The ledger

What the figures prove — page 5

Page 5 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.

• person 1 is right on exactly half the assignments ×5
• the number of people is a whole number between 3 and 12 ×2
• a choice from each of the pairs is one of two to the power of their number
• a set containing a decisive set is decisive
• after its last difference each sequence agrees with the representative for good
• all scores equal is impossible at even sizes
• and everyone after it is right
• between two and five pairs, each of two named things
• beyond voter 5 the ballots agree
• different pairs of whole numbers name different numbers a + b√2
• different spanning trees from the three listings
• each element belongs to exactly one chain
• each rational point sits on the line through (1, f(1))
• each stage of the chain is a forest
• edges kept, which a spanning tree of eight points has
• edges of each spanning tree
• every moved point lands strictly inside the open interval
• every point the map has to reach is reached
• every transitive two-voter rule is of one of the four kinds
• exactly one of each set and its complement is decisive
• f(x + y) = f(x) + f(y) on a + b√2
• in the cyclic profile each pair is won by two classes in three
• joint choices for the first five rows
• no two moved points land on the same place
• no whole number lies in every class of the chain …111
• of them always transitive
• on every one of the 2^8 assignments, everyone after the first is right
• rules for two voters that decide each pair from that pair alone
• spanning trees of a square with one diagonal
• the assembled map sends no two elements to the same place
• the chain of …111 keeps only 4,095 below 4,096 and loses it at the next halving
• the chain of voter 37 closes on voter 37 alone
• the common part of two decisive sets is decisive
• the dictator is voter 1, 2 or 3
• the empty set is not decisive, and the dictator alone is
• the function is not the line y = x
• the kinds number 6, 84, 2 and 2
• the largest denominator is a whole number between 2 and 24
• the last wrong guess is at the last place the hats differ from the representative
• the left-to-right map sends different elements to different places
• the left-to-right shift is a whole number between 1 and 4
• the maximal forests are exactly the spanning trees
• the members of a pair are named apart exactly when an order is claimed
• the number of elements drawn from each side is a whole number between 5 and 14
• the number of sequence terms drawn is a whole number between 4 and 12
• the option is one this view reads
• the pairs either carry an order or they do not
• the plotted graph meets every cell of a six-by-six grid over the window
• the range of whole coefficients is a whole number between 2 and 6
• the right-to-left map sends different elements to different places
• the right-to-left shift is a whole number between 1 and 4
• the rule fits two to the power of the pairs it cannot tell apart
• the rule names exactly one of the rows exactly when the pairs carry an order
• the rule of the chosen voter is among the survivors
• the two maps are different, or there are no chains to see
• the union of the chain is a forest
• the value given to √2 is a number of modest size
• the view is one the family draws
• the window shows at least three chains
• two regular tournaments on three points and 24 on five
• unanimous rules for three voters that decide each pair from that pair alone
• under the parity strategy all are right together or all wrong together
• voter 37 is odd, prime and a multiple of 37
• with √2 sent to √2 the function is the line y = x
• with three voters the unanimous transitive rules are the three dictators
• with two voters the unanimous transitive rules are the two dictators

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

• the 1 axioms c > 0, …, c > 0 are satisfied by 1 and not by 0 ×6
• models with 1 colours ×5
• (a + b) + c = a + (b + c) holds on every sampled triple
• (a × b) × c = a × (b × c) holds on every sampled triple
• −1 appears at the first stage
• −1 is a square, so the relation is symmetric
• √2 appears at some stage drawn
• 8 to 32 vertices drawn
• a + b = b + a holds on every sampled triple
• a × (b + c) = a × b + a × c holds on every sampled triple
• a × b = b × a holds on every sampled triple
• a < b and 0 < c imply a × c < b × c holds on every sampled triple
• a < b implies a + 1 ≤ b holds on every sampled triple
• a < b implies a + c < b + c holds on every sampled triple
• a ≠ 0 implies a = d + 1 for some d holds on every sampled triple
• a line drawn as parallel was checked and does not meet the base line inside the disc
• a triangle has three vertices
• and by the last every request is answered
• and never reaches the upper half
• and to no vertex of V
• back: a partner exists on the left
• between 4 and 9 terms
• between three and eight pieces, on a line long enough to hold them
• between two and four levels of galaxies
• each crossing point was computed and lies on both lines
• each dot is a root of its polynomial
• each lower truncation is below π and each upper one above
• each neighbourhood shows 2 to 4 steps either side
• each residue has (q − 1)/2 neighbours
• each stage adds new elements
• every angle of the triangle is positive
• every drawn polynomial belongs to the structure
• every function from A to B is listed
• every geodesic arc meets the boundary circle at a right angle
• every ordinary number is one or the other
• every other vertex is disqualified
• every partial matching preserves order both ways
• every vertex is inside the disc
• exactly one of a < b, a = b, b < a holds on every sampled triple
• exactly one of the lines through the point never meets the base line
• for size four, 89
• for size three, 29
• forth: a partner exists on the right
• halving preserves order
• heights from 2 to 7
• inside the model no function is onto
• more than one geodesic through the point misses the base line
• more vertices answer more requests
• no drawn dot is a known transcendental
• nothing lies strictly between an element and its successor
• once a prime passes, every larger prime in the table passes too
• outside, the bijections number size factorial
• sets of two to four elements
• some element is neither even nor odd
• the 17-vertex Paley graph fails some request of three
• the angles of a triangle in this model add to less than a straight angle
• the chosen vertices are joined exactly as Petersen's are
• the defect is a positive area
• the drawn elements are in increasing order
• the enumeration of the other side reaches every gap
• the even witness doubles back to the element
• the first prime Paley graph answering every request of size two has 13 vertices
• the galaxies are distinct and in order
• the graph has a vertex joined to exactly the right earlier ones
• the labels fit in four levels
• the lower ones rise and the upper ones fall
• the matching keeps every edge and every non-edge
• the matching preserves order
• the model's functions stay not-onto after relabelling B
• the number of lines tried through the point is a whole number between 3 and 9
• the number of parallels drawn is a whole number between 1 and 9
• the odd witness gives the element
• the outside point is inside the disc
• the Paley counts are more tightly bunched than the coin-toss counts
• the Paley graph on 101 answers every request of three
• the Paley graphs pass where coin-toss graphs of the same size often fail
• the parity statement fails on sampled elements
• the point is off the base geodesic
• the random graph has a vertex of the needed kind
• the request is answered
• the rule is symmetric
• the view is one of triangle, euclid, compactness, polymodel, parity, axioms, galaxies, algebraic, skolemhull, relative, sentences, supremum, backforth, bfsteps, forthonly, bfalgebraic, endpoints, gap, rado, radowitness, radorandom, radoembed, radoiso, radoresilient, paley, paleytable, paleyfail, paleyrandom, paleycounts
• the witness for "∀a∀b∀c ∃x x³ + ax² + bx + c = 0" checks
• the witness for "∀x (0 < x → ∃y y·y = x)" checks
• the witness for "∀x∀y (x < y → ∃z (x < z ∧ z < y))" checks
• the witness for "∃x (x·x = x + x ∧ ¬ x = 0 ∧ ¬ x = 1 + 1)" checks
• the witness for "∃x x·x = −1" checks
• the witness for "∃x x·x = 2" checks
• the witness for "∃x x·x·x = x + 1" checks
• the witness is joined to every vertex of U
• the witness is not among the deleted vertices
• three to nine elements of the structure
• two or three stages
• X and its whole neighbourhood exceed every ordinary number

• at 1 rounds the search agrees with the rule that says same length or both at least 1 ×3
• at depth 1 the words must be 1 letters before Duplicator survives ×3
• by the largest size drawn, Duplicator survives 2 rounds on every pair sampled ×2
• Duplicator survives 3 rounds on a cycle of 22 against two of 11 ×2
• the search agrees with the rule that says Duplicator survives 3 rounds from 5 points up ×2
• a degree-three point gives four middle points
• a star splits at the first round
• a vertex's neighbours induce different shapes in the two graphs, so they are not isomorphic
• an even number of edges is a coin toss at every size, as the flip bijection says it must be
• and at every point of the second
• and does not at the smallest
• and does not survive the three-pebble one
• and Duplicator survives at every length past it
• and every one of them swaps an even number of edges
• and it does grow
• and no pair survives three pebbles without surviving two — more pebbles never help Duplicator
• and once Duplicator survives at one length it survives at every longer one
• and so do two triangles, which is why refinement cannot separate them
• and so does the two-dimensional refinement, which separated the last pair at once
• and that object is a path
• and the group has order two
• and the shape is the same at every point of the first
• and the share rises with the size rather than wandering
• and the two graphs have the same number of edges
• and they have the same parameters
• at every depth there is a length past which the two words cannot be separated
• between fifty and four hundred graphs at each size
• between four and twelve pairs of graphs at each size
• between ten and eighty graphs are sampled at each size
• between three and six graph sizes, each of four to sixty points
• between three and six rounds shown
• between three and six sizes, each of four to forty-four points
• between three and six sizes, each of three to forty points
• between two and four depths
• between two and three round counts are tabulated
• both graphs are strongly regular
• containing aba as a factor is aperiodic but not J-trivial
• containing aba as a scattered subword is J-trivial
• Duplicator survives the two-pebble bijective game
• every J-trivial language with at most four states is decided by subwords of length at most three
• every language with a J-trivial monoid is decided by its short subwords on all words up to ten letters
• every point of both graphs has two neighbours
• exactly four of the eight sign patterns are realised
• exactly half of the 64 graphs on 4 points have an even number of edges
• large ones almost never do
• no odd pattern is realised, which is the whole of what the gadget enforces
• on average J-trivial monoids are the smallest and non-aperiodic ones the largest
• one graph is connected and the other is not, so a sentence separating them would express connectedness
• one or two round counts, each of at most three
• one state recognises exactly two languages, everything and nothing
• one-dimensional refinement gives both graphs the same colours and cannot tell them apart
• one-dimensional refinement gives them the same colours
• out of the eight subsets of its three edges
• six neighbours each
• sixteen points each
• small graphs usually fail the property
• so the count grows like a logarithm rather than like the length
• some pair survives two pebbles and not three, which is the point of a third variable
• the cycle stays one colour at every round, since every point looks alike
• the deeper game needs a larger graph before Duplicator always survives it
• the depth is between one and three
• the factor language is decided by no subword length up to three on words of ten letters
• the first-order property "every pair has a common neighbour" has settled on 1 by the largest size drawn
• the first-order property "some point joined to nothing" has settled on 0 by the largest size drawn
• the first-order property "three points all joined" has settled on 1 by the largest size drawn
• the game agrees with the neighbourhood argument about who wins
• the game is played between two and five rounds deep
• the game is played to at most three rounds, which is as far as a search finishes
• the game runs between one and four rounds
• the measured share agrees with the count of expected failures
• the number of colours never falls
• the number of rounds Spoiler needs is computed
• the other two monoids are aperiodic, so both languages are first-order definable
• the parity language's monoid contains a non-trivial group, so no first-order sentence says it
• the rounds needed against a chain one longer follow the logarithm of its length
• the sizes are listed in increasing order
• the subword language is decided by subwords of length three
• the sweep runs over between three and eight cycle lengths
• the sweep runs to between five and nine
• the table runs to between four and nine
• the transcript ends the way the exhaustive search says the game ends
• the two cycles are of different lengths
• the two cycles have three points each
• the two graphs have the same number of points and the same number of edges
• the two neighbourhoods are the same object, computed from the cycles rather than named
• the two small cycles run to between three and thirteen points
• the view is one the family draws
• two chains of between two and nine elements
• two cycles of between five and thirty points
• two-dimensional refinement does tell them apart
• with length two the number of classes has stopped growing
• with subwords of length one there are exactly four classes: which of a and b occur
• words up to between four and six letters
• words up to between six and twelve letters

• a set of 0 has more subsets than members ×13
• height 2 holds finitely many polynomials, and the enumeration produced them ×5
• −1 is among the roots the enumeration found
• √2 is among the roots the enumeration found
• 1 is among the roots the enumeration found
• a fixed column holds a vanishing share of A's points and a fixed row nearly all
• a half is among the roots the enumeration found
• A holds the pairs on or below the diagonal of the list
• a larger height admits more polynomials
• all 512 listings of 3 subsets of a set of 3 were tried
• and the second
• and yet the two images are far apart, so the weave is not continuous
• by a factor of more than a thousand at this many places
• each length contributes finitely many words
• each stage keeps two thirds of the length before it
• every fraction the grid reaches is on the list
• every kept cell got exactly one place in the list
• every piece is a genuine interval
• every point claimed to be a limit has points of the set crowding onto it
• in every square A holds roughly half the points
• more fractions pin the curve more tightly
• no fraction is listed twice
• no rational is listed twice
• no word is listed twice
• one step before, only the point 0 is left
• the column of the i-th element meets A in i + 1 places
• the complemented diagonal is on no row of the listing
• the derivatives reach the empty set
• the digit strings are as long as the drawing says
• the first number's digits are between four and ten decimal places
• the golden ratio is among the roots the enumeration found
• the largest height enumerated is a whole number between 3 and 6
• the largest sum of numerator and denominator drawn is a whole number between 3 and 9
• the number of places drawn is a whole number between 4 and 10
• the number of rungs drawn is a whole number between 3 and 5
• the pieces at each level are disjoint
• the places run from one with no gaps
• the second number has the same number of places
• the set is built from one to three reciprocal terms
• the size at which every listing is tried is a whole number between 2 and 3
• the two expansions differ by less than one unit in the last place drawn
• the view is one the family draws
• the walk's count and the sum of totients agree, having been computed independently
• the window holds enough of them to draw
• the window on the line is between one and twelve wide
• the woven number has one place from each, alternately
• unweaving recovers the first number

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

• □□p → □p is valid exactly on the dense frames
• □p → □□p is valid exactly on the transitive frames
• □p → ◇p is valid exactly on the serial frames
• □p → p is valid exactly on the reflexive frames
• □p at a world is p at every world it can reach
• ◇□p → □◇p is valid exactly on the confluent frames
• ◇p → □◇p is valid exactly on the euclidean frames
• ◇p at a world is p at some world it can reach
• 4 is valid on a frame exactly when the frame is transitive
• 5 is valid on a frame exactly when the frame is euclidean
• a formula separating the other pair exists among those searched
• a transitive model filters to a transitive quotient here, checked rather than assumed
• a world and a valuation were found at which the axiom fails
• a world that reaches nowhere makes every necessity true and every possibility false
• all thirty-two axiom sets are accounted for, thirteen of them giving S5
• and fewer worlds than the model it came from
• and it is deeper than anything the closure could ask
• and no frame with any worlds in it validates both Löb's axiom and □p → p
• and on every one of them ¬□⊥ is a fixed point of ¬□p
• and some class of the quotient can — the collapse invents a way back that the model has none of
• and the other two starting worlds are not
• and two frames agreeing on all of them disagree about the axiom
• and validates what is valid
• B is valid on a frame exactly when the frame is symmetric
• between one and five axioms are tabulated
• between two and six frames are compared
• D is valid on a frame exactly when the frame is serial
• each implication holds on every frame up to four worlds or has a small counterexample
• every frame on three worlds validates K, and exactly the five equivalence relations validate S5
• every pair of different logics has a small frame separating them
• excluded middle at a stage is exactly p there or ¬p there
• formulas up to depth two to four are compared
• Löb's axiom is valid exactly on the transitive frames with no cycles
• no condition in the catalogue matches McKinsey's axiom
• no stage forces both p and its negation
• no world of the model can be returned to
• on a frame with the property the axiom holds everywhere, under every valuation
• on that frame every deeper modality agrees with the first, not only the second
• one of the two frames collapses the tower and the other does not
• p → □◇p is valid exactly on the symmetric frames
• so every frame of the logic is transitive, which Löb's axiom implies
• some depth of looking ahead separates the chain from its quotient
• some frame this family knows has the property
• some frames validate it
• some stage forces neither p nor its negation, so p ∨ ¬p fails there
• T is valid on a frame exactly when the frame is reflexive
• the axiom is one of the five
• the bisimilar worlds agree on every formula up to the drawn depth
• the chain runs between eight and twenty-four worlds
• the deeper closure is asked for or not
• the frame is one this figure knows
• the gallery's verdict for a branch, both ends dead is the sweep's
• the gallery's verdict for a single dead end is the sweep's
• the gallery's verdict for a transitive chain is the sweep's
• the gallery's verdict for a two-cycle is the sweep's
• the gallery's verdict for a world seeing itself is the sweep's
• the gallery's verdict for one step and stop is the sweep's
• the model is the chain or the strict order on its worlds
• the model refutes □p → p, which is not valid in this logic
• the quotient has at most one world for each way of answering the closure
• the S5 frames on four worlds are exactly the fifteen equivalence relations
• the sweep runs over frames on three or four worlds
• the tower is followed to between two and four boxes
• the truth lemma holds for □□p: it is true at a world exactly when it is true at that world's class
• the truth lemma holds for □p: it is true at a world exactly when it is true at that world's class
• the truth lemma holds for p: it is true at a world exactly when it is true at that world's class
• the truth lemma holds: □□p is true exactly where it is accepted
• the truth lemma holds: □p is true exactly where it is accepted
• the truth lemma holds: p is true exactly where it is accepted
• the two starting worlds are related by the bisimulation
• the view is one the family draws
• thirty-two axiom sets give fifteen logics
• this frame is not reflexive, so the axiom has a chance of failing
• this frame is not symmetric, so the axiom has a chance of failing
• this frame is not transitive, so the axiom has a chance of failing
• two frames this family knows
• what is established at a stage stays established later
• worlds share a class exactly when no formula of the closure tells them apart

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

• the 2-chain validates G2 exactly when 2 > 2 ×30
• G3 holds in the 2-chain and fails in the 3-chain ×5
• an algebra of 2 elements validates Gödel's disjunction in 3 letters ×4
• at p = 1/2 every multiple of 1/2 occurs ×3
• at p = 1/2 every value is a multiple of 1/2 ×3
• the tent at 1/2 is defined by some formula with at most 8 connectives ×2
• ¬¬¬p → ¬p holds in every algebra searched
• ¬p → q ∨ r ⟹ (¬p → q) ∨ (¬p → r) holds in every algebra searched
• 234 assignments satisfy eight of the nine bases
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (¬p) ∧ (p ∨ q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (¬p) ∨ (p ∧ q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (¬q) ∧ (p ∨ q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (¬q) ∨ (p ∧ q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (p ∧ q) ∨ (¬(p ∧ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (p ∧ q) ∨ (¬(p ∨ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (p ∨ q) ∧ (¬(p ∧ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: (p ∨ q) ∧ (¬(p ∨ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(p ∧ (q ∨ (¬p)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(p ∧ (q ∨ (¬q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(p ∧ q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(p ∨ (q ∧ (¬p)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(p ∨ (q ∧ (¬q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(p ∨ q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(q ∧ (p ∨ (¬p)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(q ∧ (p ∨ (¬q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(q ∨ (p ∧ (¬p)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬(q ∨ (p ∧ (¬q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬p
• a formula is a classical tautology exactly when it always holds in the logic of paradox: ¬q
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (¬(p ∧ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (¬(p ∨ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (¬p)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (¬q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (q ∧ (¬(p ∧ q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (q ∧ (¬(p ∨ q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (q ∧ (¬p))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (q ∧ (¬q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (q ∨ (¬p))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ (q ∨ (¬q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∧ q
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (¬(p ∧ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (¬(p ∨ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (¬p)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (¬q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (q ∧ (¬p))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (q ∧ (¬q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (q ∨ (¬(p ∧ q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (q ∨ (¬(p ∨ q)))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (q ∨ (¬p))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ (q ∨ (¬q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: p ∨ q
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∧ (¬(p ∧ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∧ (¬(p ∨ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∧ (¬p)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∧ (¬q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∧ (p ∨ (¬p))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∧ (p ∨ (¬q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∨ (¬(p ∧ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∨ (¬(p ∨ q))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∨ (¬p)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∨ (¬q)
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∨ (p ∧ (¬p))
• a formula is a classical tautology exactly when it always holds in the logic of paradox: q ∨ (p ∧ (¬q))
• a formula that survives three values fails at four
• a set is contained in its double negation
• and fails under Łukasiewicz's
• and here it is strictly contained — the punched-out points come back
• and it keeps rising, so the classes are not exhausted by short formulas
• and r does not follow, because q may be a glut
• and so does double-negation elimination
• and the value really is not the top
• at depth four there are sixteen inequivalent formulas
• between one and four points are punched out
• both models are chain frames and the glued one is not
• both models are one-top frames and the glued one is not
• both-and-neither is false and both-or-neither is true
• classically, from ¬(p ∧ q), infer ¬p ∨ ¬q
• classically, from ¬¬p, infer p
• classically, from p ∧ q, infer p
• classically, from p ∨ q and ¬q, infer p
• classically, from p and ¬p ∨ q, infer q
• classically, from p and ¬p, infer q
• classically, from p, infer p ∨ q
• classically, inconsistent premises imply everything
• classically, infer p ∨ ¬p from nothing
• collapsing the top two values respects implication
• contraction holds under Gödel's implication
• each basis is the one left out in 26 near-solutions
• each row and column multiplies to plus or minus the identity
• ends are 0 or 1
• every assignment of true and false to the eighteen directions is counted
• every element sits below its double negation
• every formula is 0 or 1 at the ends
• every glued frame is a frame of the constructive system
• every inference the logic of paradox accepts is classically valid
• every piece has a whole-number slope
• every piece's line has a whole-number intercept
• every punched-out point is inside the set
• every smaller algebra validates (p → q) ∨ (q → p)
• every smaller algebra validates ¬¬p → p
• every smaller algebra validates ¬p ∨ ¬¬p
• every smaller algebra validates p ∨ ¬p
• every truth set is closed upwards
• exactly one product, the last column, is minus the identity
• excluded middle fails at neither
• excluded middle fails somewhere in this algebra
• excluded middle first fails in the three-chain
• excluded middle holds on each one-stage model and fails once they are glued
• formulas of up to 3 to 6 nestings
• G4 first fails in the four-chain
• Gödel–Dummett logic's axiom picks out its frames
• Gödel's implication is linear
• in the logic of paradox s is not settled, though s ∨ ¬s is
• it keeps excluded middle, which the gap logic loses
• more than four formulas in one variable are distinguished
• negation undoes itself
• neither p/2 nor the constant one half appears
• no assignment of plus and minus one satisfies all six products
• no colouring of Peres's thirty-three directions exists
• no open set larger than the implication satisfies the condition
• no two chains glue into a chain
• no two frames with one last stage glue into one
• on chains one letter gives six formulas and two give 342
• p ∨ ¬p is designated at every value
• Peres's set has 33 directions, 16 orthogonal triads and 72 orthogonal pairs
• removing any one of Peres's directions allows a colouring
• so does Peirce's law
• so is Łukasiewicz's
• some axiom holds in one algebra and fails in another
• the 49 directions with coordinates in {0, ±1, ±2} have no colouring
• the algebra has at least two elements
• the basis left out always holds an even number of true directions
• the count never falls as the formulas grow
• the counts the table reports
• the depth-two model for two letters has 265,454 up-sets
• the disjunction's value is the second-highest truth value
• the drawn near-solution makes four directions true
• the equivalence is one minus the distance
• the first model refutes the first disjunct at its root
• the first model's root forces the same things after gluing
• the formula (p ⊕ p) ⊙ ¬(p ⊙ p) is among those enumerated
• the formula ¬p is among those enumerated
• the formula p ∧ ¬p is among those enumerated
• the formula p ⊕ p is among those enumerated
• the formula p ⊙ p is among those enumerated
• the formula p is among those enumerated
• the formulas are among em, wem, dnn, gd, kp, triple
• the gap logic has no tautologies at all
• the glued valuation only ever grows
• the implication really does meet the antecedent inside the consequent
• the implication really is the largest element whose meet lies below
• the logic of paradox rejects explosion, modus ponens for ∨, and the disjunctive syllogism
• the logic of the weak law's axiom picks out its frames
• the negation of an open set is open
• the new root forces neither disjunct, so it does not force the disjunction
• the number of conditions met is always odd
• the orders are among point, chain2, chain3, anti2, v, lambda, diamond
• the pair is one of em, gd, wem, deep
• the probabilities of each basis add up to one
• the propagating search refutes Peres's set after a few dozen choices
• the refuting algebra has a valuation making (p → q) ∨ (q → p) fall short
• the refuting algebra has a valuation making ¬¬p → p fall short
• the refuting algebra has a valuation making ¬p ∨ ¬¬p fall short
• the refuting algebra has a valuation making p ∨ ¬p fall short
• the search has several algebras in it
• the search runs over orders on up to 2 to 4 points
• the search separates the formulas it refutes from the ones it cannot
• the second model refutes the second disjunct at its root
• the separating algebras run to 2 to 4 points
• the set sits strictly inside the ambient line
• the set together with its negation does not fill the line
• the shortest formulas grow by the golden ratio per step
• the shrinking finds sets of thirty-one
• the smallest set found is uncolourable and every direction in it is needed
• the thirty-three of Peres
• the three lines are distinct, so any two of them span the plane and meet only at the origin
• the union has as many pieces as the two sets between them
• the view is one the family draws
• the weak law and linearity hold in every chain
• thirty-one directions with no colouring
• three negations are the same as one
• three negations collapse in every Heyting algebra
• three negations collapse to one
• three to five letters
• two letters give four final states and eighteen states one step before them
• values stay in the interval
• which is not the top, so the formula fails
• with p both true and false, p ∧ ¬p is designated while a false q is not

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

• the machine for "x = 2y" is right on 0, 0 ×4096
• the machine for "x is a multiple of 3" is right on 0 ×256
• the machine for "x is a power of two" is right on 0 ×256
• the machine for "x is even" is right on 0 ×256
• column 2 ×59
• column 2 is empty exactly when 2 is prime ×59
• functions pairing off 1 points ×7
• one-to-one maps of 1 points missing a point ×7
• column 0 of the sum is right ×6
• Abelard's winning strategies on 2 points ×5
• slice a = 1: a solution exists exactly when the eliminated formula says so ×2
• the formula for ∀x ∃y at n = 2 ×2
• the shadow repeats every 5 ×2
• ∀x ∀y R(x, y) holds for R exactly when its dual fails for the complement
• ∀x ∀y R(x, y) implies ∃x ∀y R(x, y) with nothing between
• ∀x ∀y R(x, y) implies ∃y ∀x R(x, y) with nothing between
• ∀x ∃y R(x, y) holds for R exactly when its dual fails for the complement
• ∀x ∃y R(x, y) implies ∃x ∃y R(x, y) with nothing between
• ∀y ∃x R(x, y) holds for R exactly when its dual fails for the complement
• ∀y ∃x R(x, y) implies ∃x ∃y R(x, y) with nothing between
• ∃x ∀y R(x, y) holds for R exactly when its dual fails for the complement
• ∃x ∀y R(x, y) implies ∀y ∃x R(x, y) with nothing between
• ∃x ∃y R(x, y) holds for R exactly when its dual fails for the complement
• ∃y ∀x R(x, y) holds for R exactly when its dual fails for the complement
• ∃y ∀x R(x, y) implies ∀x ∃y R(x, y) with nothing between
• 240 labelled tournaments on seven players have the property, 5,040 / 21
• 7³ = 343 relations have a mark in every row
• a column holds a mark exactly when its values of a satisfy |a| ≥ 2
• a full column forces every row to carry a mark, so one reading implies the other
• a machine that keeps writing and never halts exists
• a machine that ran three thousand steps does not halt in the classifier
• a machine that runs six steps and leaves four ones exists
• a match is at least as long as the run it writes out, less the cells the rewriting dominoes take two or three at a time
• a matching covers every point exactly when the number of points is even
• a program computing how many times f occurs in it gets it right about itself
• a program computing its own length gets it right about itself
• a program computing its own text reversed, first 24 characters gets it right about itself
• a program computing whether its length is even gets it right about itself
• a row copied from the table is found in the table
• a shifted repeat visible within sixty steps exists
• a tournament with every pair beaten by a third is found
• a two-state machine recognises the pairs (x, largest power of two dividing x)
• almost every cell is checked against the formula
• and 512 − 343 = 169 have a full row
• and for ∃x ∀y
• and none leaves more than four ones
• and one predecessor
• at 16 points almost every relation has a mark in every row
• between 3 and 7 points
• carrying out the instruction prints the instruction itself
• each generation prints exactly the next one
• each point has one successor
• each writer's match is found and is shortest
• Erdős's expectation falls below one at 21 players
• every friendship graph searched is a windmill
• every halting machine is counted once
• every point is in exactly one colour class
• every step of the match is legal
• every step of the witness follows an edge
• every total past the largest gap can be made
• every winning pair has f = g
• every x makes it positive exactly when a² − 4b < 0
• five logics in the table
• following successors from any point visits all eight before returning
• functions from five points to five points
• in a line, every row can be answered
• in a line, the sentence holds from two points on
• M′ computes the successor exactly when M halts
• most halting machines never read all four entries of their table
• most of the unexplained machines spread like the square root of time
• no carry is left over after the last column
• no drawn row agrees with the built row everywhere it has been checked
• no edge joins two points of one class
• no halting two-state machine runs more than six steps
• no language defines truth for its own sentences
• no match is shorter than the drawn depth
• no row of the table is the Russell row
• no shorter match could have escaped the search
• no three-state machine leaves more than six 1s
• no tournament of two to six players has every pair beaten by a third
• no unexplained machine halts within two hundred thousand steps
• of which 1,379 halt
• on one point Abelard wins, since x ≠ y cannot hold
• one half begins 0.5000
• pairing six points off makes the branching sentence true
• pi minus three begins 0.14159
• six machines are chosen, halting and not
• slice a = -1: a solution exists exactly when the eliminated formula says so
• some relation satisfies ∀x ∃y R(x, y) but not ∀y ∃x R(x, y)
• some relation satisfies ∀x ∃y R(x, y) but not ∃y ∀x R(x, y)
• some relation satisfies ∀y ∃x R(x, y) but not ∀x ∃y R(x, y)
• some relation satisfies ∃x ∀y R(x, y) but not ∀x ∃y R(x, y)
• some x makes it zero exactly when a² − 4b ≥ 0
• the backtracking search never tries more than all colourings
• the backward test never refutes a machine that does halt
• the bottom row is ahead after every domino but the last
• the built row differs from row n in place n
• the chart counts table entries read or ones left
• the check in a line makes at most n⁴ plays
• the coins share no common factor, so every large enough total is reachable
• the compressor sees no pattern in pi and a strong one in the periodic string
• the count k over the root of k − 1 is a whole number only for k = 2
• the count of phrases grows no faster than the alphabet's powers
• the diagram has six arrows
• the dominoes of a halting machine have a match
• the drawn range contains the largest total that cannot be made
• the drawn window is at least as wide as it is tall
• the first number is a whole number between 1 and 63
• the friendship sentence holds exactly at the odd sizes searched
• the gaps between squares keep growing
• the labelling is one this figure knows
• the largest number is a whole number between 20 and 80
• the longest halting run is 21 steps
• the machine accepts exactly the powers of two
• the machine accepts exactly the true sums
• the machine for "x + y = z" is right
• the machine is one the family encodes
• the machine stays on the tape it was given
• the match spells one string twice
• the match writes out the machine's run, configuration by configuration
• the new number's i-th digit differs from phrase i's
• the number of columns is a whole number between 2 and 14
• the number of drawn columns is a whole number between 3 and 16
• the number of drawn rows is a whole number between 3 and 12
• the number of levels is a whole number between 3 and 9
• the number of rows is a whole number between 2 and 14
• the number of sets in the table is a whole number between 3 and 10
• the option is one this view reads
• the Petersen graph can be coloured with three colours
• the phrase itself has fewer than ninety characters
• the program's output is its own text, character for character
• the property is rare at seven players and usual at thirty, above the counting floor
• the range is a whole number between 16 and 96
• the relation is one this figure knows how to draw
• the residue tournament on seven players has every pair beaten by a third
• the row of 'does not describe itself' differs from each adjective's row on the diagonal
• the Russell row disagrees with row k at column k
• the search agrees with 256b³ ≤ 27a⁴
• the search is drawn between 4 and 12 dominoes deep
• the search stays within its budget of states
• the second machine accepts exactly x = 2y
• the second number is a whole number between 1 and 63
• the set defined by ∃y (x = y + y) is eventually periodic
• the set defined by ∃y (y + y + y ≤ x ≤ y + y + y + 1) is eventually periodic
• the set defined by ∃y ∃z (x = 3y + 5z) is eventually periodic
• the set defined by ∃y ∃z ∃w (x = 6y + 9z + 20w) is eventually periodic
• the shadow is exactly the x whose remainder a·x mod b lies within c of a multiple of b
• the share with every row marked eventually only rises
• the shortest match of the three dominoes has thirty-six of them
• the slope's rise is a whole number between 1 and 9
• the slope's run is a whole number between 2 and 9
• the slow ones grow like the logarithm of the time, as a binary counter does
• the squares are not eventually periodic with any small period on the range checked
• the three dominoes have a match
• the three-state tree has 16,549 machines
• the tournament found has every pair beaten by a third
• two coins: the largest total that cannot be made is ab − a − b
• two or three states
• two to four coin values
• winning pairs (f, g)
• with f a rotation, every column asks two different values of v

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

• the term for x1 written out has 2^(k+1) − 1 symbols ×10
• the chain on 2 variables is refuted by a search of 5 nodes ×6
• the clauses for 2 pigeons in 1 holes ×6
• at 1 instances there is still a model, so the search has not finished ×5
• the assumption h2 is open where it is discharged ×3
• at 1 instances there is still a model ×2
• the assumption h1 is discharged somewhere below it ×2
• the variable h1 is bound where it is used ×2
• ∀X. X has no closed normal inhabitant, ∀X. X → X one and ∀X. X → X → X two
• 2ⁿ is found one type up
• 2ⁿ needs the input one type up
• a conjunction is introduced from two derivations
• a conjunction is taken apart from one derivation
• a disjunction is introduced from one derivation
• a disjunction is used by considering both halves
• a function is applied to an argument of its input type
• a function that is 0 at 1 is 0 from 1 on
• A implies B on every assignment
• A implies the interpolant
• a longer cycle needs a larger search
• a negation is introduced from a derivation and its denial
• a path of two to five vertices starting at the charged one
• a term starts with a name
• a tree-shaped term has the tree type
• a variable shares a component with its negation exactly when no assignment works
• a vertex of degree d contributes 2^(d−1) clauses
• an assumption has nothing above it
• an implication is introduced from one derivation
• an introduced implication is an implication
• and both cases reach the same conclusion
• and every assumption it made has been discharged
• and it is discharged as (p -> q) & (q -> r)
• and it is discharged as p
• and it is discharged as p -> p
• and it is discharged as p & (q | r)
• and it is discharged as q
• and it is discharged as r
• and it is the conjunction of what they derived
• and its antecedent is the discharged assumption
• and its argument has the antecedent's type
• and its consequent is what was derived
• and not at the lower types
• and once it closes it stays closed
• and the count of models is positive at every stage drawn
• and the drop is sharper for more variables
• and the formula it proves is true in every row of its table
• and the goal follows from the lemma
• and the goal is true in every row of its table
• and the interpolant implies B
• and the lemma the other proof uses is not a subformula of the goal
• and the named half is the conclusion
• and the named half is what was derived
• and the negated assumption is the conclusion
• and the second is its antecedent
• and the sweep contains both kinds
• arguments close with a bracket
• between two and seven stages
• binary trees with k forks are counted by the Catalan numbers
• both cases have the same type
• cases are taken on a disjunction
• consecutive path vertices are joined by an edge
• corners first, centre last, is an optimal order
• each arrow along the chain resolves with the clause built so far
• each extra hole multiplies the search by more than the last one did
• each extra row and column multiplies the peak by more than two and a half
• each numeral has type N
• each substitution adds one symbol and removes no variable
• eliminating all three letters of the odd triangle produces the empty clause
• every assignment violates an odd number of vertices
• every column of the incidence matrix has two ones, so the rows add to zero
• every drawn step is the resolvent of the two clauses above it
• every edge of the grid lies inside some bag
• every formula here is refuted at some width
• every formula in the cut-free proof is a subformula of the goal, or its negation
• every formula of the derivation with no detour is a subformula of the goal
• every function a proof of N → N computes is an extended polynomial
• every grid formula is refuted
• every label is falsified by the assignments on its own path
• every leaf of the formula is a variable
• every level has a node that reaches the bottom
• every literal names a variable of the clause set
• every literal names a variable of the set
• every lower bound sits below the exact width and every order's width above it
• every one of the 9! orders of the 3 × 3 grid is counted
• every opening bracket in the formula is closed
• every order of the six variables
• every order refutes the odd-charged grid
• every parity formula with an odd charge ends in the empty clause
• every variable is declared
• exponentiation has type N → N → N
• flipping an edge moves the violation to its other end
• from edge probability 0.5 the bound proves fewer than half of the optimal orders optimal
• it closes exactly when the universal has been used 3 times
• least fill is optimal at least as often as the bound proves it
• maximum, minimum, subtraction, equality and powers are not
• modus ponens takes two derivations
• more clauses per letter make elimination far costlier
• n, n² and n³ are all reached
• no arrow leads from a true literal to a false one
• no assignment satisfies every vertex
• no fixed order beats the best search that chooses afresh at each node
• no label is a tautology
• no node has more children than the branching factor
• no random order beats eliminating row by row
• no variable shares a component with its negation
• on a cycle the clause count never exceeds its starting value
• on the 4 × 4 grid least fill never holds more clauses than it started with
• on the 4 × 4 grid the bounds meet, so least fill is proved optimal there
• on the 6 × 6 grid least fill forms twenty-four times fewer resolvents than the row order
• one numeral of every size, and 2ⁿ strings of n letters
• one of the six formulas
• only a disjunction is taken apart by cases
• only a pair is projected
• only a polymorphic term is applied to a type
• only a term of implication type is applied
• parity and halving need the input to iterate pairs
• removing a detour leaves the proved formula unchanged
• resolution and the truth table agree on every one of the sets
• resolution reaches the empty clause exactly when no assignment satisfies the set
• resolution refutes A together with the negation of B
• saturation stays within two million clauses
• shared as a graph it has k + 1 distinct nodes
• so the conclusion is its consequent
• so the count of unsatisfiable sets and the count of refuted sets are one number
• successor, Ackermann, predecessor and subtraction have the types N → N, N → N → N, N → N and N → N → N, and the identity applied to itself is again the identity's type
• sums, products and a test on zero are reached
• the 3 × 3 grid has treewidth 3
• the 4 × 4 grid has treewidth 4 and some random order attains it
• the algorithm refuses x = f(x) by the occurs check
• the assignment satisfies every clause
• the assumption f is discharged somewhere below it
• the assumption x is discharged somewhere below it
• the assumption y is discharged somewhere below it
• the branching factor is a whole number between 2 and 3
• the case assumption h2 is open
• the chain from x to y ends in the clause ¬x ∨ y
• the chain length is a whole number between 4 and 12
• the charges add to one
• the clashing literals unify
• the clause set is one the arrow views know
• the clause set is one the family knows
• the conclusion of →I is an implication
• the conclusion of ¬I is a negation
• the conclusion of ∨I is a disjunction
• the contraction bound never exceeds an order's width
• the cut-free proof closes, so the formula is a theorem
• the depth drawn is a whole number between 3 and 7
• the derivation ends at its goal
• the derivation ends at the stated goal
• the derivation had at least one detour to remove
• the derivation is one of contraposition, transitivity, distribute
• the derivation is one of contraposition, transitivity, distribute, identity, swap
• the derivation is one of identity, swap
• the derivation uses only implication, conjunction and disjunction, where every step has a term
• the derivation's term has the goal as its type
• the exact width lies between the bounds
• the expansion does close, at some stage this figure reaches
• the first derivation gives a disjunction
• the first is an implication
• the first resolvent keeps one variable
• the formula and its disjunctive normal form agree on every row
• the formula has between one and five variables
• the formula is a non-empty string
• the function is computed by some term
• the graph is one the family knows
• the graph is small enough to check every assignment
• the identity applied to itself reduces in one step to the identity
• the interpolant uses only atoms the two formulas share
• the largest bag holds treewidth + 1 vertices
• the last term contains no detour
• the lemma is itself a theorem, so it may be cut in
• the literals resolved have opposite signs
• the most holes is a whole number between 3 and 6
• the narrowest refutation has the width found before
• the option is one this view reads
• the pair is one the family knows
• the path passes through one node per level
• the predecessor is computed with the input two types up
• the predecessor needs it two types up
• the propagating search only runs on unsatisfiable sets
• the pruning parameter is a whole number between 2 and 9
• the question is one of reaches, never
• the reduction finishes within a dozen steps
• the root's label is the empty clause
• the rounds drawn is a whole number between 2 and 5
• the row order refutes it too
• the search only runs on unsatisfiable sets
• the second resolvent is the empty clause
• the set has a variable in its negation's component
• the sets with a variable in its negation's component are the unsatisfiable ones
• the short form agrees with the extracted interpolant on every shared assignment
• the smallest unsatisfiable set of two-literal clauses needs four of them
• the specific substitution is also a unifier
• the specific unifier factors through the most general one
• the state answers k − 1 to nothing and k to the successor
• the strongest interpolant implies the one read off the refutation
• the subset search is run only on graphs small enough to tabulate
• the substitution found makes the two terms identical
• the sweep is over three variables
• the table has one row per assignment
• the tableau and the truth table reach the same verdict
• the tableau stays small enough to draw
• the tableau tests validity or satisfiability
• the term computes Ackermann's function
• the term computes mⁿ
• the term computes subtraction cut off at nought
• the term computes the predecessor
• the term's type at this node is the formula the derivation has there
• the tree drawn actually reaches the bottom row
• the trials per point is a whole number between 20 and 400
• the truth table, resolution and the components agree on every set
• the two derivations are a formula and its denial
• the two formulas share at least one atom
• the two formulas use at most six atoms between them
• the two unit clauses resolve to the empty clause
• the variable f is bound where it is used
• the variable h is bound where it is used
• the variable k is bound where it is used
• the variable x is bound where it is used
• the variable y is bound where it is used
• the view is one the family draws
• the walk always has a surviving child to step to
• the whole formula is consumed by the parser
• the whole string is one term
• the whole term is closed, because every assumption was discharged
• there are twelve two-literal clauses on three variables
• this pair unifies
• this view draws a refutation, so the set has to be unsatisfiable
• two to four numbers of variables from 10 to 2000
• unification finishes
• which derived a conjunction
• which implies the weakest
• with every edge false only the charged vertex is violated
• with few clauses nearly every random set can be satisfied
• with half as many clauses again as variables, the largest sets almost never can
• x is still there after every round

• 222 classes, sizes dividing 768, adding to all 65,536 tables
• a restriction shrinks the formula by at least the share of letters it frees
• all 256 counted
• and 114 functions of four letters need sixteen
• and at three letters the construction is the true minimum, 10
• and neither does its best value over subsets
• and no formula for it has fewer
• and only parity and its negation need ten
• as it is at four, 16
• at powers of two the construction meets it
• Burnside's count matches the known number of classes
• direct enumeration and Burnside's lemma agree
• every corner meets one edge per variable, each edge counted once
• every function of three letters has a circuit of at most four gates
• every leaf of the formula is a variable
• every one of parity's twelve edges joins a true corner to a false one
• every opening bracket in the formula is closed
• every set of more than half the corners has a corner of degree at least √n
• exactly 56 three-letter functions express everything alone, the same 56 Post's criterion picks out
• exactly one true needs exactly 4 gates
• fixing the best letter always leaves at most (3/4)^(3/2) of the formula
• for parity the bound is 16
• if a then b else c needs exactly 3 gates
• Khrapchenko's quantity never exceeds the true formula size
• leaves of the parity formula
• leaves surviving the restriction x3 = 0, x4 = 1
• majority of p, q, r is a weighted vote
• majority of three needs exactly 4 gates
• majority's middle levels give ((n + 1)/2)² for odd n
• no choice of corners can pass n², since each corner has at most n edges
• no construction beats the count
• no function has degree above its sensitivity squared
• no function shrinks more slowly than the share of letters freed
• odd number true needs exactly 2 gates
• on majority the rule stops making mistakes
• on parity it never does
• one to four named functions
• only nand and nor express everything alone
• p, or both q and r is a weighted vote
• parity is sensitive to every letter everywhere
• parity needs the most letters of any function
• parity of four letters needs exactly sixteen
• parity of three letters needs exactly ten
• parity reaches n² exactly
• parity: an odd number of p, q, r is not a weighted vote
• Post's criterion agrees with the closure
• Post's criterion agrees with the closure on every pair
• relabelling, negating letters or negating the answer keeps the count at four
• the bound never exceeds the true size
• the circuit drawn for exactly one true computes it
• the circuit drawn for if a then b else c computes it
• the circuit drawn for majority of three computes it
• the circuit drawn for odd number true computes it
• the constants and the parity of four letters each form a class of two
• the construction never beats Khrapchenko's bound
• the count lands within a small factor of 2ⁿ/n
• the counts are 4, 14, 104 and 1,882
• the cube is drawn for two, three or four variables
• the fixed tables add up to fourteen times the group's 96 elements
• the formula and its disjunctive normal form agree on every row
• the formula has between one and five variables
• the formula is a non-empty string
• the function is one this figure knows
• the highlighted rows are the true ones, the false ones, or neither
• the kinds account for every function that needs four gates
• the matrix squared is n times the identity
• the measure of how many tables have no symmetry climbs towards one
• the nonzero entries are the cube's edges
• the number of letters is a whole number between 2 and 4
• the option is one this view reads
• the search is over the 3-cube or the 4-cube
• the search reaches every function
• the signed matrix is drawn for one to four letters
• the table has one row per assignment
• the table-and-selector construction is within a constant factor of the count
• the top corner and its neighbours prove n for all true
• the tree computes the parity of four letters
• the two ends of an edge differ in exactly one variable
• the two formulas hold
• the value Andreev's function reads in the drawn instance
• the view is one of cube, threshold, thresholdcount, perceptron, sensitivity, huang, signed, degree, circuits, costs, shannon, build, explicit, hardest, khrapcube, formulahist, khrapscatter, paritytree, paritygrowth, khrapfamilies
• the weights account for all 7,581 functions
• the weights and the cut give the function
• the whole formula is consumed by the parser
• three letters give fourteen classes up to permuting, negating letters and negating the output
• twenty pairs f ≤ g among the six functions of two letters
• with sixteen letters at its leaves

• 3 circles realise every one of the 8 patterns ×2
• 4 circles cannot realise all 16 patterns ×2
• a successful try is found
• a try that walks several steps and succeeds
• adding clauses satisfied by the plant eventually leaves it the only solution
• after erasing the middle terms, the conclusion's region is empty
• and each of the sixteen is a single connected region
• and it finds the one solution
• cap is read only by the collapse view
• cap, the most things kept in one region, is 1 to 3
• each letter comes last in a critical clause at least a third of the time
• each step flips exactly one letter, so crosses exactly one curve
• every invalid classical form fails with at most one thing in each region
• every invalid form has a counterexample with at most four in each region
• every leaf of the formula is a variable
• every letter has a critical clause, because flipping it must break some clause
• every letter in the set expression names a drawn curve
• every region the drawing uses is present in it
• exactly two premises mention the first erased class
• fifteen classical forms are valid
• for every clause length the walk's base is below this method's, and both are below 2
• guessing right every time reaches the solution
• most M are P and most M are S together force some S to be P
• most S are M and most M are P do not make most S P
• ninety-nine per cent of M are P and no S is P
• ninety-nine per cent of S are M
• no form with 'most' in both premises and conclusion is valid
• no occupancy the premises allow refutes the conclusion
• no pattern occupies more pieces than the arrangement has
• puzzle is read only by the sorites, erase, clauses views
• resolving on the erased classes derives the conclusion's clause
• sentence is read only by the model, collapse, search views
• sets is read only by the circles view
• the average number of guesses is at most two thirds of the letters
• the bound is (3/4)ⁿ up to a factor of the walk's length
• the chance from distance j is at least 2^(−j) up to a factor the length of the walk
• the clauses leave exactly one satisfying assignment
• the collapsed model gives the sentence the same truth value as the large one
• the drawn counts satisfy both premises and deny the conclusion
• the drawn counts satisfy both premises and the conclusion
• the forced overlap is a + b − 1
• the formula has exactly one satisfying assignment
• the formula is a non-empty string
• the four ellipses realise all sixteen patterns
• the last resolvent is the conclusion's clause
• the linear programme's solution scales to a whole-number counterexample
• the measured rate is at least the pessimistic bound
• the measured success rate is at least the bound 2^(−2n/3)
• the number of sets is a whole number between 2 and 5
• the pieces the circles cut the plane into match Euler's count
• the premises empty every region the conclusion needs empty
• the puzzle is one this family knows
• the search is over one-per-region models, so the sentence has no ≠
• the sentence has a model to draw
• the sentence is one this family knows
• the success rate matches the average of 2 to the minus guesses
• the two clauses clash on the erased letter
• the view is one the family draws
• the walk's base at three letters is 4/3
• the whole formula is consumed by the parser
• with ≠ in the sentence, one thing per region is too few

• the integers modulo 2 have a swap-proof map exactly when 2 is odd ×10
• 2363 passes and its swap 2633 fails ×2
• 385 scramblings catch every spoken slip, and none of them more than 828 jump swaps
• a scheme that weights every position the same catches no transposition
• and every single-digit error too
• Damm catches every single error
• Damm catches every swap of neighbours
• Damm's table is not a group's
• descending weights over a prime modulus catch every transposition
• every jointly best scrambling is an 8-cycle with a 2-cycle
• every row and column holds each digit once
• every single-digit error is caught exactly when the modulus is at least ten and coprime to every weight
• every symmetry can be undone
• exactly one check digit completes a number under Damm
• exactly one check digit completes a number under Luhn
• exactly one check digit completes a number under Verhoeff
• exactly one check digit completes a number under weights 3,1 mod 10
• no pair of weights modulo ten catches every single error and every swap
• no running value lets a swap through
• the best a pair of weights catching every single error can do on swaps is eight in nine
• the count for eleven is the known 37,851
• the diagonal is zero
• the error kinds are single, transpose or twin
• the number is between four and fourteen digits
• the number is between six and fourteen digits
• the pentagon's symmetries have swap-proof maps
• the scheme is one of sum-9, sum-10, flat-11, weighted-10, isbn-11
• the sums modulo ten each miss some swaps
• the table is associative
• two collisions, two values missed
• Verhoeff catches every single error
• Verhoeff catches every swap of neighbours
• Verhoeff's permutation never lets a swap through

• the 1-th spoke has length √1 ×16
• the 1-th division of the height gives 1/3 of the angle ×6
• after 1 steps the shortfall is θ/(3·4^1) ×5
• the first 18 binary digits of 1/3 are right ×5
• the angle is between 12 and 174 degrees ×4
• 2cos(2π/11) is a root of it ×3
• √(2 + √2) is a root of its stated polynomial ×2
• √2 is a root of its stated polynomial ×2
• the circle about P1 has a centre and a radius ×2
• the circle of radius 1/3 of OP meets the spiral on the arm at 1/3 of the angle ×2
• the intersection named P1 is the one meant ×2
• the objects of step P1 meet ×2
• (1 + √5)/2 is a root of its stated polynomial
• ∛2 is a root of its stated polynomial
• 2 cos(2π/17) is a root of its stated polynomial
• ⁴√2 is a root of its stated polynomial
• a collapsing compass cannot carry a length from one place to another
• a length between 0.5 and 3
• a number built from square roots has degree a power of two
• a number the table knows
• a rational is a whole numerator over a non-zero whole denominator
• a right angle can be trisected
• a root found by the theorem really is a root of the cubic
• a square with side √π has the area of the unit circle
• a straight angle can be trisected
• a straight line goes to a straight line
• a third of AB takes seven circles with the compass alone
• a third repeats 01 forever
• a zero angle can be trisected
• AL equals BC, as Euclid's second proposition promises
• all eight conjugates of 2 cos(2π/17) are real
• and a number of degree three is not built from them
• and above it
• and at least one is out of reach of both
• and exactly half way from the other end too
• and falls short by about (π/6)·y²
• and five with the compass alone
• and from the other
• and is not half way along after it
• and it fails when the given point is not the midpoint, so the midpoint is what is used
• and it is a different line from the segment
• and it is one radius beyond B
• and it lies on the segment
• and it sits over the midpoint
• and neither centre stays where it was
• and on the segment
• and that crossing and the marked point are both on the circle
• and the centre to a point inside it
• and the diameter really is twice the radius
• and the map undoes itself
• and their images are not equally spaced
• and they are the numbers 2 cos(2πk/17)
• and with the compass alone
• at every positive height the curve stops short of 2/π
• at least one number the compass reaches and the dividers do not
• at least two polygons separate the two instrument sets
• at the end of one turn that length is the whole circumference of the circle through P
• at two different points
• between 4 and 48 sectors
• between three and eight polygons, each with 3 to 24 sides
• between two and eight steps
• between two and seven equal parts
• both crossings exist
• CB′ equals AB: reflecting in the line XY carries A to C and B to B′
• copies further out look smaller
• copying a length with a rigid compass costs four operations
• corner A is at 0°
• corner B is at 60°
• corner C is at 120°
• corner D is at 180°
• corner E is at 240°
• corner F is at 300°
• cos(θ/3) is a root of the triple-angle cubic
• Dinostratus: the quarter arc is to the side as the side is to the foot of the quadratrix
• divisors are taken of a positive whole number
• dropping the ruler makes the midpoint dearer
• each circle is carried onto itself
• each crossing lies on the polar x·P = 1
• each line through the point cuts the circle twice
• each map takes the circle to itself
• each move uses points already known
• each step divides the shortfall by four
• each step lands short of the far end
• each step meets the circle
• each step of the compass round the circle meets it
• Euclid's route to copying a length is dearer than a reflection with the same compass
• every arc in a class has the same price
• every coefficient of the minimal polynomial is a whole number
• every computed conjugate is a root of the polynomial
• every image lands on the second line
• every point of the circle lands on the circle
• every point of the first round is where two drawn objects cross
• everything the classical pair reaches, a conic reaches too
• Gauss's condition says the degree is a power of two
• Gauss's criterion and the degree test agree about every polygon
• Gauss's criterion and the degree test agree about the tripled polygon
• in both coordinates
• in every row the sum stands the same fraction of the window short of the third
• nineteen of the polygons up to sixty can be drawn
• no cheaper construction exists for this arc
• no construction of five circles or fewer finds the centre, at a generic arc
• no construction of the far corners of the square on AB with six circles or fewer
• no construction of the point one third of the way from A to B with six circles or fewer
• one radicand has a negative reading and the other has none
• one round puts two points off the line
• one unit square to the spoke makes the next spoke √(k + 1) — the step √(1 + x²)
• points far away are carried to the line between the circles
• six circles do
• six points on the circle
• six steps of the radius come back exactly to the start
• sixty degrees cannot be trisected
• some angle in the table the two instruments cannot cut, and the marked one can
• the angle comes from a polygon that can be drawn
• the angle the placed straightedge makes is a third of the original
• the apex is near the segment
• the apex is the carried length from one end
• the apex sees the diameter at a right angle
• the boost is between 0.1 and 1.5
• the centre and the reflected point are inverse in the first circle
• the centre is carried to tanh b along the axis
• the centre is equidistant from the ends of the diameter
• the chords cross inside the circle
• the circle about A has a centre and a radius
• the circle about B has a centre and a radius
• the circle about C has a centre and a radius
• the circle about D has a centre and a radius
• the circle about E has a centre and a radius
• the circle about F has a centre and a radius
• the circle about O has a centre and a radius
• the circle about X has a centre and a radius
• the circle about Y has a centre and a radius
• the circle has a sensible radius
• the classical six circles land on the centre
• the compass alone never needs fewer moves than both instruments
• the constructed line is parallel to the diameter
• the construction is one of those scored
• the cross ratio is the same on both lines
• the crossing is equidistant from the two ends of the remaining piece
• the crossing point sits on the turning radius at that instant
• the curve meets the radius at angle θ at height θ/90°
• the degree of the cosine is half of n minus one
• the degree's smoothness and the Pierpont condition agree
• the denominator the cosines are taken over is a whole number between 2 and 16
• the diameter is tilted between 5 and 175 degrees
• the distance from the centre is the fraction of the turn completed
• the fewest moves for the midpoint with ruler and compass
• the fewest moves for the square's far corners with ruler and compass
• the fewest moves with compass and straightedge
• the fixed opening is fixed
• the fourth proportional of 2/π, 1 and 1 is π/2
• the free point sits inside the segment PM
• the gap between them is one sector of the full circle
• the half-chord at the split is the geometric mean √(ab)
• the hendecagon is beyond the trisector
• the hexagon's other four corners take five moves with ruler and compass
• the image of a crossing is the crossing of the images
• the intersection named B is the one meant
• the intersection named B' is the one meant
• the intersection named C is the one meant
• the intersection named D is the one meant
• the intersection named E is the one meant
• the intersection named F is the one meant
• the intersection named G is the one meant
• the intersection named L is the one meant
• the intersection named M is the one meant
• the intersection named P is the one meant
• the intersection named Q is the one meant
• the intersection named X is the one meant
• the intersection named Y is the one meant
• the inverting circle meets the circle of radius AB
• the largest polygon considered is a whole number between 12 and 40
• the largest polygon tested is a whole number between 12 and 300
• the line from the point to each touching point is square to the radius — a tangent
• the line through them crosses the common chord at its midpoint
• the long side is √(1 + x²)
• the marked segment between the line and the circle is the radius
• the marked straightedge cuts every angle in the table
• the marks start equally spaced
• the middle mark is half way along before the projection
• the new height is √3/2
• the nine-gon is not — 3 is a Fermat prime but 9 repeats it
• the number being factorised is a whole number between 1 and 1000000
• the number of columns in the grid is a whole number between 6 and 20
• the number of compass steps is a whole number between 2 and 12
• the number of construction rounds is a whole number between 1 and 2
• the number of equally spaced marks is a whole number between 3 and 9
• the number of marks is odd so one of them is the midpoint
• the number whose square root is constructed is between 0 and 12
• the objects of step B meet
• the objects of step B' meet
• the objects of step C meet
• the objects of step D meet
• the objects of step E meet
• the objects of step F meet
• the objects of step G meet
• the objects of step L meet
• the objects of step M meet
• the objects of step P meet
• the objects of step Q meet
• the objects of step X meet
• the objects of step Y meet
• the perpendicular at the join has height √n
• the Pierpont condition and the degree's smoothness agree
• the point found is exactly half way along
• the point found is halfway along
• the point is outside the circle
• the polar of an outside point cuts the circle
• the polygon has a prime number of sides, so the degree is (n − 1)/2
• the polygon is a whole number between 5 and 23
• the polynomial has no rational root, so it does not factor off a linear piece
• the polynomial has the degree the totient predicts
• the region lies between the inscribed and circumscribed sectors
• the remaining cheap arcs are the multiples of 45°
• the search found the same pattern of moves at every generic arc
• the second line is tilted between 0.15 and 0.75
• the second round is shown or not
• the segment cut off is longer than the radius far out and shorter close in
• the segment is between 0.4 and 6 units long
• the segment is between half a unit and two units long
• the segment really is longer than the opening can span in one go
• the seven-gon is not
• the seventeen-gon is constructible
• the sliding point and the centre are the same distance from the near crossing
• the spiral runs to √4 … √17
• the tangent cuts off, on the perpendicular through the centre, a length equal to the arc of radius OP through the angle turned
• the tangent is drawn a quarter, half, three quarters or a whole turn out
• the totient of n above two is even
• the triangles to √17 go round less than once, so none overlaps
• the two arcs about D and E meet
• the two arcs of the fixed opening meet
• the two crossings sit at the same height, so the line is parallel
• the two readings multiply to the norm
• the two tangents of each circle at the crossing points meet on the line of centres
• the view is one of root, trisect, polygons, compass, hexagon, straightedge, neusis, reach, parallel, poncelet, rusty, conics, hendecagon, lemoine, lemoinetable, quadratrix, quaddivide, quadsquare, quadfoot, quadreach, cauer, cauerorbit, polar, pair, theodorus, dividers, conjugates, embed, seventeen, mean, spiral, spiraldivide, spiraltangent, spiralarea, bisectlimit, bisecterror, binaryratio, cheapmid, cheaptable, cheapsquare, cheapthird, cheapgrowth, cheaphex, centredial, centreclasses, centresix, centreinversion, centresearch, centresame, centreruler, centreprices, centrecheap, centresteps
• the walk was built
• three steps of the radius reach the point twice as far away
• three steps reach the far side, which is the doubling
• twenty-four of the polygons up to a hundred can be drawn
• two circles, apart or meeting, not touching
• two points and one round of drawing give four more
• two positive lengths
• two to five boosts up to 2.5
• what is left is short enough for the fixed opening to span
• whatever compass and straightedge reach, a trisector reaches too
• which is not the centre
• which makes it that piece's midpoint

• the tour of 6 choose 2 steps only along edges ×11
• the tour of 6 choose 2 visits every vertex once ×11
• step 1 changes exactly one place ×7
• graphs with smallest degree 3 that have a tour ×3
• the dimension of the cube is a whole number between 2 and 4 ×3
• closed tours of the 3 × 10 board ×2
• 100 to 1,000 processes a size
• 8, 10, 12 or 14 points
• a board from 5 to 12 squares each way
• a board that carries a closed tour
• a closed tour is found
• a code is more than one word and fewer than all of them
• a knight on an outer row always moves to an inner row
• a path through every point with unjoined ends exists
• a process whose graph at τ₂ has a tour is found
• a tour visits every point once
• a tour visits every square once
• a tour with changes 4, 4, 4, 4 exists and the search found it
• a tour with changes 8, 6, 6, 6, 6 exists and the search found it
• an open tour of the 5 × 5 board exists
• and its shortest cycle has seven edges
• and its worst single step changes all 4 places at once
• and none twice
• and the changes add up to the number of steps
• and the last corner is one step from the first, so the walk closes
• and the same number of them whichever vertex is removed, by symmetry
• and visits none of them twice
• but it has an open one
• closed tours of the 4 × 6 board, by exhausting every path
• counting up in the ordinary way changes more places than that
• Coxeter's graph has no closed tour, by exhaustive search
• Coxeter's graph has twenty-eight vertices
• each cycle was met once in each direction
• each step changes one place
• each step of the tour is a knight's move
• each step of the tour is an edge
• each step, the last back to the first included, swaps one element out and one in
• each with three neighbours
• every board the theorem allows has a tour found and checked
• every codeword has the length the cube has dimensions
• every corner meets one edge per coordinate
• every graph meeting Dirac's condition has a tour
• every graph meeting Ore's condition has a tour
• every knight's move changes colour
• every other Kneser graph tried has a closed tour
• every place changes an even number of times
• every step of the code changes exactly one place
• every subset appears
• every vertex's removal leaves a closed tour of the other nine
• every walk is enumerated for at most the 4-cube
• every word within 1 of a codeword decoded back to it
• labelled graphs on six points
• neither graph has a closed tour
• Petersen's graph has exactly six perfect matchings
• Petersen's graph has no closed tour
• removing a perfect matching leaves two pentagons, never one ten-cycle
• seed is read only by the path and tour views
• ten to eighteen points
• the 3-cube carries 6
• the 3-cube carries exactly 6 such walks
• the 4-cube carries 1,344 tours
• the 4-cube carries exactly 1,344
• the balls partition the cube exactly, so the code is perfect
• the board size is read only by the tour view
• the code is a word list or one of repetition, parity, none, even2
• the counting finds a crossing pair
• the cycle visits every word of the two levels once
• the drawn walks are the first of the ones counted
• the graph has a closed tour
• the half-length k is a whole number between 1 and 4
• the length of the words is a whole number between 2 and 4
• the levels hold every word
• the measured waiting time matches the Poisson prediction at each size
• the number of ones is a whole number between 2 and 4
• the number of places is a whole number between 3 and 11
• the one exception is Petersen's graph
• the prediction lies above γ and is coming down towards it
• the radius of the balls drawn is a whole number between 0 and 2
• the rule usually succeeds and sometimes fails
• the search closes a tour of the odd graph
• the search found a cycle through the middle two levels
• the share of processes with a tour at τ₂ rises with the number of points
• the size of the set is a whole number between 3 and 7
• the size of the subsets is a whole number between 1 and 5
• the smallest degree of the complete bipartite graph
• the smallest degree of the two cliques
• the three-row exceptions have no closed tour, by search
• the tour is one of reflected, balanced
• the tour of the full graph of thirty-five steps only along edges
• the tour of the full graph of thirty-five visits every vertex once
• the tour steps only along edges
• the tour visits every corner once
• the tour visits every vertex once
• the vertex removed is a whole number between 0 and 9
• the view is one the family draws
• the walk visits all 8 corners
• three-row boards of length 4, 6 and 8 have no closed tour, by exhausting every path
• turning the places one step sorts the twenty-eight words into four rings of seven

• the linear factors of x³ − 2 mod 5 are its roots ×15
• the linear factors of x³ − 3x + 1 mod 5 are its roots ×15
• φ^0 is 1 + 0φ ×10
• φ^1 has norm ±1 ×9
• the highest power is a whole number between 5 and 12 ×2
• √2 + ∛3 has degree six
• √2, √5 and 1/3 are distances already; the golden ratio is not yet; ∛2 and π are not
• ∛2 × ∛2 lands on a multiple of ∛2²
• ∛2 × ∛2² lands on a multiple of 1
• ∛2 × 1 lands on a multiple of ∛2
• ∛2² × ∛2 lands on a multiple of 1
• ∛2² × ∛2² lands on a multiple of ∛2
• ∛2² × 1 lands on a multiple of ∛2²
• 1 × ∛2 lands on a multiple of ∛2
• 1 × ∛2² lands on a multiple of ∛2²
• 1 × 1 lands on a multiple of 1
• a bigger subgroup names a smaller field, never the other way round
• a number missing from the line is missing
• a rational is a whole numerator over a non-zero whole denominator
• a subgroup of order two fixes exactly one root
• all three roots are real
• and grow without bound
• and the product's at √2 · φ
• and the sign alternates
• at least one of 2, 3 and 6 is a square modulo every prime
• between twenty and fifty-two primes are drawn
• both are monic
• divisors are taken of a positive whole number
• each ±√2 + ω^j∛3 is a root of the polynomial
• each field's points are symmetric about the line through the starting points and about their perpendicular bisector
• each new square root here is of a rational number
• each number under a root is a whole number between 2 and 40
• each number under a root is square-free, or the step is not a real step
• each pattern's share is its cycle type's share of the group
• each trace step divides exactly, as it must for an integer matrix
• every coefficient is a whole number under 400
• every coefficient of the class polynomial is a whole number to thirty places
• every degree divides the degree of the field, six
• every factor pattern of x⁷ − 7x + 3 is the cycle type of a collineation
• every factor pattern that occurs is the cycle type of an element of the group
• every measured share is within a percentage point of its share among the collineations
• every measured share is within a percentage point of its share in the group
• every pattern that occurs is a cycle type of the Galois group
• every point is rational, in ℚ(√3), or needs one more square root
• every positive unit is a power of ∛2 − 1
• every product of basis elements is a multiple of a basis element
• every rational root found by sweeping is one the theorem listed
• every unit found is ± a whole-number combination of the two
• for p ≡ 2 mod 3, x³ − 2 has exactly one root
• four to ten witnesses are listed
• it splits completely exactly when 2 and 3 are both squares
• j at the one reduced form is minus the cube
• modulo a prime leaving 3 on division by 8, 2 and −1 are non-squares and −2 is a square
• modulo every prime, x³ − x − 1 has as many roots as the class polynomial of −23
• no integer polynomial in the searched range vanishes at π
• no power of two is a multiple of three
• no whole number the rational-root theorem allows is a root of x³ − m
• some collineation has cycle type 1 + 2 + 4
• some combination has a smaller degree than the bound
• the cell of ℤ[√5] has area 2√5, twice as large
• the cell of ℤ[φ] has area √5
• the census runs to a thousand, ten thousand or a hundred thousand
• the census runs to ten thousand or a hundred thousand
• the characteristic polynomial of the sum's matrix vanishes at √2 + φ
• the closest miss is a genuine miss
• the coefficient bound is a whole number between 2 and 8
• the coefficient range is a whole number between 2 and 6
• the computed counts are 6 and 203
• the control polynomial really vanishes at its algebraic number
• the cube root really cubes to it
• the degree equals the number of reduced forms
• the degree is at most the product of the two degrees
• the degree of a tower of square roots is a power of two
• the degree of the tower is the size of its basis
• the denominators of the non-integer's powers never shrink
• the discriminant has three classes
• the discriminant has two or three classes
• the earlier powers are independent, so the solve is unique
• the element has the same cycle type on both sets of roots
• the element is k ↦ uk + b with u odd and b from 0 to 7
• the exact value and the decimal one agree
• the expansion begins 1/q + 744 + 196,884 q + 21,493,760 q²
• the factor degrees add to the degree
• the factor degrees add up to the degree
• the field has one class
• the field is one the family knows
• the fifteen points on the line are the ones named
• the five cycle types account for all 168 collineations
• the grid half-width is a whole number between 3 and 8
• the group of six permutations has exactly six subgroups
• the largest degree searched is a whole number between 1 and 4
• the largest prime listed is a whole number between 30 and 120
• the leading coefficient is invertible modulo p
• the leading coefficient is not zero
• the list of patterns covers every cycle type of the group
• the monic test agrees with a and b having the same parity
• the number being factorised is a whole number between 1 and 1000000
• the number being searched for is one of pi, sqrt2, cbrt2, phi
• the number of roots modulo p matches the forms that take p
• the number whose cube root is taken is between 2 and 30
• the polynomial found vanishes at √2 + ∛3
• the polynomial has degree between one and four
• the polynomial is named for the caption
• the polynomial vanishes at the number
• the powers of two are checked out to between 6 and 20 steps
• the product is the coefficient times the basis element it lands on
• the range of primes is a whole number between 2000 and 50000
• the rank is never negative
• the same search does find a polynomial for an algebraic number
• the search bound is a whole number between 2 and 7
• the searched number is a root of one of them
• the shares settle on 1/3 and 1/6
• the shortfall is close to 196,884 e^(−π√d)
• the size of the subgroup times the degree of its field is the degree of the whole
• the three logarithms of a unit add to zero, because its norm is ±1
• the tower has one basis element per subset of its roots
• the tower is built from one, two or three square roots
• the two chosen units are independent
• the value of j is a perfect cube
• the view is one the family draws
• two rounds give 6 and then 203 points, in floating point and exactly
• x³ − 3x + 1 splits exactly when p ≡ ±1 mod 9
• x⁴ − 10x² + 1 is never irreducible modulo a prime
• x⁸ − 3 and x⁸ − 48 factor alike modulo every prime not dividing 6
• x⁸ − 48 has as many roots modulo p as x⁸ − 3
• φ^k lies in ℤ[√5] exactly when k is a multiple of 3

• the point 0.041, 0.068, 0.026 lies in exactly one of the six pieces ×729
• the left piece keeps its 1th edge at 0° ×21
• the right piece keeps its 1th edge at 0° ×21
• the left piece still touches its pin at 0° ×7
• the right piece still touches its pin at 0° ×7
• the dihedral angle 90.000000° is one rational part of a half turn plus a whole multiple of the tetrahedron's angle ×5
• every one of the 2 triangles has area exactly 1/2 ×4
• 3 triangles ×3
• a side of 52° is longer than twice the leg, so the half-turn point exists ×2
• the number of sides is a whole number between 3 and 8 ×2
• 13 × 8 takes six squares, the Fibonacci numbers
• a grid whose spacing divides the common denominator
• A is on the circle at the legs' distance beyond the midline
• A is on the sphere
• a kite: and vertically
• a kite: the outward normals weighted by edge length cancel horizontally
• a larger quadrilateral has a larger fourth angle
• a parallelogram: and vertically
• a parallelogram: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a parallelogram: the outward normals weighted by edge length cancel horizontally
• a polygon of 5 sides falls into 3 triangles
• a polygon of n sides falls into n − 2 triangles
• a quadrilateral's corners add to 360°
• a rectangle: and vertically
• a rectangle: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a rectangle: the outward normals weighted by edge length cancel horizontally
• a regular hexagon: and vertically
• a regular hexagon: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a regular hexagon: the outward normals weighted by edge length cancel horizontally
• a tiling is found only for shapes whose partner is positive
• a tiling of the square is found
• a trapezium: and vertically
• a trapezium: the outward normals weighted by edge length cancel horizontally
• a triangle: and vertically
• a triangle: the outward normals weighted by edge length cancel horizontally
• a triangle's quantity does not vanish, so no number of cuts slides it into a rectangle
• A′ is on the circle at the legs' distance beyond the midline
• A′ is on the sphere
• A′′ is on the circle at the legs' distance beyond the midline
• A′′ is on the sphere
• A′′BC has the area of ABC
• A′BC has the area of ABC
• ABC has the area of ABC
• all six pieces have the same edge lengths
• an even number of triangles
• an L: and vertically
• an L: the outward normals weighted by edge length cancel horizontally
• and at Q
• and equal bases, so they are congruent
• and exactly one step up
• and so is its area from its corners
• and so is the angle at P
• and that current is the segment's length
• and the arc from C
• and the floor is well above nought
• and the ledger balances: 4 × 6 − 12 = 12
• and the other side at its midpoint
• and the same two tile the square
• and the spread is (5√5 − 11)/8
• and through the point opposite C
• and together they are the triangle's whole angle sum
• and volumes summing to 2
• at half a turn the three pieces tile the rectangle
• B and C are the same distance from the midline
• between one and five swing angles, each between 0 and 180 degrees
• between one and three side lengths, each under 150 degrees
• between two and five shapes
• between two and six sizes for the table, each up to 60 degrees
• both the areas and their conjugates add up to one
• each corner is a latitude and a longitude, both within 80 degrees
• each corner tetrahedron has edge 1
• each numerator over its power of three is the cosine of that multiple
• each odd step cuts the best spread by a factor of five or more
• each piece has a sixth of the cube's volume
• each shape is a named list of between three and ten finite points
• each small one's is 6 ⊗ θ
• each square is a whole block of grid cells
• every angle of each piece is a rational part of a half turn, so its Dehn invariant is nought
• every conjugate area is positive when the partner root is
• every drawn point is well on the side of the sphere facing the reader
• every edge of a closed solid belongs to exactly two faces
• every inner segment passes on exactly the current it receives
• every labelled point is on the side of the sphere facing the reader
• every square joins two segments
• every three-coloured triangle has area of 2-adic size at least 2
• every tile is similar to the given rectangle
• every turned corner is a point
• M is the midpoint of AB
• N is the midpoint of AC
• no edge has zero length
• no line carries all three colours
• no multiple of the tetrahedron's angle is a whole number of half turns
• no numerator in the sequence is a multiple of three
• no small move of the vertex does better than the best found
• no square appears among golden-rectangle tilings of up to nine tiles
• no two pieces overlap at any of 361 positions of the swing
• no two squares overlap
• on an irrational rectangle the squaring is still going when the squares are too small to draw
• on whole-number sides the squaring ends
• P is the midpoint of A′′B
• P is the midpoint of A′B
• P is the midpoint of AB
• six rectangles of ratio φ² tile a square, and no fewer side by side and stacked
• so does the arc from B
• some triangle is larger than 1/n and some smaller
• Sperner: the number of three-coloured triangles is odd
• the 2-adic valuation of zero is not a number
• the amount by which the fourth angle exceeds a right angle is the quadrilateral's area
• the angle at O is a right angle
• the apex sits over the base, away from its two ends
• the apex sits over the base, between its two ends
• the arc from A meets the midline at a right angle
• the area of A′′BC from its angles agrees with its area from its corners
• the area of A′BC from its angles agrees with its area from its corners
• the area of ABC from its angles agrees with its area from its corners
• the best is reached from several starts, and most starts settle elsewhere
• the bottom edge takes in the whole width
• the circle passes through the point opposite B
• the common quadrilateral's summit angles are half the angle sum
• the cube's dihedral angle is a quarter turn exactly
• the cut is the square's side, and the two perpendiculars make the other side
• the Dehn invariant of the cube, edge 1, added up edge by edge, is 0
• the Dehn invariant of the one sixth of a cube, added up edge by edge, is 0
• the Dehn invariant of the regular octahedron, edge √2, added up edge by edge, is −12√2 ⊗ θ
• the Dehn invariant of the regular tetrahedron, edge √2, added up edge by edge, is 6√2 ⊗ θ
• the Dehn invariant of the triangular prism, added up edge by edge, is 0
• the directions are sampled between 90 and 2000 times
• the drawn side is between 2 and 60 degrees
• the exact lengths match the drawing
• the fan of triangles tiles the polygon
• the four corners and the octahedron fill the large tetrahedron's volume
• the four pieces fill the triangle exactly
• the fourth angle is obtuse, so the four corners cannot all be right angles
• the half-turn about M carries B to A
• the large tetrahedron has edge 2
• the large tetrahedron's Dehn invariant is 12 ⊗ θ
• the midpoint of A′′C is on the midline too
• the midpoint of A′C is on the midline too
• the midpoint of AC is on the midline too
• the network is connected
• the network's equations have a unique solution
• the network's total current over its voltage is the rectangle's width over its height
• the number of multiples tested is a whole number between 6 and 30
• the number of risers in the staircase is a whole number between 1 and 5
• the number of spanning trees is a whole multiple of the width
• the octahedron has edge 1
• the octahedron has the volume of four small tetrahedra
• the octahedron's is −12 ⊗ θ
• the option is one this view reads
• the pieces are pulled apart by between nought and one and a half
• the pieces are pulled apart by between nought and one cube
• the polygon is turned by at most a half turn
• the polygon's radius is between 0.5 and 1.5
• the polygon's radius is between a half and two
• the quadrilateral's area from its angles is the triangle's
• the quadrilateral's two legs are equal
• the quadrilateral's two summit angles are equal
• the rectangle and the square have the same area
• the rectangle has the triangle's area
• the rectangle's shape is one of sqrt2, 13:8, phi
• the same three pieces tile the rectangle
• the shape's conjugate is positive
• the side A′′B has the length asked for
• the side A′B has the length asked for
• the side of the square is a whole number between 2 and 24
• the slice meets one side at its midpoint
• the slide is exactly one step across
• the smallest of the five areas is (3 − √5)/4
• the solid has Euler characteristic two, so its faces close up
• the solved current, scaled, is exactly the square's side
• the spread falls from a quarter to a few thousandths
• the square lies inside the range the cuts reach
• the square's side is divisible by n(n+1) so every corner lands on a whole number
• the squared rectangle is one of moron, duijvestijn
• the squared rectangle is one this family knows
• the squares tile the rectangle
• the squares' areas add up to the rectangle's
• the strips fall on the common grid
• the three pieces tile the triangle
• the tiles' areas add up to the square's
• the top edge sends out the whole width
• the triangle fits well inside one hemisphere, where midpoints and perpendiculars are unique
• the triangle is between 0.4 and 1.6 times as tall as its base is long
• the triangle's area from its angles agrees with its area from its corners
• the triangles exactly cover the square
• the triangles fill the square exactly
• the triangles' areas add up to the polygon's
• the turned pieces do not overlap
• the two pieces tile the rectangle
• the view is one of steps, polygon, dehn, hinge, translate, count, sydler, orthoscheme, tetocta, saccheri, lambert, lexell, squared, network, solve, greedy, grid, monskygrid, monskytri, monskycount, evencut, monskylines, oddbest, oddspread, oddlandscape, oddslice, oddareas
• the volume of the cube, edge 1 measured from its corners is 1
• the volume of the one sixth of a cube measured from its corners is 1/6
• the volume of the regular octahedron, edge √2 measured from its corners is 4/3
• the volume of the regular tetrahedron, edge √2 measured from its corners is 1/3
• the volume of the triangular prism measured from its corners is 1/2
• three pieces per triangle
• triangles ADM and BEM have equal legs
• triangles ADN and CFN have equal legs
• two tetrahedra and an octahedron of the same edge have Dehn invariants summing to nought
• two to four angles from 0° to 180°

• y² = x³ + 0x + 1 over GF(43) has a count inside p + 1 ± 2√p ×1806
• a curve over GF(101) has exactly 82 points ×41
• the 1 × 1 kernel has the dimension of gcd(p(x), p(x + 1)) ×40
• in GF(7) the sum of the 0th powers is 0 ×33
• every count the bound allows over GF(5) occurs ×16
• every cubic over GF(5) is inside the bound ×16
• the 9 zeros of a sum of 3 squares over GF(3) are divisible by p to at least the promised power ×12
• over GF(7) no cubic misses p by more than 2√p ×11
• there is a field with 4 elements only if 4 is a prime power ×7
• in GF(7) the sum of the 6th powers is −1 ×5
• A is a set of residues modulo 13 ×3
• B is a set of residues modulo 13 ×3
• the largest cap in dimension 1 ×3
• the monomial count at n = 4 ×3
• the prime is between 7 and 61 ×3
• the size of the field is a whole number between 2 and 9 ×3
• 2 zeros and 2 ones contain no 3 with a sum divisible by 3 ×2
• boards up to between 10 and 60 on a side ×2
• every 5 residues mod 3 contain 3 with a sum divisible by 3 ×2
• primes to 500 or 1,000 ×2
• the largest prime is a whole number between 13 and 97 ×2
• 1 − f² is 1 at a zero of f and 0 elsewhere
• A and B are read only by the sumset and cdgrid views
• a board is solvable exactly when every quiet pattern covers an even number of its lights
• a conic y² = x² + b misses p by exactly one
• a cycle can be lit exactly when its length is a multiple of four
• a graph on 21 vertices is never invertible
• a path can be lit unless its length leaves remainder 1 on division by 4
• a prime from 101 to 20,011
• a progression's own operation gives exactly 2k − 1
• a quiet pattern changes nothing
• a ring of composite size has a pair of non-zero elements multiplying to zero
• adding a vertex changes the dimension of the kernel by exactly one, up or down
• addition and multiplication are commutative
• an irreducible polynomial of the right degree was found
• an m × n board and an n × m board have the same kernel
• and every one of them can be lit
• and it is exactly the composite ones that leave an element without a reciprocal
• and no curve y² = x⁵ + ax + b by more than 4√p
• and so is the mean fourth power, an eighth
• and stays below the limit
• and that variable's power sums to 0 over GF(3)
• and the fewest is the smallest
• and the parabola construction is the smallest there is
• between 3 and 8 variables
• chasing leaves only the bottom row lit
• chasing the lights agrees with full elimination
• dim is read only by the dvir view and the cap views that take a dimension
• each quiet pattern of presses changes nothing
• each walk ends within 2√p of zero
• every constructed array appears in the search
• every constructed array is a Costas array
• every graph has an even number of odd-degree vertices
• every group without the false share still finds the secret
• every K(a, 1; p) is within 2√p
• every light sees an even number of the witness set
• every monomial has a variable whose power is not a positive multiple of p − 1
• every non-zero element has a reciprocal
• every other point is the third point of some pair of the cap
• every point of every chosen line is in the set
• every quiet pattern presses an even number of lights
• every random graph has an all-on press pattern
• every solution turns the board off
• every window of ten columns of the strip is a Costas array
• failures are exactly the boards with equal powers
• for every graph to seven vertices the kernel's law is exactly the chain's
• for sampled graphs the kernel's law matches the chain's within sampling error
• for the exact sizes the gap to the prediction shrinks with every vertex
• from order nine the search grows by a factor between four and five and a half per order
• from order seven on, the constructions account for under one array in twenty
• given a kernel of dimension k, about one graph in 2ᵏ can be lit
• Golomb's arrays from the field of thirteen are Costas arrays
• graphs on two vertices that can be lit
• K(1, 1; p) is within 2√p
• mod is read only by the cdmin view
• modulo a composite number the bound fails somewhere
• modulo a prime the smallest sumset is exactly min(p, a + b − 1)
• multiplication distributes over addition
• multiplication is associative
• no Kakeya set beats the counting bound
• no Kloosterman sum for any prime up to the bound exceeds 2√p
• no line holds three points of the cap
• no random cap beats the largest possible
• no row of the difference triangle repeats a value
• no set is small under both
• no two non-zero elements multiply to zero
• no two points of the cap have their third point in it
• odd primes up to 11
• on six vertices 2^10 = 1,024 labelled graphs have every degree odd
• on six vertices every graph with all degrees odd is litable
• one to three prime-power fields from 3 to 16
• over a cap, x + y + z = 0 only on the diagonal
• over a set with a line, it happens off the diagonal too
• q is an odd prime from 3 to 13
• q is read only by the tables, the log view and the kakeya view
• random graphs on an even number of vertices can be lit more often than on an odd number
• so the number of zeros is 0 mod 3
• so, since (0, 0, 0) is one, there is another
• some one-way network cannot be turned all on
• some part of every split is at most two thirds
• some row is divisible by exactly the higher power Ax and Katz promise, and no more
• the 3 × 3 press matrix is invertible over F₂
• the 5 × 5 board's press matrix has a kernel of dimension two
• the adjacency matrix of a graph has even rank over the two-element field
• the board is solvable exactly when the two-power rule says
• the board passes every parity check before it is solved
• the cap drawn is a largest one
• the cap has twenty points
• the cap's share of the space falls with every dimension
• the constant is 2.7551
• the dimension is 2, 3 or 4
• the drawn points and the count agree
• the drawn presses turn every light on
• the expansion is drawn over GF(3), where it fits on a page
• the exponent of a product is the sum of the exponents
• the field is closed
• the four-by-four board can be lit and the five-by-five cannot
• the full sum is real and within 2√p
• the largest cap blocks every point outside it
• the largest cap in the plane of nine points has four
• the largest cap in the space of twenty-seven points has nine
• the largest power summed is a whole number between 8 and 20
• the lower bound never exceeds what a construction achieves
• the m × n kernel has the dimension of gcd(pₘ(x), pₙ(x + 1))
• the mean square of the normalised error is the semicircle's, a quarter
• the modulus is 5, 7, 8, 9, 11 or 12
• the modulus is 7, 11 or 13
• the modulus is 7, 11, 12, 13 or 17
• the monomials at or below two thirds of the top degree are a minority
• the monomials of every degree number 3ⁿ
• the most points any curve has is the largest whole number the bound allows
• the number being factorised is a whole number between 1 and 1000000
• the number of primitive elements is φ(q−1)
• the number of random caps is a whole number between 200 and 5000
• the option is one this view reads
• the plane of nine points has twelve lines
• the possible secrets all share a parity
• the powers of a primitive element close back onto one
• the press pattern lights every light
• the prime is 1,009, 4,001 or 10,007
• the prime is 101 or 1009
• the rate climbs with n
• the sampled shares agree with Σ P(k)/2ᵏ
• the search finds the published count of Costas arrays
• the set has q(q + 1)/2 + (q − 1)/2 points
• the set holds a line in each of the q + 1 directions
• the size of the ring drawn beside it is a whole number between 0 and 12
• the sum of K(a, 1; p)² over a is exactly p² − p − 1
• the sum of K(a, 1; p)⁴ over a is exactly 2p³ − 3p² − 3p − 1
• the sumset is at least as large as Cauchy and Davenport promise
• the theorem is checked for n = 3 and n = 5
• the three-variable form is drawn over GF(3), GF(5) or GF(7)
• the tuples are all counted
• the two limiting shares add up to one
• the two-variable form is drawn over a prime from 3 to 11
• the view is one the family draws
• the walk ends at the count less p + 1
• the zeros of x² + y² + z² over GF(5) number a multiple of 5
• Welch's array from the prime 11 and the primitive root 2 is a Costas array
• with self-toggling every board can be lit (Sutner)
• with α = β the array is symmetric about its diagonal — Lempel's case
• x² + y² over GF(7) has only the zero at the origin, since −1 is not a square

• GF(7): 1 has exactly one inverse ×31
• the search agrees with the conjecture at q = 2, k = 2 ×27
• the number of codewords of weight 0 matches the formula for any MDS code ×9
• class 000 has as many words as the code has ×8
• class 000 has exactly one lightest word ×8
• lengths 6 to 12 ×2
• words of 8 to 14 bits ×2
• 1122 has the smallest worst class among the five patterns, 256
• 759 octads
• a = 0 gives the largest of the codes
• a ball of radius three holds 2,048 words
• a length of 8 to 14 and two positions within it
• a random code of the same size has some dependent triples
• a row called exact really divides to a power of two
• a word with r runs leaves between C(r − 1, 2) and C(r + 1, 2) words
• above capacity it rises
• above it they take over
• an exhaustive search for lengths 3 to 9
• and always reaches what Gilbert and Varshamov guarantee
• and it weighs no more than one, which is what perfection means
• and none of them is zero
• and the climb is not monotone — some length is worse than the one before it
• at distance three the bound is only ever exact one short of a power of two
• below capacity the error falls with length
• below the limit failures vanish with length
• each doubling gives 2n + k + 1 coins
• each of the 42,504 five-element sets lies in exactly one octad
• each span is a distance between three and five and a longest length up to eight
• each vertex is not joined to seven vertices, itself included
• every 3 columns of G are independent
• every 7 columns of the dual's generator are independent
• every codeword passes all three parity checks
• every Helberg class corrects two deletions
• every one of the 2²³ words lies within distance three of exactly one codeword
• every one of the single deletions is repaired to the codeword
• every pair of erasures is recoverable, since the distance is 3
• every pattern of three or fewer errors is corrected
• every rate is between zero and one
• every row of H is orthogonal to every row of G
• every single deletion of every codeword of VT₀(n) is decoded correctly
• every three columns of the extended Reed–Solomon code are independent
• every two rows share an even number of ones
• every two words of the code found are at least the distance apart
• every vertex of the icosahedron has five neighbours
• every word in the space is repaired to a codeword by its own syndrome
• exactly one way of putting two bits back lands in the codeword's class, and it is the codeword
• four data bits give sixteen codewords
• four doublings give fifteen weighings for thirty-two coins
• four errors always lead to a wrong codeword, three away
• Knuth's rule never needs more than five guesses
• more repeats, fewer errors
• most of the probability lies within two spreads of np
• no single-deletion code of length up to 7 is larger than VT₀(n)
• no word of length n − 1 is one deletion from two codewords
• nor Plotkin's, where it applies
• nor the Singleton bound
• one deletion turns a word into exactly as many words as it has runs
• one rate below 1 − ε and one above
• some length meets the sphere-packing bound exactly
• some pair of words in the one-deletion code shares a two-deletion remnant
• some triples are and some are not
• the 128 words fall into eight classes
• the block length is a whole number between 20 and 400
• the chance of failure is below 2 to the −m
• the count for k points matches the design formula
• the count of VTₐ(n) matches Ginzburg's divisor sum
• the divisor sum is exact for every length
• the eight lists of distances are all different
• the exact answer never beats the sphere-packing bound
• the field order is one of 2, 3, 4, 5, 7, 8, 9
• the flip probability is between 0 and a half
• the Golay code of length 23 has 253 words of weight 7
• the Hamming code's rate is above capacity at p = 0.1
• the highlighted codeword is a whole number between -1 and 15
• the largest arc in the plane of order 8 has 10 points
• the lightest non-zero codeword weighs what the closest pair are apart
• the longest word length in the table is a whole number between 5 and 23
• the longest word length searched is a whole number between 5 and 8
• the mean of the distribution is the sum
• the minimum distance is a whole number between 3 and 5
• the minimum distance over all 120 pairs is three
• the minimum distance the bound is taken at is a whole number between 3 and 7
• the number of cosets shown across is a whole number between 4 and 8
• the number of message bits is a whole number between 4 and 30
• the option is one this view reads
• the rate at the longest length beats the rate at the shortest, at every distance drawn
• the received word is seven bits
• the repetition code and the Hamming code both meet the bound exactly
• the repetition code meets the bound at n = d
• the search finished rather than running out of budget
• the search reaches the published value
• the set of words found corrects two deletions
• the seven columns of the parity-check matrix are distinct
• the seven single errors give seven different syndromes
• the smallest distance between two codewords is eight
• the sum settles on the constant
• the syndrome of a single error is the column of the position it hit
• the table is symmetric
• the trials agree with the exact product
• the view is one of syndrome, cosets, bound, best, rate, capacity, typical, repetition, random, rank, erasures, becsim, overhead, mdserasure, mdstable, mdsprofile, mdsdual, mdsweights
• the weights are 0, 8, 12, 16, 24 with counts 1, 759, 2576, 759, 1
• the word is 8 to 14 bits
• the word the syndrome repairs really is a codeword

• blocks 0 and 1 meet in exactly one point ×210
• points 0 and 1 lie on exactly one block ×210
• points 0 and 1 share 1 blocks ×156
• the pair 0,1 appears once ×105
• GF(9): 1 has exactly one inverse ×31
• point 0 is in 4 blocks ×20
• the difference 1 occurs exactly once ×20
• the diagonal of the Gram matrix is the block count at point 0 ×13
• a prime power 2 is never excluded by Bruck–Ryser ×11
• order 2 has a recorded status ×11
• the generator of GF(4) has order 3 ×11
• the powers of the generator of GF(4) do not repeat early ×11
• PG(2, 5) has n² + n + 1 lines ×7
• PG(2, 5) has n² + n + 1 points ×7
• PG(2, 5): every two points lie on a line ×7
• PG(2, 5): two points lie on only one line ×7
• the parabola y = x² closes to an arc in order 3 ×6
• in odd order 3 no other monomial graph is an oval ×4
• the plane of order 3 has n² + n + 1 lines ×4
• the plane of order 3 has n² + n + 1 points ×4
• the plane of order 3: every two points lie on a line ×4
• the plane of order 3: two points lie on only one line ×4
• in PG(2, 3) the ovals number q⁵ − q², the count of non-degenerate conics ×2
• no set of q + 2 points of PG(2, 3) avoids three on a line ×2
• PG(2, 2) has arcs of q + 2 points ×2
• the power of two is a whole number between 3 and 11 ×2
• a construction is known here for this many points
• a design on this many points was actually found by search
• a failing configuration was found
• a hyperoval has no tangent line
• a number is a sum of two squares exactly when no prime 3 mod 4 divides it an odd number of times
• a point lies on n + 1 lines and two points share exactly one
• a prime power is never excluded
• a projective plane over GF(q) has q² + q + 1 points
• a sum of two squares is divisible by 3 only when both are
• after the elimination only the last variable is left
• and distributes from the right
• and each is a conic with its nucleus
• and fails in the nearfield plane
• and no pair differs by nothing
• and none larger
• and the remaining lines miss it
• and the two squares it hands back add to n
• and therefore there are at least as many blocks as points
• any two lines meet in exactly one point
• Bruck–Ryser rules out 6 but not 10 or 12
• but not from the left, and it is not commutative
• Desargues never fails in the plane over GF(9)
• each point is in more blocks than it shares with any other
• each point lies in (v−1)/2 triples
• each vertical line holds one affine point of the curve
• enough configurations were drawn
• every block's size times the block count is the point count times r
• every class found belongs to one of the known families
• every exponent equivalent to one that works also works
• every line carries q + 1 points
• every line has q + 1 points
• every member of the base block is a residue modulo n
• every minimum respects Bruen's bound
• every other point off the oval lies on exactly one
• every oval of PG(2, 5) lies on a conic
• every pair of its q + 2 points spans its own line
• every pair of points appears
• every point lies on q + 1 lines
• every tangent passes through the horizontal direction
• every whole number is a sum of four squares
• every word here has weight 0, 8 or 16
• exactly one line — the line at infinity — passes every test
• exactly one line passes through any two points
• in every other position it fails on a large share
• in order eight the exponents 2, 4 and 6 give hyperovals
• least weight matches the flats
• no fraction has a zero denominator
• no line meets the curve in three points
• no point off the oval lies on exactly one tangent
• no random minimal set is smaller than the searched minimum
• no three of the points are collinear
• one point lies on every tangent: the nucleus
• q(q+1)/2 points lie on two tangents
• q(q−1)/2 points lie on none
• seven points meeting seven lines in three and the rest in one: a Baer subplane
• so the Gram matrix is invertible and the incidence matrix has full column rank
• the base block is between three and six distinct whole numbers
• the chosen sign leaves the variable solvable
• the count through one point times the points, divided by the size, is whole
• the design drawn is one that exists
• the design has v(v−1)/6 triples
• the design is one the family knows
• the determinant computed by elimination is the closed form
• the divisibility test and the residue rule agree
• the drawing is the projective plane over GF(2), up to relabelling
• the elimination ends at n x² = 1 + w², exactly
• the elimination is for orders that leave one or two over when divided by four
• the field order is one of 2, 3, 4, 5, 7, 8, 9
• the field plane never fails (axis at infinity, centre off it)
• the field plane never fails (axis at infinity, centre on it)
• the field plane never fails (finite axis, centre off it)
• the field plane never fails (finite axis, centre on it)
• the field size is 2, 3, 4, 5, 7 or 8
• the first orders Bruck–Ryser excludes are 6, 14, 21, 22 and 30
• the forced corners are new points off the axis
• the largest correlation is at the word sent
• the largest number of points in the table is a whole number between 9 and 25
• the largest number of points tested is a whole number between 9 and 40
• the largest order shown is a whole number between 20 and 120
• the largest power of two searched is a whole number between 5 and 12
• the last point-variable is not nought, so n is a ratio of two squares' sum
• the line at infinity is the last line built
• the majority over parallel flats finds the coefficient despite the error
• the minimum is 3(q + 1)/2 in a plane of prime order
• the nearfield plane has n² + n + 1 lines
• the nearfield plane has n² + n + 1 points
• the nearfield plane: every two points lie on a line
• the nearfield plane: two points lie on only one line
• the nearfield's multiplication is associative
• the number of points in the design drawn is a whole number between 7 and 15
• the number of points in the design is a whole number between 3 and 21
• the number written as four squares is a whole number between 1 and 99
• the option is one this view reads
• the order is 3, 4, 5 or 7
• the order is 3, 4, 5, 7 or 8
• the order is an odd prime up to 7
• the orders run are among 2, 5 and 9
• the plane drawn is of order 2 or 3
• the plane is built over a prime between 2 and 5
• the plane is built over a prime field between 2 and 5
• the plane of order 2 has no blocking set without a line
• the plane of order 4 holds 360 Baer subplanes
• the plane over GF(9) has n² + n + 1 lines
• the plane over GF(9) has n² + n + 1 points
• the plane over GF(9): every two points lie on a line
• the plane over GF(9): two points lie on only one line
• the point-variables and one more fill blocks of four exactly
• the points are labelled or not
• the prime is one that leaves three over when divided by four
• the residue rule and the divisibility rule agree at every size
• the rows are perpendicular and each has squared length n
• the search agrees with the arithmetic
• the shifts give n distinct blocks
• the size of the permuted set is a whole number between 1 and 8
• the third meeting point is off the line through the other two
• the three meets of corresponding sides are collinear
• the three midpoints are the same distance from the centre, so one circle holds them
• the two divisibility conditions are exactly v ≡ 1 or 3 mod 6
• the view is one of matrix, steiner, sizes, fisher, difference, desargues, nearfield, planecheck, desfail, affineconic, ovals, tangents, monomials, orders, axes, translation, gram, ordersmap, foursquare, brrun, twosq, monocensus, monoclasses, hypergrid
• there are exactly q + 1 tangent lines, one at each point of the oval
• with the axis at infinity and the centre on it, the nearfield plane never fails

• the squares for 1 and 2 produce every ordered pair once ×15
• and once in each column of the square for 1 ×10
• every symbol appears once in each row of the square for 1 ×10
• positions 1 and 2 together name every word ×10
• a field of order 4 has 3 non-zero multipliers ×8
• the count at order 1 is a positive number ×8
• there is a field with 4 elements only if 4 is a prime power ×8
• between 1 and 3 squares ×7
• order 3: the field's 2 squares give a code of length 4 with every two words differing in 3 places ×7
• a symbol was found for column 1 ×6
• column 1 repeats no symbol ×6
• column 1 was given a symbol it could take ×6
• relabelling left the pair 1, 2 orthogonal ×6
• symbol 0 is missing from exactly n−k of the columns ×6
• the cyclic square of order 3 has 3 transversals ×5
• the relabelled square for 1 starts 0…3 along its first row ×5
• each word has exactly 15 others at distance 4, as the MDS weight formula requires ×4
• row 1 uses every symbol once ×4
• every point lies on 4 lines, one from each class ×3
• the number of rows already placed is a whole number between 1 and 4 ×3
• and once in each column of the square for 2α+1 ×2
• and once in each column of the square for α+1 ×2
• every symbol appears once in each row of the square for 2α+1 ×2
• every symbol appears once in each row of the square for α+1 ×2
• the 3 available symbols are all used, so an extra square would repeat one ×2
• the code has 16 words ×2
• the order enumerated in full is a whole number between 1 and 4 ×2
• the order of the squares is a whole number between 3 and 8 ×2
• 48 top halves can be finished in exactly 2 ways
• a square laid over itself produces only its own diagonal of pairs
• a top half finishes 4 ways when both boxes stack the same pairs in their columns, and 2 ways otherwise
• A₄: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• A₄: every row of the table is a permutation
• A₄: the operation is associative
• addition and multiplication are commutative
• an associative Latin square has an identity
• an even order from 4 to 8
• an irreducible polynomial of the right degree was found
• an odd order from 3 to 9
• and 288 of them fill every 2×2 box
• and 9,408 of order 6
• and every grid has four or eight others that differ from it in exactly four
• and every table whose elements are all self-inverse has one
• and exactly half of the larger family does
• and fifty-six of order five
• and four have every element its own inverse
• and its largest partial transversal has 5 cells
• and nine thousand four hundred and eight of order six
• and no pair of points is met by two lines
• and none of them carries the symbol already standing in that column
• and once in each column of the first square
• and once in each column of the second square
• and once in each column of the square for 2α
• and once in each column of the square for α
• and once in each column of the square for α²
• and once in each column of the square for α²+1
• and once in each column of the square for α²+α
• and once in each column of the square for α²+α+1
• and sixteen of them associate
• and the other 48 in exactly 4
• and three of them associate
• and twelve have an element of order four
• any two coordinates already tell every word apart
• any two words differ in all but at most one place — the most a code this size can manage
• at order 11 the formula is within five per cent
• D₄: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• D₄: every row of the table is a permutation
• D₄: the operation is associative
• D₅: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• D₅: every row of the table is a permutation
• D₅: the operation is associative
• D₆: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• D₆: every row of the table is a permutation
• D₆: the operation is associative
• Dic₃: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• Dic₃: every row of the table is a permutation
• Dic₃: the operation is associative
• each class holds n lines
• each column may still take n−k symbols
• each region has four cells
• each square either associates or does not
• every forced entry is the square's own
• every grid of the smaller family stacks matching pairs in its top boxes
• every line holds n of the points
• every non-zero element has a reciprocal
• every ordered pair of symbols appears exactly once
• every ordered pair of symbols occurs exactly once
• every pair of points is met by some line
• every set of columns can still take at least as many symbols as it has columns
• every square of order 5 has a transversal
• every square of order 6 has a partial transversal of 5 cells
• every symbol appears once in each row of the first square
• every symbol appears once in each row of the second square
• every symbol appears once in each row of the square for 2α
• every symbol appears once in each row of the square for α
• every symbol appears once in each row of the square for α²
• every symbol appears once in each row of the square for α²+1
• every symbol appears once in each row of the square for α²+α
• every symbol appears once in each row of the square for α²+α+1
• every symmetry carries a grid to a grid
• exactly 1 of 4, 6 of 56 and 0 of 9,408 reduced squares of orders 4, 5, 6 have mates
• fifty-six reduced squares of order five
• forced moves alone fill the square from the critical set
• four of the sixteen have every element its own inverse
• multiplication distributes over addition
• multiplication is associative
• no entry of the two triangles can be removed
• no square without a mate holds n − 1 disjoint transversals
• no table with an element of order four has a transversal
• no three given cells ever force a unique grid
• no two non-zero elements multiply to zero
• no two pieces share a wrapped diagonal
• no two standardised squares carry the same entry under the corner
• one shift per row already placed
• one to three named partitions
• order nine almost never, order ten mostly
• Q₈: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• Q₈: every row of the table is a permutation
• Q₈: the operation is associative
• relabelling left the pair 1, α orthogonal
• relabelling left the pair 1, α+1 orthogonal
• relabelling left the pair α, α+1 orthogonal
• relabelling the symbols of a grid gives a grid
• removing any entry of the critical set loses uniqueness
• S₃: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• S₃: every row of the table is a permutation
• S₃: the operation is associative
• six entries determine the cyclic square of order five and no five do
• six of them have a mate
• sixteen tables of order four associate
• some four do
• some sampled square of order ten has a mate
• some square of order 6 has none
• squares with mates have more transversals
• stopping at the first proper square is biased; reading at fixed times is not
• the 288 grids fall into exactly two classes under the symmetries
• the alphabet size is a whole number between 3 and 9
• the box partition gives the 288 grids
• the broken partition admits none
• the cyclic square has transversals exactly at odd orders
• the cyclic square of even order 6 has no transversal
• the cyclic square of order 4 is Latin and not a grid
• the drawn order agrees with the table
• the field is closed
• the four self-inverse tables have the same number of transversals
• the grids have 96 different top halves
• the independence estimate lands within one per cent of the true count
• the largest order enumerated is a whole number between 3 and 6
• the Latin square view is one of transversal, mols, bound, net, count, extend, quasi, mates, partial, ryser, growth, groups, queens
• the multiplier of the second square is a whole number between 1 and 7
• the neighbouring orders are shown or not
• the new row uses every symbol once
• the number being factorised is a whole number between 1 and 1000000
• the number of columns is a whole number between 3 and 6
• the number of parallel classes drawn side by side is a whole number between 2 and 5
• the number of squares drawn side by side is a whole number between 1 and 4
• the order of the field the squares are built from is a whole number between 3 and 8
• the order of the plane is a whole number between 2 and 4
• the order of the square is a whole number between 3 and 7
• the pair is built from a prime order, where the construction works
• the parallel classes are the rows, the columns and one per square
• the placements are exactly the transversals
• the puzzle drawn has exactly one completion among the 288 grids
• the quasigroup view is one of counts, tables
• the quoted orders are orders between 7 and 11 and larger than the ones enumerated
• the radius of the points is a whole number between 3 and 8
• the relabelled square for α starts 0…3 along its first row
• the relabelled square for α+1 starts 0…3 along its first row
• the second square uses a multiplier that makes it a different square
• the share at the ceiling rises with the order
• the size of the field is a whole number between 2 and 16
• the size of the permuted set is a whole number between 1 and 8
• the smallest defining set over all squares of order n is ⌊n²/4⌋
• the smallest defining set reveals a cell of every intercalate
• the squares for 1 and α produce every ordered pair once
• the squares for 1 and α+1 produce every ordered pair once
• the squares for α and α+1 produce every ordered pair once
• the three transversals of each square without a mate share a cell
• the total at order four agrees with every square of order four, listed
• the word is a whole number between 0 and 15
• there are 56 reduced Latin squares of order 5
• there are 576 Latin squares of order 4
• there are 576 Latin squares of order four
• there are four reduced squares of order four
• there are twelve Latin squares of order three
• those with mates have fifteen transversals, the rest three
• twelve of them have an element of order four
• two associative squares were found to draw
• two corner triangles of ⌊n²/4⌋ entries force the cyclic square
• two different grids always differ in at least four cells
• two distinct positions kept
• with any two symbols kept, exactly one word fits
• with its first box fixed, the top band can be finished in 56 × 6⁶ ways
• with nothing revealed every one of the 161,280 squares of order five is possible
• with one kept, q words fit
• ℤ₁₀: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₁₀: every row of the table is a permutation
• ℤ₁₀: the operation is associative
• ℤ₁₂: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₁₂: every row of the table is a permutation
• ℤ₁₂: the operation is associative
• ℤ₂ × ℤ₂ × ℤ₂: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₂ × ℤ₂ × ℤ₂: every row of the table is a permutation
• ℤ₂ × ℤ₂ × ℤ₂: the operation is associative
• ℤ₂ × ℤ₂: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₂ × ℤ₂: every row of the table is a permutation
• ℤ₂ × ℤ₂: the operation is associative
• ℤ₂: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₂: every row of the table is a permutation
• ℤ₂: the operation is associative
• ℤ₃ × ℤ₃: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₃ × ℤ₃: every row of the table is a permutation
• ℤ₃ × ℤ₃: the operation is associative
• ℤ₃: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₃: every row of the table is a permutation
• ℤ₃: the operation is associative
• ℤ₄ × ℤ₂: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₄ × ℤ₂: every row of the table is a permutation
• ℤ₄ × ℤ₂: the operation is associative
• ℤ₄: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₄: every row of the table is a permutation
• ℤ₄: the operation is associative
• ℤ₅: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₅: every row of the table is a permutation
• ℤ₅: the operation is associative
• ℤ₆ × ℤ₂: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₆ × ℤ₂: every row of the table is a permutation
• ℤ₆ × ℤ₂: the operation is associative
• ℤ₆: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₆: every row of the table is a permutation
• ℤ₆: the operation is associative
• ℤ₇: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₇: every row of the table is a permutation
• ℤ₇: the operation is associative
• ℤ₈: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₈: every row of the table is a permutation
• ℤ₈: the operation is associative
• ℤ₉: a transversal exists exactly when the Sylow 2-subgroup is trivial or not cyclic
• ℤ₉: every row of the table is a permutation
• ℤ₉: the operation is associative

• shift register, 17 bits rank at size 24 ×17
• the sampled yield at bias 0.2, depth 1, matches the recurrence ×12
• the XOR of 1 flips is 1 with probability ½(1 − (1 − 2p)^k) ×12
• and output 4, which was not ×11
• xorshift, 32 bits rank at size 40 ×10
• bit 0 of the state repeats every 2ᵏ⁺¹ steps ×8
• all 4 windows of 2 bits occur somewhere in the period ×7
• at 2 digits the middle square lasts a fraction of a random map's run ×4
• seed 6239 repeats after the step the census says ×4
• the recovered rule matches output 0, which was given ×4
• the modulus is a whole number between 16 and 65536 ×3
• the register of 4 bits visits every nonzero state before repeating ×3
• xorshift128+, lowest bit rank at size 144 ×2
• 32 bits cover 12 cards, 64 bits 20, and 128 bits fall short of 52
• 52! is about 2^225.58
• a 32-bit seed reaches at most one deck in 10^58
• a coin that can land either way
• a lone 1 spreads to a handful of bits in the first ten twists and to about half the state within eight hundred
• a multiplier consistent with the outputs exists
• a multiplier is a whole number between 2 and 1048576
• a sample with two repeated gaps turns up quickly
• a short vector the modulus annihilates exists
• a state holding one small number starts in zeroland as a single bit does
• an even number of flips drawn, so they pair up
• and arrives at p(1 − p) bits per flip
• and it is a de Bruijn sequence: every window of that width, exactly once
• and its neighbouring output bits are correlated
• and never exceed the Chor–Goldreich bound
• and so is the second
• and the all-zero window never appears
• and the outputs cover as many values as random draws would
• and the plain generator's
• and they occur equally often, but for the one the all-zero state would have supplied
• and visits every sixteen-bit state once
• and visits none of them twice
• applying the filter twice returns the word
• at k = 16m nearly every run of the generator lands in the Poisson tail
• at k = 16m nearly every run rejects, for every period from 2^12 to 2^30, with multipliers of merit above 0.97
• below its state size the generator gives nearly full rank
• between 32 and 256 steps
• bit b of the high byte is state bit b + 8, repeating every 2ᵇ⁺⁹ steps
• bits above the lowest few are nonlinear and give nearly full rank
• each card below the top is most likely to end one place higher than it started
• each card ends somewhere
• each extra level adds, and none passes the entropy
• each item ends somewhere
• each of them exactly once
• each order's probability is a whole number of runs over 256
• each output bit is the sum of the bits five and two places before it
• each window is between one bit and the register's width
• every generator's points fall on a genuine family of parallel lines
• every one of the 200 starts escapes within two million outputs
• every order Sattolo's shuffle produces is a single cycle
• every output bit of the permuted generator has the full period
• every point of the sequence satisfies the relation exactly
• every random rectangle obeys Lindsey's bound
• every recurrence escapes within the outputs computed
• every seed ends in exactly one cycle
• Fisher–Yates gives every order of six cards the same chance
• for decks of three to eight cards the unshuffled order is never the most likely
• four outputs pin the permuted generator's state down to one
• from the low bits only 22,282 orders appear, from the high bits a little more than a random assignment would give
• how many high bits are predicted is a whole number between 1 and 8
• inside a single row of cells, the occupied cells have at most six different gaps
• Lindsey's lemma: the rectangle's sum is at most √(|A||B|N)
• middle-square seeds repeat in less than half the steps a random map's would
• no filter passes the ceiling
• no four-digit cycle is longer than four
• no matrix from a linear generator exceeds the state size in rank
• no more rows than state bits
• no short whole-number relation holds between consecutive outputs
• on the two perpendicular sets every entry is the same
• only the 32-bit linear generator fails at forty
• pairs, triples and quadruples all match λ at k = m and leave it a few doublings later
• random points and the generator at k = m/16 both match the Poisson law closely
• rank of the 192 × 192 grid from xorshift128+'s lowest bit
• rank of the 48 × 48 matrix from the 32-bit xorshift
• rank of the nine-by-nine grid from a five-bit register
• raw, beyond one bit, only three outputs in a row are evenly spread — the three words of state
• raw, the patterns that occur number 2^rank
• Sattolo's shuffle of six cards produces exactly the 120 orders that are a single cycle
• six cards identify every one of the 2²⁰ seeds
• six digits: seventeen cycles, the longest of 210
• some card-and-place pairs are a third more likely than fair and some a quarter less
• sources on coordinate blocks give bias ½·2^−(overlap)
• tempered the rows are independent and raw they are not
• tempered, every pattern occurs equally often but zero, which is one short
• tempered, likewise
• tempered, the rank is full
• tempering never makes the spread worse at any depth
• the 256 runs of four steps are shared among the 24 orders
• the 65,536 states reach 62,616 of the 362,880 orders of nine cards
• the all-zero state is the one the register cannot leave
• the all-zero window is the one that is short, by exactly one
• the bare rule from this seed visits only a handful of values
• the bias grows with the deck
• the catching sample is the cube root of 128m
• the criterion and the measured period agree
• the dimension is a whole number between 2 and 3
• the drawn steps compute the generator's output
• the example word is a whole number between 0 and 255
• the fifteen gaps add up to the span and two of them repeat an earlier gap
• the filter sends the 256 words to 256 different words
• the first outputs are almost all zeros and those after 400,000 are not
• the first prime is a whole number between 3 and 4000
• the first prime is three modulo four, which the construction requires
• the first seed is one of the longest-lived
• the flips drawn is a whole number between 8 and 48
• the high byte's triples fail it by far
• the highest tap is the register's own width, or it is a shorter register
• the increment is a whole number between 0 and 1048576
• the inner product is closer to fair than either simple rule
• the larger and sparser the recurrence, the later it escapes
• the last item stays last about half the time
• the least likely order is the last card moved to the top, every other card one place down
• the linear solve, run on this generator, predicts the next output wrongly
• the low bits put cards in some positions far more often than others
• the modulus is small enough that the arithmetic stays exact
• the multiplied generator gives a nearly full rank
• the multiplier 229 passes the two-dimensional spectral test well
• the multiplier is a whole number
• the multiplier is a whole number between 2 and 1048576
• the multipliers really do differ, by more than a factor of two in spacing
• the multiplying generator passes the rank test at 32
• the number of outputs it then predicts is a whole number between 2 and 16
• the number of outputs the solver is given is a whole number between 3 and 8
• the number of points is a whole number between 16 and 4096
• the number of predictions is a whole number between 50 and 4000
• the number of triples is a whole number between 100 and 4000
• the occupied hyperplanes are consecutive, so the family has no gaps
• the output is balanced to within sampling error
• the output of the sticky source is balanced
• the output shows every nonzero window of the register's width
• the outputs drawn is a whole number between 16 and 96
• the patched sequence is as long as there are windows
• the pattern is eight bits, so it fits a 16 × 16 grid
• the permuted output leaves the share of empty cells a random sequence would
• the permuted output's triples pass the chi-squared test
• the plain recurrence matches λ up to k = m and leaves it from k = 16m on
• the plain recurrence reaches 256 cells with ten gap lengths, the scrambled one with over forty
• the plain recurrence's reachable cells never have more than forty different gaps; the scrambled ones have over a thousand
• the predictor gets essentially every output right
• the rectangle is one of random, subspace
• the register does not repeat before its full period
• the register visits every non-zero state before repeating
• the register width is a whole number between 5 and 14
• the relation involves the middle coordinate, so an edge-on view exists
• the relation involves the second coordinate
• the relation's normal lies in the screen, so the planes are seen edge on
• the scrambled states stay at λ however fine the cells
• the second prime is a whole number between 3 and 4000
• the solver is given only a handful of outputs
• the state is a sixteen-bit number
• the state returns to its start after exactly 2¹⁶ steps
• the state's own bytes leave far more empty
• the taps are positions inside the register
• the top card is equally likely to end anywhere, and every card is equally likely to end at the bottom
• the triples lie on a small number of planes
• the twisted generator runs through every non-zero state before repeating
• the twister reproduces its reference outputs for the seed 5489
• the twister's own seeding puts out half ones from the first block
• the two multipliers give different numbers of lines
• the vector really does annihilate every point
• the view is one of compare, period, planes, lfsr, spectral, recover, nextbit, bbs, equi, vneumann, rates, sticky, xor, kdist, patterns, rankgrid, temper, raster, twosourcegrid, lindsey, twosourcebias, twosourcesim, pcgbits, pcgpairs, pcgmachine, pcgtriples, pcgrecover, pcgperiods, mswalks, mstails, mscycles, msscale, msweyl
• the width of the register is a whole number between 3 and 6
• three to eight multiplier-and-increment pairs
• three to six bits
• two multipliers below the modulus
• two taps
• which is enormously better than guessing the leading bits
• while the independent coin's output shows no correlation
• with the counter, no state repeats in the run

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

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