
the coefficient of x^0 in the product is the one in xⁿ − 1 ×16
degree 30: the complex roots come in conjugate pairs ×13
and that number is the Möbius function of 1 ×12
and the average over the 1th roots of ∏(1 + ζᵏ) gives the same ×12
degree 30: the polynomial changes sign across each real root ×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
and whole-number coefficients (index 3, modulo 13) ×9
c₁₅(1) has no imaginary part, because the primitive roots come in conjugate pairs ×9
c₁₅(1) is a whole number ×9
the period polynomial of index 3 modulo 13 has real coefficients ×9
c₃₀(1) has no imaginary part, because the primitive roots come in conjugate pairs ×8
c₃₀(1) is a whole number ×8
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
degree 30: each root found is a root ×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 sampled average at degree 2 agrees with the Kac–Rice value ×7
the walk along the image of the circle of radius 0.4 is fine enough to see which way it turned ×7
c₁₅(1) agrees with von Sterneck's formula μ(q/g)·φ(q)/φ(q/g) ×6
c₃₀(1) agrees with von Sterneck's formula μ(q/g)·φ(q)/φ(q/g) ×6
normal 4: the complex roots come in conjugate pairs ×6
normal 4: the polynomial changes sign across each real root ×6
signs 4: the complex roots come in conjugate pairs ×6
signs 4: the polynomial changes sign across each real root ×6
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
with binomial variances the expected count is exactly √4 ×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
c₁₃(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₁₃(2) is a whole number ×4
c₁₄(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₁₄(2) is a whole number ×4
c₁₆(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₁₆(2) is a whole number ×4
c₁₇(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₁₇(2) is a whole number ×4
c₁₈(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₁₈(2) is a whole number ×4
c₁₉(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₁₉(2) is a whole number ×4
c₂₀(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₀(2) is a whole number ×4
c₂₁(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₁(2) is a whole number ×4
c₂₂(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₂(2) is a whole number ×4
c₂₃(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₃(2) is a whole number ×4
c₂₄(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₄(2) is a whole number ×4
c₂₅(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₅(2) is a whole number ×4
c₂₆(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₆(2) is a whole number ×4
c₂₇(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₇(2) is a whole number ×4
c₂₈(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₈(2) is a whole number ×4
c₂₉(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₂₉(2) is a whole number ×4
c₃₁(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₁(2) is a whole number ×4
c₃₂(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₂(2) is a whole number ×4
c₃₃(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₃(2) is a whole number ×4
c₃₄(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₄(2) is a whole number ×4
c₃₅(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₅(2) is a whole number ×4
c₃₆(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₆(2) is a whole number ×4
c₃₇(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₇(2) is a whole number ×4
c₃₈(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₈(2) is a whole number ×4
c₃₉(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₃₉(2) is a whole number ×4
c₄₀(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₀(2) is a whole number ×4
c₄₁(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₁(2) is a whole number ×4
c₄₂(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₂(2) is a whole number ×4
c₄₃(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₃(2) is a whole number ×4
c₄₄(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₄(2) is a whole number ×4
c₄₅(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₅(2) is a whole number ×4
c₄₆(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₆(2) is a whole number ×4
c₄₇(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₇(2) is a whole number ×4
c₄₈(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₈(2) is a whole number ×4
c₄₉(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₄₉(2) is a whole number ×4
c₅₀(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₀(2) is a whole number ×4
c₅₁(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₁(2) is a whole number ×4
c₅₂(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₂(2) is a whole number ×4
c₅₃(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₃(2) is a whole number ×4
c₅₄(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₄(2) is a whole number ×4
c₅₅(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₅(2) is a whole number ×4
c₅₆(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₆(2) is a whole number ×4
c₅₇(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₇(2) is a whole number ×4
c₅₈(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₈(2) is a whole number ×4
c₅₉(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₅₉(2) is a whole number ×4
c₆₀(2) has no imaginary part, because the primitive roots come in conjugate pairs ×4
c₆₀(2) is a whole number ×4
c₇(1) agrees with von Sterneck's formula μ(q/g)·φ(q)/φ(q/g) ×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 partial sums for n = 2 close in on σ(n)/n ×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
at degree 25 most roots lie within 5/n of the circle ×3
c₃₆₀(1) has no imaginary part, because the primitive roots come in conjugate pairs ×3
c₃₆₀(1) is a whole number ×3
c₉₇(1) has no imaginary part, because the primitive roots come in conjugate pairs ×3
c₉₇(1) is a whole number ×3
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 Barker sequence of length 7 is among the extremes of its length ×3
the partial sum for n = 1 has come close to nought by 3000 terms ×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
|P|² + |Q|² is 2n at every point of the circle
271441 is 521 squared
a ±1 polynomial: each root found is a root
a classical quartic has its known group
a coset containing −1 with each root has a real period
a cubic x³ + px + q with small whole coefficients and three real roots
a cyclic quartic has a positive discriminant
a polynomial of degree 3 has 3 roots
A4 quartics that are translates of even ones
all 12 roots add to zero
an even quartic's group follows the square-class rule
an irreducible cubic has distinct roots
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 combination reproduces every product
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 degree 1000 the expected count is (2/π) ln n + 0.6257 to two places
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
between two and six powers from 1 to 60
box is read only by the census view
c is read only by the cube-root view
c₁(1) is the Möbius value μ(1)
c₁(1) is φ(1), every root raised to 1
c₁(n) repeats with period 1
c₁₀(1) is the Möbius value μ(10)
c₁₀(10) is φ(10), every root raised to 1
c₁₀(n) repeats with period 10
c₁₁(1) is the Möbius value μ(11)
c₁₁(11) is φ(11), every root raised to 1
c₁₁(n) repeats with period 11
c₁₂(1) is the Möbius value μ(12)
c₁₂(12) is φ(12), every root raised to 1
c₁₂(n) repeats with period 12
c₂(1) is the Möbius value μ(2)
c₂(2) is φ(2), every root raised to 1
c₂(n) repeats with period 2
c₃(1) is the Möbius value μ(3)
c₃(3) is φ(3), every root raised to 1
c₃(n) repeats with period 3
c₄(1) is the Möbius value μ(4)
c₄(4) is φ(4), every root raised to 1
c₄(n) repeats with period 4
c₅(1) is the Möbius value μ(5)
c₅(5) is φ(5), every root raised to 1
c₅(n) repeats with period 5
c₆(1) is the Möbius value μ(6)
c₆(6) is φ(6), every root raised to 1
c₆(n) repeats with period 6
c₇(1) is the Möbius value μ(7)
c₇(7) is φ(7), every root raised to 1
c₇(n) repeats with period 7
c₈(1) is the Möbius value μ(8)
c₈(8) is φ(8), every root raised to 1
c₈(n) repeats with period 8
c₉(1) is the Möbius value μ(9)
c₉(9) is φ(9), every root raised to 1
c₉(n) repeats with period 9
clustered: each root found is a root
cubics in the largest box
cyclic cubics whose discriminant has neither a prime 1 mod 3 nor the factor 9
cyclic cubics with every coefficient at most 2
cyclic quartics whose discriminant's squarefree part has a prime 3 mod 4
D4 is the commonest exception and C4 the rarest
each example lands in the group it was chosen for
each hull is no larger than the one before
each nested square root is a root of x⁴ − 4x² + 2
each pair of periods adds to the period it came from
each pairing's value is a root of the resolvent cubic, computed from the coefficients alone
each point cubes to the number
each root found really is a root
each slice runs to a whole number of size 8 to 30
each step of the division gives a whole number
each sum u + v is a real root
each third of the angle projects onto a root
each trigonometric value is a root
each u times its partner v is −p/3
eight of its ten roots are on the unit circle
Erdős and Turán's inequality holds, with their constant 16
even quartics in the slice with group S4 or A4
even quartics with group S4 or A4
every cyclic cubic's discriminant has a prime 1 mod 3 or the factor 9
every discriminant leaves remainder 0 or 1 on division by 4 (Stickelberger)
every no is a splitting field whose degree is not a power of two
every period down to the cosines is real
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 a polynomial with coefficients ±1 lies strictly between radius 1/2 and radius 2
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)
every shifted overlap of the Barker sequence is 0 or ±1
every smaller group grows more slowly than the reducible ones, and C4 slowest of all but V4
every yes with all roots real has degree a power of two
in both coordinates
in both degrees H times the exceptional share has levelled
in the middle of the range the density is close to 2/π per unit of the logarithmic coordinate
irreducible cubics x³ + px + q in the box
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
n runs to between 8 and 30
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
normal: each root found is a root
of them with a square discriminant
of them with three real roots
one fewer root of p′ than of p
one is outside
one real root and a complex pair
one step off the even plane, most quartics have the full group
one to four values of n
one to three of the named root sets
one to three of the named triangles
one to three values of n
outside the hull the pushes never cancel
p and q are read only by the Cardano and trisection views
q is from 3 to 36
q runs to between 4 and 16
quartic is read only by the pairings view
quartics in the largest box
random signs rise with the length
real roots of the 11-gon quintic
real roots of x³ − 4x + 2
real roots of x⁴ − 4x² + x + 1
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
Rudin–Shapiro stays under √2 at every length
set is read only by the vanish view
signs: each root found is a root
six roots, then five, four, three, two and one
square discriminants run at three to six times the random-number expectation
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 24 permutations fall into 6 classes of 4 by their effect on the pairings
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 average square on the circle is the number of coefficients
the bar chart shows a set of 3 to 9 numbers inside the table
the carried density integrates to the expected count
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 classical cyclic cubics are found
the coefficients run to between 10 and 40 each way
the complex pair lies inside the unit circle
the cubic has three distinct real roots
the cyclic count grows like H to the three-halves
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 expectation grows at nearly the same rate as the true count
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 a prime from 5 to 23
the modulus is small enough for exact products in a double
the nearest polynomial outside the disc is well clear of the tolerance
the normalised counts have settled from H = 100
the number is a whole number from 2 to 9
the number of periods divides p − 1 and is less than it
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 pairings that come out whole are the resolvent's rational roots
the parts add up
the period polynomial is monic
the periods add to −1, the sum of all the non-trivial roots
the permutations fixing every pairing are the identity and the three double swaps
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 polygon has 5 to 16 sides
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 polynomial: each root found is a root
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 product of each pair is a whole-number combination of the level above
the pushes cancel at each root of p′
the quartic has no whole-number root, so its pairings are not trivially rational
the quartic is one the family names
the radii drawn are between 2 and 5 positive numbers no larger than 4
the real radical expression is a root
the real root is the plastic number, 1.3247…
the reducible count grows like H²
the reducible quartics grow like H³
the remainders are taken on division by 2 to 12
the resolvent of an irreducible quartic has zero, one or three rational roots
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 Rudin–Shapiro polynomial never exceeds √2 times √n
the search stays small enough to finish
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 seven-term sum splits into turned regular polygons
the seventh root returns to 1 at the seventh power
the share with smallest root r falls like 1/r²
the sign of the discriminant says how many roots are real
the six-term sum is minimal and is no union of turned polygons
the slice holds cyclic cubics
the slice runs to a whole number of size 8 to 40
the smallest measure at degree 10 is Lehmer's polynomial
the square root in Cardano's formula is of a negative number
the squared size is the sum of the overlaps as cosines
the sum is one the family names
the sum is zero, exactly, modulo the cyclotomic polynomial
the sum runs to between 200 and 6000 terms
the sum runs to Q from 10 to 200
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 tower is drawn for the Fermat primes 5 and 17
the two chains drawn are for moduli in the table
the vanish view takes its n from the named sum
the view is one the family draws
three to seven real roots
with an even number of sides some diagonals meet three or more at a point
with an odd number of sides no three diagonals meet, so there are C(n, 4) crossing points
with nothing imaginary left over
with variances 1/k! the count over √n falls towards 2/π
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
x³ − 2 is not cyclic
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`

375 kinds of claim · 112 placements

the hex ball of radius 0 ×13
the king ball of radius 0 ×13
the square ball of radius 0 ×13
the pieces of Z/2 ∗ Z/2 outside radius 0 that reach the outer shell do not change when it moves out ×10
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
D4: a class's size times its centraliser is the group ×6
D4: a group that does not commute reaches at most 5/8 ×6
D4: the centralisers of one class add up to the group ×6
D4: the centre is at most a quarter of the group ×6
D4: the commuting pairs are the size times the number of classes ×6
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
the pieces of F₂ outside radius 0 that reach the outer shell do not change when it moves out ×5
the pieces of Z × Z/2 outside radius 2 that reach the outer shell do not change when it moves out ×5
the pieces of Z outside radius 2 that reach the outer shell do not change when it moves out ×5
the pieces of Z/6 outside radius 0 that reach the outer shell do not change when it moves out ×5
the pieces of Z² outside radius 2 that reach the outer shell do not change when it moves out ×5
the pieces of Z³ outside radius 0 that reach the outer shell do not change when it moves out ×5
every motion sending corner 1 to 0 is one of 2 ×4
P2: a class's size times its centraliser is the group ×4
P2: a group that does not commute reaches at most 5/8 ×4
P2: the centralisers of one class add up to the group ×4
P2: the centre is at most a quarter of the group ×4
P2: the commuting pairs are the size times the number of classes ×4
the Pauli group on 1 has 4^1 + 1 classes ×4
the Pauli group on 1 has a centre of two ×4
the Pauli group on 2 has 2·4^2 elements ×4
S3: the classes are the partitions of 3 ×3
S4 has the number of elements it should ×3
S4: a class's size times its centraliser is the group ×3
S4: a group that does not commute reaches at most 5/8 ×3
S4: the centralisers of one class add up to the group ×3
S4: the centre is at most a quarter of the group ×3
S4: the commuting pairs are the size times the number of classes ×3
the elements reachable in 1 steps are exactly those at distance 1 or less ×3
the radius runs to between 3 and 8 ×3
A4 has the number of elements it should ×2
A4: a class's size times its centraliser is the group ×2
A4: a group that does not commute reaches at most 5/8 ×2
A4: the centralisers of one class add up to the group ×2
A4: the centre is at most a quarter of the group ×2
A4: the commuting pairs are the size times the number of classes ×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 drawing of Z/2 ∗ Z/2 shows as many outward pieces as are counted ×2
the radius is between 2 and 6 ×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 finite group has nothing left outside a large enough ball
a pair names two different corners
a point reached in d steps lies inside d·P
a prime-cycle witness is found for the drawn pair
a shape the group lacks never occurs as a pattern
a subgroup whose blocks differ has an element that shows it
a tree walk is back after two steps with chance 1/4
a tuple no turn changes repeats one element
a turned tuple still has product e
A5 to the 19th is two-generated and to the 20th is not
above the threshold the diagonal time is the minimum, below it more
all four pieces are non-empty
all graphs are drawn for at most four points
and a finite stretch between the cuts
and divides the index
and does not fall away
and it is already below three fifths
and nearer at the larger radius
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 every square c = k² the central element is exactly 4k steps away: the commutator of xᵏ and yᵏ
at most 20,000 tuples are enumerated
at radius 44 the ball rises r²/16 over its centre and r²/8 over a = ±r/2
at the largest radius the Heisenberg ball holds several times as many elements as the three-dimensional lattice's
axis + one diagonal: |B(r)|/r² is near the polygon's area
axis steps: |B(r)|/r² is near the polygon's area
between one and five of the quintics this family knows
between one and four groups this family knows
between one and six of the quintics this family knows
between one and six things are tabled
between one and three generating sets among square, hex, king, knight
between one and three generators are named
between one and three groups this family knows
between three and eight groups this family knows
between two and four generating sets
both maps are of the same group
box is read only by the trinomial view
classes of generating pairs of A5 under automorphisms
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
conjugate subgroups share a Möbius value
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 local exponent approaches the degree Bass and Guivarc'h's formula gives
each pattern's share is its shape's share of the group
each root found makes the polynomial vanish
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 of the tree of triangles is placed
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 factor pattern that occurs is a cycle shape in the group
every generator names an element of the group
every group drawn has a picture in the plane
every irreducible x⁵ + b has the group of twenty
every map ax + b is in the group
every named element is in the group
every point of the box is reached
every point of the diamond is the shadow of something in the ball
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
F₂'s count rises at every radius
faces times the rotations holding one still is every rotation
from twelve letters most failures are a shared fixed letter
generating pairs of A5 up to automorphism
graphs on three to five points
Hall's Möbius sum equals the direct count of generating pairs
in a line a second cut leaves two outward pieces
in a tree the second cut adds outward pieces
king's moves: |B(r)|/r² is near the polygon's area
knight's moves: |B(r)|/r² is near the polygon's area
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
near radius thirty the Heisenberg ball grows with exponent close to four and the lattice in three dimensions with exponent close to three
no map ax + b has a 3-cycle or a single swap
no missed point is more than three polygon-steps inside the edge
no point is reached in fewer steps than its norm
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
on the tree the distance grows at about half a step per step
one block for each place in the orbit
one rotation for each way of putting down a directed edge
pairs of A5 that generate it
pairs of S4 that generate it
pairs that generate A4
polys is read only by the resolvent and census views
Q8: a class's size times its centraliser is the group
Q8: a group that does not commute reaches at most 5/8
Q8: the centralisers of one class add up to the group
Q8: the centre is at most a quarter of the group
Q8: the commuting pairs are the size times the number of classes
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
S4 is generated by three-eighths of its pairs
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 automorphisms act freely on generating pairs
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 ball of radius one holds the identity and the four generators
the ball of radius two holds 17 elements, two fewer than the lattice in three dimensions would give if xy and yx were the same
the ball's shadow lies inside the diamond |a| + |b| ≤ r, since x and y are the only steps that move a and b
the bars are drawn for at most 24 elements
the block a product lands in does not depend on which elements were picked
the blocks between them cover the group exactly once
the bound is drawn for a group that does not commute
the box runs to between 6 and 16
the chance falls and stays above one half
the classes are drawn for at most 64 colourings
the classes hold every element once
the classes hold every graph exactly once
the coefficients run to between 10 and 40 each way
the commutator xyx⁻¹y⁻¹ is four steps from the identity
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 discriminant is a square exactly when every symmetry is even
the drawing has extent in both directions
the drawing of F₂ shows as many outward pieces as are counted
the drawing of Z × Z/2 shows as many outward pieces as are counted
the drawing of Z shows as many outward pieces as are counted
the drawing of Z² shows as many outward pieces as are counted
the drawing reaches at least two steps past the cut
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 element types account for every element
the elements are closed under composition
the estimate falls as the distance grows
the even symmetries of five letters have no normal subgroup in between
the exhaustive search is drawn for at most 5 corners
the factor degrees add to five
the fibres add up to the ball
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 generating set is one this family knows
the generators are distinct
the grid is drawn for at most twelve elements
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, ZZ2, C6
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, ZZ2, C6
the groups drawn do not all grow alike
the Heisenberg ball and the lattice in four dimensions grow at the same degree with different constants: the lattice's is the larger
the highest point of the whole ball of radius 44 is r²/4
the identity is a class of its own
the knight misses points inside its polygon
the labelled trees number n to the power n − 2, as Cayley's formula says
the lattice counts agree with direct enumeration
the lattices' roots approach 1
the m₁ and m₂ generate the whole group
the measured rate matches 2√(k − 1)/k
the motions sending the thing to one place are one coset of the stabiliser
the named elements are closed under composition
the outer sphere is (k − 2)/(k − 1) of a large ball
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 primes run to between 1,000 and 20,000
the r and m₁ generate the whole group
the r generate the whole group
the radius removed is between 0 and 3
the real roots of x⁵ + -5x + -12
the real roots of x⁵ + -5x + 12
the real roots of x⁵ + 20x + -32
the real roots of x⁵ + 20x + 32
the removed radius runs to between 2 and 5
the rings hold every tuple once
the sampled share follows Dixon's expansion from nine letters
the search reaches central elements beyond c = 100
the second is too
the series never grows
the sets run to size parameter 4 to 12
the sextic has a whole-number root exactly for the solvable groups
the sextic's coefficients are whole numbers
the share of the polygon reached does not fall as r grows
the singletons are exactly the elements whose p-th power is e
the six values are distinct, so the test applies as stated
the solid is one of the five
the solvable trinomials with a ≠ 0 in the box are exactly these six
the sphere divided by r³ levels off
the spread grows like a power near one third
the subgroup is named by between one and eight element labels
the Sylow subgroups are exactly the conjugates of one of them
the three-dimensional count agrees with enumeration
the translate's centre is inside the drawing
the tree's root approaches √3/2
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 two tests and the primes place each quintic in its group
the view is one the family draws
the word e lands on e
the word length is never more than three above the norm
the word length never decreases as c grows
the word length of (0, 0, c) stays within four of 4√c
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
three or four increasing radii up to 24
two affine maps compose to the affine map with multiplied slopes
two generating sets are compared
two motions with the same target differ by a motion holding the thing still
two of Z, F2, T and D are compared
under a prime turn every ring has 1 or p tuples
upTo is read only by the census view among the quintic views
Z × Z/2 settles on 0, 1 or 2
Z does not
Z settles on 0, 1 or 2
Z/2 ∗ Z/2 does not
Z/2 ∗ Z/2 settles on 0, 1 or 2
Z/2 ∗ Z/3's count rises at every radius
Z/6 settles on 0, 1 or 2
Z² does not
Z² settles on 0, 1 or 2
Z³ does not
Z³ settles on 0, 1 or 2

`linear-map`

321 kinds of claim · 175 placements

the closed walks of length 1 are the 1th power sum of the eigenvalues ×8
there are 2 graphs on 2 points up to relabelling ×5
and the 5th power agrees with 5 multiplications ×2
every determinant is a multiple of 2^(n−1) = 4 ×2
the matrix is square, 2×2 to 7×7 ×2
48 of the 880 are symmetric about their centre, opposite cells adding to 17
a centre neither collapses nor escapes
a component made of k discs holds exactly k eigenvalues
a degenerate sink carries a generic point to the origin
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 point outside every disc is not an eigenvalue
a power of the matrix is between 2 and 24
a quoted maximum is a multiple of the unit and under the cap
a real 2×2 map has two eigenvalues counted with sign
a saddle carries a generic point away
a sink (node) carries a generic point to the origin
a sixth of the closed walks of length three is the number of triangles
a skew matrix exponentiates to a rotation, first entry
a source (node) carries a generic point away
a spiral sink carries a generic point to the origin
a spiral source carries a generic point away
a symmetric matrix's eigen-directions are perpendicular
a touching pair and a free pair were both found
a window of 32 average gaps holds a nearly fixed count of eigenvalues, as Dyson and Mehta predicted
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 exp L has determinant one
and exponentiating the two eigenvalues gives the same matrix
and in some column disc
and it is at most the smaller of the two counts
and it keeps area exactly
and its discs obey the same count
and its signed area is about the determinant times the square's
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 none passes the next one's starting value
and reaches it on the smallest patch drawn
and so are every two columns
and so are their images
and so does the triple product of the columns
and some non-embeddable matrices break the inequality
and the angle shrinks by the ratio of the two eigenvalues
and the boundary's image winds that many times round the origin
and the direction it occurs in is an eigen-direction, to within one sample
and the identity leaves the unit square alone
and the largest is the largest multiple of the unit under the bound
and the lowest at the second
and the lowest is the smaller
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 the two are a right angle apart
and there is a direction it sends to the origin
and where it has got to is what that rate predicts
any plane containing the top eigenvector reaches the top eigenvalue
at n = 10,000 the edge matches the Tracy–Widom mean and spread
at n = 1000 the histogram follows the semicircle
AᵀA is symmetric and has two real eigenvalues
Av is λv in the first coordinate
Av is λv in the second
bisection and QL agree on the largest eigenvalue
both eigenvalues are positive, so the level set is a closed curve
closer to staying put more tables pass, and fewer pass with every state added
critical damping never overshoots
D⁻¹AD has the same eigenvalues as A
Dürer's square has rank three
Dürer's square is one of the 880
each coefficient of the characteristic polynomial is a whole number
each eigenvalue makes A − λI singular, by a determinant the polynomial never saw
each eigenvalue of the smaller block sits between two of the larger's
each exponentiates back to the same transition matrix
each Jacobi pair satisfies Av = λv
each marked matrix has determinant one
each matrix in a row is four finite entries under 8 in size
each of the three basis squares is magic
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 eigenvalue lies in some row disc
every eigenvalue of a square and a lone point is a root of its characteristic polynomial
every eigenvalue of a star with four arms is a root of its characteristic polynomial
every eigenvalue of another graph on 6 points and 7 edges is a root of its characteristic polynomial
every eigenvalue of one graph on 6 points and 7 edges is a root of its characteristic polynomial
every embeddable matrix has 0 < det P ≤ p₁₁p₂₂p₃₃
every entry is a sign
every matrix drawn has a real logarithm
every matrix lands inside the plane drawn
every one has 5 in the centre
every one of these is a logarithm of the same matrix
every pair was classified once
every plane's maximum lies between the middle and top eigenvalues
every point drawn satisfies xᵀAx = 1
every point of that line lands on the origin
every preimage maps back onto its target
every row of Dürer's square is killed by (−1, 3, −3, 1)
every sign matrix is counted once
every singular square has its pairs adding to 17 placed by one symmetry of the rows and columns
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
every triangle has three different corners
every two rows are perpendicular
exactly eight of the 362,880 arrangements of 1 to 9 are magic
fourth
half of every size have a positive determinant; Goodman's condition and the exact test thin them much faster
half the closed walks of length two is the number of edges
heavier damping settles more slowly
in both coordinates
inside the unit circle the series converges
near doing nothing, more matrices are embeddable than among matrices drawn at random
near staying put no table fails by a negative eigenvalue; drawn at random most do
no angle's sine exceeds one
no determinant exceeds Hadamard's bound
no eigenvalue goes down
no exponential of a traceless matrix has trace below −2
no invertible square has its pairs placed by such a symmetry
no square has its pairs placed by two of the symmetries at once
no triangle is listed twice
on a closed surface every edge is shared by exactly two triangles
on five points exactly two graphs share an adjacency spectrum: the star and the square with a lone point
one dimension survives and one is lost, and they add to the two started with
one eigenvalue is strictly the largest
one entry of v per row
one square on each side of the fold
one to three graphs
one to three spaces
one variant at a time
outside it the partial sums run away
P D P⁻¹ is the map it started from
Paley's matrix has size q + 1 for a prime q ≡ 3 mod 4
reached at the top eigenvector itself
scaling one column scales the area by the same factor
second
size five stops at forty-eight
size four reaches the bound exactly
size three stops at four
small gaps are rare among eigenvalues and common among independent points
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
Sylvester's matrices have sizes 2, 4, 8, 16
t runs up to at most 40
the alternating sum of the holes is the alternating sum of the cells
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 boundary of a triangle's boundary is nothing
the camera angles are within a half turn
the census finds a cospectral pair at this size
the census runs to between four and six points
the census walks every sign matrix of size 2, 3 or 4
the change of basis is invertible
the classes' orbits cover every labelled graph once
the column is scaled by between one and three
the complementary pairs fall into exactly twelve patterns, Dudeney's twelve groups
the construction is Sylvester's or Paley's
the correspondence is drawn at size 3, 4 or 5
the count holds at every stage of the continuation
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 bound exactly
the determinant is the number of spanning trees, counted
the determinant is the product of the lengths times the sine
the determinant of the product is the product of the determinants
the determinant over the disc is the degree times the disc's area
the diagonal family crosses and the coupled family does not
the drawn area is the determinant
the edges split into what the boundary kills and what it keeps
the eigenvalues add to the trace
the eigenvalues add up to the trace
the eigenvalues of a three-state rate matrix lie within 30° of the negative axis
the eigenvalues rise in total by exactly t|v|²
the eigenvalues' gaps vary less than independent points' gaps
the eight symmetries of the square cut them to 880, Frénicle's number
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 flat drawing is for the small spaces
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 gap's mean is 2√(π/2)
the graph is one the family knows
the highest point sits at the first eigen-direction
the highest value on the circle is the larger eigenvalue of the symmetric part
the holes in dimension nought are the connected pieces
the image line has a length to normalise
the image of the unit square has the determinant's area
the image turns the way the determinant's sign says
the incidence matrix's rank is the number of vertices less the number of components
the integral of the determinant equals the integral of the signed count
the iteration runs between 2 and 40 steps
the Laplacian has one zero eigenvalue for each connected piece
the largest eigenvalue approaches 2√n from below and its spread shrinks
the lattice is drawn far enough out to fill the counting window
the lattice is drawn out to between 3 and 12 steps
the line crosses the cusp region
the Lo Shu square is 5J − 3U + V
the logarithm comes out real
the logarithm exponentiates back to the matrix
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 lowest of the plane-maxima is the middle eigenvalue
the magic squares of order n form a space of dimension n² − 2n
the magic squares of order three form a three-dimensional space
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 of the three the family folds
the map is one the family knows
the maps drawn have different ranks
the marked point is a complex number as a pair
the matrix drawn here collapses the plane onto a line
the matrix handed to the symmetric solver is symmetric
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 invertible
the matrix is square, 2×2 to 6×6, entries under 40
the matrix is the adjacency or the Laplacian
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 minmax view is for a 3×3 matrix
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 numbers are the rationals (0) or the two-element field (2)
the parabola through a row's first three entries passes through its fourth
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 points split into what the boundary kills and what it keeps
the QL iteration converges
the rate from 1 to 3 turns negative near s = 2·D₁₃ ÷ (D²)₁₃, and the table stops having steady rates there
the rate matrix found exponentiates back to the transition matrix
the rates found reproduce the transition probabilities
the raw count jumps only where the line crosses the cusp
the residual falls by four at every halving of the patch
the right angle reaches the product of the lengths exactly
the roots add to the trace
the rotation method and the polynomial agree
the row operation adds one row to a different one
the row operation leaves the determinant exactly where it was
the scaling is one positive weight per row
the search never beats the divisibility cap
the search runs to size 3, 4 or 5
the second is too
the semi-magic squares of order n form a space of dimension (n − 1)² + 1
the series has converged
the share falls with every state added, at every distance from doing nothing
the shifted determinant and λ² − (trace)λ + determinant agree
the sign matrix's determinant is (−2)^(n−1) times the bit matrix's
the signed count is one all the way along
the signed count of preimages is the same at every target drawn
the signed sum is the determinant
the smallest ellipse holding a lattice point has area at most four times the determinant
the space is one the family knows
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 target is a rotation or minus the identity
the targets drawn include one-sheet and three-sheet points
the three basis squares are independent
the three columns are not flat, or there is no solid to draw
the trace of exp L is 2 cos √det L (or cosh), whatever L's entries
the triangles split into what the boundary kills and what it keeps
the tridiagonal model and full matrices give the same largest eigenvalue
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
there are 7,040 normal magic squares of order four
third
this transition matrix has exactly two logarithms that are rate matrices, among the branches tried
three well-separated eigenvalues
two column lengths between 0.4 and 2
two different roots give two independent directions
two equal columns leave no area at all
two finishing points, on a small grid
two of these logarithms do not commute
two rows agree in exactly half their places
two starting points, on a small grid
two to eight matrices
two to four angles strictly between 5° and 175°
two to four of the maps the figure knows
two valid logarithms need the complex eigenvalues to be tiny
unfolded spacings have mean one
walk lengths up to between 3 and 10
what survives and what is lost add to the source's dimension
what survives and what is lost add to two
while the raw count takes both one and three
within each pattern either every square is singular or none is

`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`

122 kinds of claim · 39 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 coordinate of a uniform point on the three-sphere has mean square one quarter
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
a uniform angle averages 90°
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
both paths end at the same orientation
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 set is closed under multiplication
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
slerp turns at constant speed
slerp turns through exactly the angle of the single rotation
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 binary groups have 24, 48 and 120 unit quaternions
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 Euler path turns through more angle
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 histogram follows the semicircle
the icosahedral grid leaves no rotation more than 45° away and beats sixty random rotations
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 mean trace after k steps is 3((1 + 2 cos ε)/3)ᵏ
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 caps hold their predicted shares
the two orders give different quaternions
the two rotations move at least one axis a long way apart
the uniform rotations' mean angle is π/2 + 2/π, 126.48°
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`

114 kinds of claim · 42 placements

exactly half the arrangements of the 3×3 tray are reached ×5
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
the search on θ, inner paths 2, 2, 2 agrees with Wilson's theorem ×2
a single closed tour moves the tiles in one cycle
a tour of 4 squares cycles 3 tiles
a tray of four to nine squares, so its far end can be drawn
a tray of four to ten squares
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 exchanging two is an odd permutation
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 follows an edge
each step of the tour moves the blank one square
every board one legal move from a reachable one has the same parity
every distance fits in a byte
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
exactly two tokens have moved
exchanging two tiles puts the board on the other value of the invariant
from every arrangement some move goes one step closer
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 bars are split by nothing or by the blank's square
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 empty vertex is home again
the empty vertex is where the tour says
the even classes together are exactly half the group
the far end holds one to four arrangements
the farthest arrangement chosen exists
the graph is one this family knows
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 moves has the parity of the blank's displacement
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 search on a house agrees with Wilson's theorem
the search on a six-cycle agrees with Wilson's theorem
the search on a six-cycle with one chord agrees with Wilson's theorem
the search on a wheel of six agrees with Wilson's theorem
the search on K₂,₃ agrees with Wilson's theorem
the search on the 2×3 tray agrees with Wilson's theorem
the search on θ₀, inner paths 1, 2, 2 agrees with Wilson's theorem
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 taxicab sum and the distance have the same parity
the taxicab sum never exceeds the true distance
the tour starts and ends at the empty vertex's home
the tour starts from the blank's home square
the two components account for every arrangement
the view is one the family draws
the walk reaches solved in exactly 31 moves
the whole graph is drawn only for the two-by-two board
there are exactly as many irreducible characters as conjugacy classes
three to ten graphs
three to ten vertices, so the search fits in memory
two decompositions of one permutation have the same parity
two pieces exchanged breaks its own invariant
two pieces exchanged breaks only its own invariant
two to six trays
with nothing odd among them

Discrete

12 families

`bipartite`

106 kinds of claim · 51 placements

the branch giving a₀ the partner b1 dies when a1 is reached ×5
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
a sliding domino lands on a free square of the diamond
a typical random order matches more
an order as bad as the adversary's is rare
and each level has at most two nodes, one of them about to die
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 stays above it, up to sampling
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
arrivals choosing by rank and places choosing by arrival time give the same matching
between four and nine vertices a side are drawn
between four and twelve vertices a side
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
dominoes in an Aztec diamond of this order
each of which is reached trivially
every board missing a corner and a square of the other colour can be tiled
every edge gets exactly one colour
every edge of the best matching touches a greedy edge
every finite set of left vertices has at least as many neighbours
every hole is a two-by-two block
every level of the finite-degree tree is nonempty
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
four to seven levels
frozen squares and the outside of the circle mostly coincide at order 100
greedy in the adversary's order matches half
greedy, taking the first available, matches one
in the adversary's order the rule matches exactly half
jobs: and it is the size of the largest matching
jobs: the cover the search leaves is a smallest one
minus signs round a unit face
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
no graph found beats the proved floor for RANKING in random order
no order does worse than the adversary's half
no ranking does worse than half, as any greedy rule guarantees
no signing of K3,3 makes the determinant 6
nor the floor 1 − 1/e for greedy in random order
plain determinant terms that come out positive
random-order greedy and adversarial RANKING both approach 1 − 1/e
RANKING in random order climbs well above both on this graph
RANKING's share on the triangle approaches 1 − 1/e
rectangles whose plain determinant is 0 or 1
regular: and it is the size of the largest matching
regular: the cover the search leaves is a smallest one
signings of K3,3 giving 0 or 4
so exactly one vertex is left out
so greedy has at least half as many edges
star: the matching's shortfall is the worst odd excess
tail: the matching's shortfall is the worst odd excess
the attempted partner for a₀ is one of the drawn b's
the best matching matches both
the column rule alone is wrong exactly when the hole is interior
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 determinant with the cut agrees with the listed count
the diamond has 2 to the 1/4 tilings per square at every order
the disagreement shrinks as the diamond grows
the formula agrees with the listed count
the graph is one the family knows
the graph is the counterexample or the finite-degree one
the largest matching is exactly the left side less the worst deficiency
the matching drawn is made of edges of the graph
the order is drawable
the per-square rate climbs to its limit from below
the plain determinant of the 4 by 4 board
the search starts at every left vertex the matching missed
the search starts from the triangular graph and keeps anything worse
the set built from the failed search touches every edge of the graph
the signs count the 4 by 5 board
the step drawn deletes a colliding pair
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 triangular graph has a perfect matching: arrival t to place t
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
tilings of the 4 by 4 board
tilings of the board missing two opposite corners
tilings of the order-2 Aztec diamond
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
with the signs every tiling contributes alike

`clock`

48 kinds of claim · 30 placements

for odd 3 not dividing 1, the count is 0 or 2 at every power ×13
the cycle through 0 has the length of 2's order modulo 21/gcd(0, 21) ×11
x² ≡ 2 holds modulo 7^1 ×8
the residues that 2 leaves in place number gcd(1, 180) ×6
modulo 8, an odd a has 4 square roots when a ≡ 1 mod 8 and none otherwise ×3
multiplying by 2 reaches N/gcd(2, 199) of the residues ×3
the walk reaches m × n / gcd cells on a 3 by 5 grid ×3
and that length divides 2's order modulo 21 ×2
3 to 12 digits
a dial needs at least three positions
a dial of 5 to 60
a dial of 6 to 400 places
a multiplier from 2 to N − 1
a multiplier sharing no factor with the dial
a row is a permutation exactly when its multiplier is coprime to the modulus
a small prime
an odd prime
each root reduces to a root one level down
every cycle of units has the full order
every Newton step at least doubles the number of correct digits
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
exactly one digit extends the root, and it is the one the formula gives
it fills the whole grid exactly when the moduli are coprime
multiplying `order` times returns to 1
the arithmetic and the walk agree
the dial and the complex circle agree on where each residue sits
the dial view is one the family draws
the drawn entry is the arithmetic
the drawn walk passes the top as often as the division says
the envelope touches each chord along its own direction
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 slope is invertible at every step
the starting digit is a square root of 2 mod 7
the walk is between one and 144 steps
the walk runs at least one full period
two to eight multipliers
two to five levels, the top modulus under 20,000
two to five primes
two to six Newton steps
two to six values of a
with p odd and not dividing a, every level has as many roots as the first — each lifts in exactly one way

`complete-graph`

269 kinds of claim · 121 placements

a tree with a cherry has a symmetry, so the shares are ordered at 4 ×13
the branch sets of 0 and 1 are joined by an edge ×13
asymmetric trees on 1 points ×12
the labellings on 1 points add to Cayley's count ×12
the number of trees on 2 points ×12
every tree on 2 points is graceful ×11
the 2+1-part family meets the formula at edge density 0.560 ×9
the 2-point witness has no four-cycle ×8
functions with 1 cyclic points and marked trees with 1 points on the path ×7
on 3 points, avoiding a complete graph on 3 allows Turán's count ×7
at the Turán density 1 − 1/2 the curve touches Goodman's bound ×6
K5: a subdivision of K5 is also a minor ×6
2^2 functions, 2^0 trees, 2² pairs of marks ×5
measured cyclic points agree with the sum at 10 ×5
no two functions on 2 points give the same marked tree ×5
Ramanujan's sum is √(πn/2) − 1/3 to within a fraction at 10 ×5
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
every marked tree on 2 points reads back to its function ×4
on 3 points no two trees share a code ×4
the search reaches the known fewest crossings for 7 points ×4
10 to 120 points ×3
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
every vertex has degree 7 ×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
the table runs to between 8 and 12 points ×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
— the midpoint of the two corners they hang from
12 edges on 7 points can avoid every triangle
175 edges: 50 × 7 / 2
2 to 4 cars
a caterpillar has a graceful labelling
a named graph
a path has a graceful labelling
a plane over the field with 2, 3 or 5 elements
a plane small enough to draw its pairs
a point's degree is one more than its count in the Prüfer sequence
a spider has a graceful labelling
a star has a graceful labelling
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
almost every large random tree has a cherry
an asymmetric tree has no cherry
and each exactly once
and every pair gets one colour either way round
and far above Euler's
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 meets the Moore bound
and no move creates the forbidden clique
and no tree is listed twice
and none has fewer than the minimum curve allows
and on this many points it is the most any such graph has
and respects the cherry bound
and so are its point labels
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 second has d(d − 1), every one reached exactly once
and the trees, built one by one, number the same
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
between three and seven nested triangles
branch sets are disjoint
cars is read only by the parklist view
counting every tree gives the binomial law for the degree
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
curved drawings never need more crossings than straight ones
each clique size counted is between 3 and 8
each clique size is a whole number between 3 and 8
each is triangle-free and not bipartite
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
each triangle is a fixed fraction of the one outside it
enough well-spaced points were placed
EVERY colouring of six people contains a monochromatic triangle
every count of labellings is whole
every crossing of a straight drawing is the pair of diagonals of four points in convex position, and every such four gives one
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 lower bound is below the known value
every move strictly adds edges
every pair has at most one common neighbour
every pair is coloured exactly once
every point at the average of its neighbours, with the outer face convex, gives a drawing with no crossings
every point gets an image
every point has three neighbours
every ratio lies inside the drawn range
every tree has at least one graceful labelling
every tree is drawn for between 3 and 5 points
every tree on eight points is graceful
every triangle-free graph is bipartite up to four vertices; at nine fewer than half are
every vertex is within two steps of every other, checked from another vertex too
every vertex is within two steps of the first
exactly one of the n + 1 rotations of a list empties the last spot
fewer removed points than rows leave free rows and a free column
five edges in the pentagon
five to nine points
for large complete graphs the sampling bound is within a factor of about eight of the conjectured value
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
Mantel: at most 16 edges on eight vertices, reached only by the 35 balanced complete bipartite graphs
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 cage is smaller than the Moore bound
no graph has more triangles than a clique of the same edge density
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
non-bipartite triangle-free graphs need at least five edges
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
on eight vertices every non-bipartite triangle-free graph has a shortest odd cycle of five or seven
on seven vertices every non-bipartite triangle-free graph becomes bipartite on deleting one edge
on the circle each preference is one of the n + 1 spots
on the circle every list parks every car and leaves exactly one spot empty
one edge more forces ⌊n/2⌋ triangles — Rademacher's theorem
one, one and three asymmetric trees on seven, eight and nine points
only degrees 2, 3, 7 and 57 give whole-number multiplicities
Otter's formula agrees with the shapes found from every Prüfer code
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
points minus edges, plus one for a flipped edge, is one
prefs is a list of 3 to 7 preferred spots, each between 1 and the number of cars
prefs is read only by the parking and parkcircle views
q + 1 points are orthogonal to themselves
Razborov's curve never dips below Goodman's bound
sampled trees have the predicted share of leaves
six to fifteen points
so the cherries are at most the pairs
so the graph has exactly 1 + d + d(d − 1) = d² + 1 vertices, the most a diameter of two allows
so the parking functions are (n + 1)^n / (n + 1) of the lists
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
t(n) α⁻ⁿ n^(5/2) approaches 0.53495
the alkane counts begin 1, 1, 1, 2, 3, 5, 9, 18, 35, 75
the averaging converges
the bijection is checked up to between 3 and 6 points
the bipartite count's exponent approaches n²/4 from above
the bound is of the order of 2 to the k over 2
the cherries' swaps are symmetries, so their count divides the total
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 complete graph has between 5 and 9 points
the converged drawing has no crossings
the decoded tree has one edge fewer than points
the dial has a radius between 40 and 400
the drawing has no crossings
the drive and the sorting test agree on whether every car parks
the end is Turán's graph and Turán's count
the enumeration and the generating function agree on the bipartite graphs
the enumeration matches Rényi's count of trees with k leaves
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 first layer is the neighbours
the forbidden clique has three to five points
the forbidden complete graphs have between three and five points
the free-tree counts begin 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, …
the function permutes its cyclic points
the function sends every point to a point
the graph has q(q + 1)²/2 edges
the graph is connected
the graph is one of the three-connected ones the view knows
the graph is small enough for bitmask search
the graph searched is the Petersen graph, the split K5 or the Wagner graph
the growth constant is 2.95576528565…
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 labelling's edge differences are all different
the labellings of all shapes on 7 points add to Cayley's count
the largest count is the balanced split's
the last spot stays empty exactly when the list is a parking function on the line
the majority colour is one of the two
the mean is Ramanujan's sum Σ n!/((n − k)! n^k)
the mean number of leaves is n(1 − 1/n)^(n − 2)
the pairs of rooted halves come in twos
the Paley colouring is drawn on 5, 13 or 17 points
the parking functions number (n + 1)^(n − 1)
the parking functions of length n number the labelled trees on n + 1 points
the paths share no interior point, and pass through no branch point
the Petersen graph has girth 5
the Petersen graph has K5 as a minor and not as a subdivision
the Petersen graph's shortest cycle has five edges
the Petersen or the Hoffman–Singleton graph
the plane beats the random graph and stays under the bound at every size
the plane has q² + q + 1 points
the process ends near n^(3/2)√(ln n)/(2√2) edges, far from bipartite
the random tree is graceful
the ratio with Otter's power removed is close to α long before the plain ratio is
the reading gives n − 1 edges
the recursion is shallow enough to colour
the result is complete multipartite
the rooted-tree recurrence divides exactly
the rotated copies reach every edge of K₇
the scallops rise at most a few hundredths above the parabola
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 shortest cycle has five edges
the stages are two or three sweep counts
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 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 tree, read back, is the same function
the triangle bound allows it and the girth bound does not
the two points hung between opposite corners are pulled to the same place
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
there are 23 tree shapes on eight points, by search and by Otter's formula
there are 23 trees on eight points
they are exactly the lists whose sorted form never exceeds its positions
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
whitening the point set leaves every four in the position it was
with at most one part fewer than the forbidden clique

`euler-path`

126 kinds of claim · 61 placements

edge 0 lies in exactly two cycles of the cover ×30
the final circuit: step 1 walks along the edge it names ×28
every random bridgeless cubic graph on 10 points has a cover ×6
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
no naive cycle for 3 ×2
the shorthand cycles found by search are the Euler circuits counted by the BEST theorem, once per starting point (3) ×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 cycle exists with one extra symbol
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
and there are n! of them
and two arrive
at n = 4 the shortest length is 33 and the string is unique up to renaming
each face's side is an edge
each ordering appears exactly once
each word is smaller than all its rotations
every edge borders exactly two faces
every order-isomorphic cycle is found once from each start
every vertex drawn is a vertex used
every window has a different relative order
every window of two consecutive symbols is a different pair
every window stands for a different permutation
every word of the right length appears in the cycle
exactly one bridge, the middle edge
Fleury's walk uses every edge
Fleury's walk: one edge between each pair of consecutive stops
Fleury's walk: the walk closes
from 1234 exactly one visiting order gives 33, and none gives less
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 cycle of six on three symbols has six different permutations as windows
no cycle passes through the bridge
no cyclic sequence of 6 symbols shows every pair from 4 exactly once
no Euler walk exists
no prefix of the classical string beats the best possible count, and 33 symbols hold all 24
nor any cycle of twenty-four on four symbols
pairing the closest two first costs 6
the 24 orderings fall into six rings of four rotations
the alphabet has two or three letters
the best pairing costs 4
the circuit uses every edge once
the circuit uses every pair once
the classical string is never more than one ordering behind the best
the concatenation is a de Bruijn sequence
the construction and the greedy rule both give 1! + … + n!
the costs add up to 1! + … + n!
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 fewest cycles that cover each edge twice is five
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 flower snark has a cycle double cover
the flower snark has no bridge
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 order-isomorphic cycle is drawn for three or four
the patch appears at exactly one position of the array
the patch is taken from inside the sheet drawn
the Petersen graph has 57 cycles
the ring is drawn for three or four symbols
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 shortest string steps eighteen times within a ring, four times to a neighbouring ring and once by three symbols
the shorthand graph is drawn for three or four symbols
the sixteen positions give sixteen different blocks
the spliced tour uses every edge
the string of 33 contains all 24 orderings
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 edges leave every vertex
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`

97 kinds of claim · 26 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
a dearer flow leaves a cycle of negative cost
a flow is cheapest exactly when it leaves no cycle of negative cost
a road cheaper than the price difference is full
a road dearer than the price difference carries nothing
a road used but not full has reduced price nought
A's isolating cut is no dearer than the boundary of its side
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 is within 2 − 2/k of it
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
B's isolating cut is no dearer than the boundary of its side
between 50 and 400 networks
C's isolating cut is no dearer than the boundary of its side
each extra unit costs at least as much as the one before
each road is priced at one half
each route continues until it reaches the sink
every network falls in one bar
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 usable road pays its way at these prices
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 network breaks the 4/3 guarantee
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
pushing round the cycle lowers the cost by its price times the amount
some flow delivers 4
stars with 3 to at most 8 spokes
the amount sent is a whole number between 1 and 5
the cheapest flow leaves no cycle of negative cost to push round
the cheapest set of roads separating every pair has 2 roads
the cut is either drawn or it is not
the drawn flow is a flow
the drawn roads add up to the cut's value
the fifteen cheapest cuts take at most five values
the linear program is bounded
the network is one this family draws
the number of networks is a whole number between 200 and 5000
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 shortcut never beats the cheapest cut
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 three sides' boundaries count every cut road twice
the two cheapest isolating cuts together separate all three places
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`

91 kinds of claim · 46 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
claw-free graphs on 3 vertices with a non-real independence root ×7
graphs on 3 vertices whose matching polynomial has a non-real root ×7
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 5 and 10 terms ×2
a chromatic coefficient sequence rises to one peak and falls (Read's unimodality)
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 exact sign search does not find them all on the negative axis
and the marked coefficient counts the combinations written out beside it
as many sign changes on the negative axis as the degree: every root is real and negative
at k = 1 the counts are 1, m and C(m, 2), exactly binomial
at least one coefficient exceeds one, so the polynomial is not merely a string of ones
at some coefficient the ratio falls below what real roots would require
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
each root of unity is a root of the inversion polynomial
each root's reciprocal is a root
equality holds only where every set of k − 1, k and k + 1 edges is a forest
every combination of choices is listed
every root lies within 2√(Δ − 1) = 2√2 for a graph of largest degree three
every sampled graph's edge count lies on the drawn axis
every small graph satisfies all three of Mason's inequalities
its coefficients are nevertheless log-concave
no real chromatic root is negative, between 0 and 1, or between 1 and 32/27
on six points some forest polynomials have roots off the real line
one root for each unit of degree
random graphs on nine points satisfy the strongest inequality
so the chromatic roots of this graph are not all real
some graph with a claw has an independence polynomial with non-real roots
the bound is exact at k = 2 exactly when the graph has no triangle
the checked size is between 3 and 6
the chromatic coefficients are log-concave on every graph
the coefficient at size 4 counts a deal of the labels together with a structure on each side
the coefficients are log-concave
the coins give mean (n − 1)/2 and variance (n + 1)/12
the complete graph on four points has 1, 6, 15 and 16 forests with 0 to 3 edges
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 descent row satisfies every one of Newton's inequalities
the distance to the bell falls as n grows
the drawn graph's matching polynomial is real-rooted
the drawn matching roots are real and negative
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 inversion row fails every one
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 largest matching root is at most 2√(d − 1)
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 matching sizes are exactly a sum of independent coins
the mean grows with the size, so the reading is not a constant
the Petersen graph has claws and a real-rooted independence polynomial all the same
the recurrence agrees with counting descents directly
the roots of the graph with a vertex removed interlace the graph's roots
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 3 and 10 terms (20 for the root views)
the six-vertex wheel has non-real chromatic roots
the sizes drawn are between 3 and 5
the sizes drawn are ones this statistic is enumerated at
the spider's matching polynomial is real-rooted and its independence polynomial is not
the statistic is descents or cycles
the statistic is one this family tracks
the sum of independent coins has exactly the counted distribution
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 155 graphs with at least one edge on six points and 33 on five, up to isomorphism
there are C(4, 2) ways to deal the labels between the two pieces
though it is still log-concave in the plain sense
two different statistics this family tracks

`map-colour`

66 kinds of claim · 38 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
a greedy colouring is a proper colouring
a planar graph always has a vertex of degree at most five left
after: no edge joins two regions of one colour
alternating between the rows needs four
an odd rim needs four and an even rim three
and has no triangle
and it has one with four
and so is its short diagonal
and twenty edges
and two is the true answer
and two points of one hexagon are less than one apart
before: no edge joins two regions of one colour
between 3 and 6 rows are drawn
both are triangle-free
Brooks: at most Δ colours
colours are counted up to between 3 and 7
every edge has length exactly one
every greedy colouring uses between χ and Δ + 1 colours
every one of the seven regions touches every other
every other connected graph needs at most Δ
four colours are enough
greedy in the reverse order uses at most one more colour than the largest remaining degree
Grötzsch's graph has eleven vertices
inside the window both conditions hold
it has 3·2^(k−2) − 1 vertices
it has no triangle
its chromatic number is four
Mycielski's k-th graph needs exactly k colours
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
outside it one fails
removing any single edge lets one colour fewer suffice
same-coloured hexagons' centres are √21 sides apart
so points of two same-coloured hexagons are more than one apart
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 circle of radius √3 holds two points one unit apart
the colourings come in families of 24, one for each relabelling of the four colours
the construction shown is the seven-region one
the exceptions need Δ + 1
the five neighbours use five different colours, which is the only hard case
the fractional chromatic numbers begin 2, 5/2, 29/10
the Golomb graph has no proper colouring with three colours
the graph is one the family draws
the graph is one this family knows
the graph is the spindle or Golomb's
the hexagons are small enough and large enough
the icosahedron has thirty edges
the map can be coloured with at most five
the map has between 4 and 24 regions
the Moser spindle has no proper colouring with three colours
the rhombus's sides are one unit
the vertical step is 2, 3 or 4
the view is one the family draws
the wheel has between 3 and 23 rim regions
the window is not empty
their squares stay between 2k and 2k + 2
three colours do not suffice and four do
top row first needs two colours
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`

89 kinds of claim · 48 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
3003 = C(78, 2) = C(15, 5) = C(14, 6)
3003 appears eight times in Pascal's triangle
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
below 10¹² exactly one number appears eight times
below 10¹² exactly six numbers appear six times
below 10¹² no number appears five or seven times
between 1 and 6 steps up
between 1 and 8 steps across
between three and six control points, each inside the unit square
C(F₂ᵢ₊₂F₂ᵢ₊₃, F₂ᵢF₂ᵢ₊₃) = C(F₂ᵢ₊₂F₂ᵢ₊₃ − 1, F₂ᵢF₂ᵢ₊₃ + 1)
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
far fewer numbers sit deeper in the triangle than on its second diagonal
for m from 6 to 2,000 the only m with C(m, 3) triangular give 120, 1540 and 7140
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 family begins with 3003
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`

16 kinds of claim · 17 placements

the pairing of order 3 puts k numbers between the two ks ×3
a Langford pairing exists exactly when the parity allows
and a Skolem sequence exactly when its parity allows
each case really does overflow its boxes
each is a Langford pairing
each order multiplies the search by between five and ten
each puts the two ks k places apart
every item is in a box
more items than boxes forces a box with two
six Skolem sequences of order 4
the fullest box holds ⌈items/holes⌉
the Langford counts are 1, 1, 26, 150, 17,792 and 108,144
the view is one the family draws
there are at least as many things as boxes
there is at least one box
twenty-six pairings of order 7, up to reversal

`progression`

52 kinds of claim · 23 placements

and has no 3-cup and no 3-cap ×15
the construction for (3, 3) has C(k + l − 4, k − 2) points ×15
nothing survives at 9 either ×7
between 5 and 30 points ×2
its longest cup has 3 points and its longest cap 3 ×2
a point set in general position was found for every seed
although many fours are
and all three sizes of hull turned up in the test
and climbs toward it
and equals the construction's count where known
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 first column is the longest decreasing one
and the surviving lengths run right up to it with no gap
between 8 and 26 cards
between one and four seeded point sets are drawn
each entry is the sum of the one above and the one to the left
eight times the cards, about half the gap
every five points in general position hold four in convex position
every known value lies between the construction and the bound
every term carries a different pair of counters
its longest climb is n
k and l between 4 and 6
k and l run to between 4 and 7
no five of the eight are in convex position
one more term and a climb or a fall of n+1 appears
one point more, anywhere, makes a k-cup or an l-cap
the cap turns downward at every point
the claim is tested on between 200 and 20,000 point sets
the colouring drawn contains no such pattern
the cup turns upward at every point
the exhaustive search runs to between 5 and 14 numbers
the extremal sequence has n² terms
the first row of the shape is the longest increasing subsequence
the longest chain found by search matches the number of patience piles
the lower-left block has no (k − 1)-cup
the number of piles is the longest increasing run in the deal
the panels drawn show different hull sizes
the pattern is a progression or a sum
the ratio stays below 2
the rescaled lengths centre near −1.77 with spread near 0.9
the scaled shape lies close to the limit curve
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 set has C(k + l − 4, k − 2) points
the set is small enough to escape a longer cup and a longer cap together
the shape has n boxes
the upper-right block has no (l − 1)-cap
the view is one the family draws

`triangulation`

72 kinds of claim · 48 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
each diameter matches the published value
Euler's relation holds on the solid these faces make
every diagram pairs off all the points
every length in between occurs
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
for every triangulation of the decagon, the distance to the fan at vertex 0 is m − 3 minus its diagonals at vertex 0
four nodes give fourteen binary trees
no two chords cross
no two chords of the triangulation cross
no two triangulations are more than 2m − 6 flips apart
one flip at a time reaches every triangulation
rotating towards the right reaches exactly the trees whose vectors are componentwise larger
so the good ones are the difference of two binomial coefficients
successive interval counts grow by a factor approaching 256/27
the 13-gon's diameter is 2m − 10 = 16
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 hexagon's flip graph has 21 edges, each one covering relation
the lattice has 68 intervals
the longest has n(n − 1)/2 steps
the number of comparable pairs is 2(4n + 1)!/((n + 1)!(3n + 2)!)
the number of small triangles carrying all three colours is odd
the octagon's flip graph has diameter seven
the path found has exactly that many flips
the place-by-place maximum is not always a tree's list
the place-by-place minimum of any two trees' lists is again a tree's list
the reflection is one-to-one on the bad paths
the second corner the second
the shortest maximal chain has n − 1 steps
the successive ratio climbs at every step
the triangle is cut into a grid of side between 3 and 12
the triangulations number the Catalan number
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
two random triangulations are typically more than half the diameter apart, and well short of it
two triangulations at the greatest distance share no diagonal
whenever two triangulations share a diagonal, their distance is the sum of the distances on its two sides

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