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
• after its last difference each sequence agrees with the representative for good
• and everyone after it is right
• between two and five pairs, each of two named things
• different pairs of whole numbers name different numbers a + b√2
• each element belongs to exactly one chain
• each rational point sits on the line through (1, f(1))
• every moved point lands strictly inside the open interval
• every point the map has to reach is reached
• f(x + y) = f(x) + f(y) on a + b√2
• no two moved points land on the same place
• on every one of the 2^8 assignments, everyone after the first is right
• the assembled map sends no two elements to the same place
• the function is not the line y = x
• 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 members of a pair are named apart exactly when an order is claimed
• the number of elements drawn from each side is a whole number between 5 and 14
• the number of sequence terms drawn is a whole number between 4 and 12
• the pairs either carry an order or they do not
• the 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 two maps are different, or there are no chains to see
• 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
• under the parity strategy all are right together or all wrong together
• with √2 sent to √2 the function is the line y = x

• 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
• (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
• √2 appears at some stage 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
• 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 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 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
• heights from 2 to 7
• inside the model no function is onto
• more than one geodesic through the point misses the base line
• no drawn dot is a known transcendental
• nothing lies strictly between an element and its successor
• outside, the bijections number size factorial
• sets of two to four elements
• some element is neither even nor odd
• the angles of a triangle in this model add to less than a straight angle
• the defect is a positive area
• the drawn elements are in increasing order
• the even witness doubles back to the element
• the galaxies are distinct and in order
• the lower ones rise and the upper ones fall
• 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 parity statement fails on sampled elements
• the point is off the base geodesic
• the view is one of triangle, euclid, compactness, polymodel, parity, axioms, galaxies, algebraic, skolemhull, relative, sentences, supremum
• 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
• 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
• Duplicator survives the two-pebble bijective game
• 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
• one graph is connected and the other is not, so a sentence separating them would express connectedness
• one or two round counts, each of at most three
• one-dimensional refinement gives both graphs the same colours and cannot tell them apart
• 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 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 sweep runs over between three and eight cycle lengths
• the sweep runs to between five and nine
• the table runs to between four and nine
• the transcript ends the way the exhaustive search says the game ends
• the two cycles are of different lengths
• the two cycles have three points each
• the two graphs have the same number of points and the same number of edges
• the two neighbourhoods are the same object, computed from the cycles rather than named
• the two small cycles run to between three and thirteen points
• the view is one the family draws
• two chains of between two and nine elements
• two cycles of between five and thirty points
• two-dimensional refinement does tell them apart
• words up to between four and six letters
• words up to between six and twelve letters

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

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

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

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

• ¬¬¬p → ¬p holds in every algebra searched
• ¬p → q ∨ r ⟹ (¬p → q) ∨ (¬p → r) holds in every algebra searched
• a set is contained in its double negation
• 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 so does double-negation elimination
• and the value really is not the top
• between one and four points are punched out
• every element sits below its double negation
• 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
• excluded middle fails somewhere in this algebra
• formulas of up to 3 to 6 nestings
• more than four formulas in one variable are distinguished
• no open set larger than the implication satisfies the condition
• some axiom holds in one algebra and fails in another
• the algebra has at least two elements
• the count never falls as the formulas grow
• the formulas are among em, wem, dnn, gd, kp, triple
• the implication really does meet the antecedent inside the consequent
• the implication really is the largest element whose meet lies below
• the negation of an open set is open
• the orders are among point, chain2, chain3, anti2, v, lambda, diamond
• 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 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 union has as many pieces as the two sets between them
• the view is one the family draws
• three negations are the same as one
• three negations collapse in every Heyting algebra
• three negations collapse to one

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

• column 2 ×59
• column 2 is empty exactly when 2 is prime ×59
• slice a = 1: a solution exists exactly when the eliminated formula says so ×2
• the formula for ∀x ∃y at n = 2 ×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
• 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
• almost every cell is checked against the formula
• and 512 − 343 = 169 have a full row
• and for ∃x ∀y
• at 16 points almost every relation has a mark in every row
• every x makes it positive exactly when a² − 4b < 0
• no drawn row agrees with the built row everywhere it has been checked
• no row of the table is the Russell row
• 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 built row differs from row n in place n
• the diagram has six arrows
• the drawn window is at least as wide as it is tall
• the labelling is one this figure knows
• the largest number is a whole number between 20 and 80
• the number of columns is a whole number between 2 and 14
• the number of drawn columns is a whole number between 3 and 16
• the number of drawn rows is a whole number between 3 and 12
• the number of rows is a whole number between 2 and 14
• the number of sets in the table is a whole number between 3 and 10
• the relation is one this figure knows how to draw
• the Russell row disagrees with row k at column k
• the search agrees with 256b³ ≤ 27a⁴
• the share with every row marked eventually only rises

• 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 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
• 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 implies B on every assignment
• A implies the interpolant
• a negation is introduced from a derivation and its denial
• a variable shares a component with its negation exactly when no assignment works
• 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 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
• between two and seven stages
• both cases have the same type
• cases are taken on a disjunction
• 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
• every drawn step is the resolvent of the two clauses above it
• every formula in the cut-free proof is a subformula of the goal, or its negation
• every formula of the derivation with no detour is a subformula of the goal
• 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 opening bracket in the formula is closed
• every order of the six variables
• it closes exactly when the universal has been used 3 times
• modus ponens takes two derivations
• no arrow leads from a true literal to a false one
• 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 variable shares a component with its negation
• only a disjunction is taken apart by cases
• only a pair is projected
• only a term of implication type is applied
• 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
• so the conclusion is its consequent
• so the count of unsatisfiable sets and the count of refuted sets are one number
• 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 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 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 expansion does close, at some stage this figure reaches
• the first derivation gives a disjunction
• the first is an implication
• the formula and its disjunctive normal form agree on every row
• the formula has between one and five variables
• the formula is a non-empty string
• the interpolant uses only atoms the two formulas share
• the last term contains no detour
• the lemma is itself a theorem, so it may be cut in
• the most holes is a whole number between 3 and 6
• the path passes through one node per level
• 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 search only runs on unsatisfiable sets
• 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 strongest interpolant implies the one read off the refutation
• 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'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 term is closed, because every assumption was discharged
• there are twelve two-literal clauses on three variables
• this view draws a refutation, so the set has to be unsatisfiable
• two to four numbers of variables from 10 to 2000
• which derived a conjunction
• which implies the weakest
• with few clauses nearly every random set can be satisfied
• with half as many clauses again as variables, the largest sets almost never can

• every corner meets one edge per variable, each edge counted once
• every leaf of the formula is a variable
• every opening bracket in the formula is closed
• every set of more than half the corners has a corner of degree at least √n
• majority of p, q, r is a weighted vote
• no function has degree above its sensitivity squared
• on majority the rule stops making mistakes
• on parity it never does
• p, or both q and r is a weighted vote
• parity is sensitive to every letter everywhere
• parity: an odd number of p, q, r is not a weighted vote
• the counts are 4, 14, 104 and 1,882
• the cube is drawn for two, three or four variables
• the formula and its disjunctive normal form agree on every row
• the formula has between one and five variables
• the formula is a non-empty string
• the function is one this figure knows
• the highlighted rows are the true ones, the false ones, or neither
• the matrix squared is n times the identity
• the nonzero entries are the cube's edges
• the search is over the 3-cube or the 4-cube
• the signed matrix is drawn for one to four letters
• the table has one row per assignment
• the two ends of an edge differ in exactly one variable
• the view is one of cube, threshold, thresholdcount, perceptron, sensitivity, huang, signed, degree
• the weights and the cut give the function
• the whole formula is consumed by the parser

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

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

• 2cos(2π/11) is a root of it ×3
• the circle about P1 has a centre and a radius ×2
• the intersection named P1 is the one meant ×2
• the objects of step P1 meet ×2
• a collapsing compass cannot carry a length from one place to another
• a rational is a whole numerator over a non-zero whole denominator
• a right angle can be trisected
• a root found by the theorem really is a root of the cubic
• a straight angle can be trisected
• a zero angle can be trisected
• AL equals BC, as Euclid's second proposition promises
• and above it
• and at least one is out of reach of both
• and exactly half way from the other end too
• and is not half way along after it
• and it fails when the given point is not the midpoint, so the midpoint is what is used
• and it is a different line from the segment
• and it is one radius beyond B
• and it lies on the segment
• and it sits over the midpoint
• and on the segment
• and that crossing and the marked point are both on the circle
• and the diameter really is twice the radius
• and their images are not equally spaced
• at least two polygons separate the two instrument sets
• between three and eight polygons, each with 3 to 24 sides
• both crossings exist
• CB′ equals AB: reflecting in the line XY carries A to C and B to B′
• 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
• divisors are taken of a positive whole number
• dropping the ruler makes the midpoint dearer
• 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 coefficient of the minimal polynomial is a whole number
• every image lands on the second line
• 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
• nineteen of the polygons up to sixty can be drawn
• one round puts two points off the line
• six steps of the radius come back exactly to the start
• sixty degrees cannot be trisected
• some angle in the table the two instruments cannot cut, and the marked one can
• the angle comes from a polygon that can be drawn
• the angle is between 12 and 174 degrees
• the angle the placed straightedge makes is a third of the original
• the apex is near the segment
• the apex sees the diameter at a right angle
• the centre is equidistant from the ends of the diameter
• the circle 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 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 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 fixed opening is fixed
• the free point sits inside the segment PM
• 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 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 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 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 totient of n above two is even
• the two arcs about D and E meet
• the two arcs of the fixed opening meet
• the two crossings sit at the same height, so the line is parallel
• the view is one of root, trisect, polygons, compass, hexagon, straightedge, neusis, reach, parallel, poncelet, rusty, conics, hendecagon, lemoine, lemoinetable
• the walk was built
• three steps of the radius reach the point twice as far away
• three steps reach the far side, which is the doubling
• twenty-four of the polygons up to a hundred can be drawn
• two points and one round of drawing give four more
• what is left is short enough for the fixed opening to span
• which makes it that piece's midpoint

• step 1 changes exactly one place ×7
• the dimension of the cube is a whole number between 2 and 4 ×3
• a code is more than one word and fewer than all of them
• 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
• 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 visits none of them twice
• counting up in the ordinary way changes more places than that
• each cycle was met once in each direction
• each step changes one place
• each step, the last back to the first included, swaps one element out and one in
• every codeword has the length the cube has dimensions
• every corner meets one edge per coordinate
• every place changes an even number of times
• every step of the code changes exactly one place
• every subset appears
• every walk is enumerated for at most the 4-cube
• every word within 1 of a codeword decoded back to it
• the 3-cube carries 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 code is a word list or one of repetition, parity, none, even2
• the cycle visits every word of the two levels once
• the drawn walks are the first of the ones counted
• 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 number of places is a whole number between 3 and 11
• the radius of the balls drawn is a whole number between 0 and 2
• the search found a cycle through the middle two levels
• 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 tour is one of reflected, balanced
• the tour visits every corner once
• the view is one the family draws
• the walk visits all 8 corners

• 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 × ∛2 lands on a multiple of ∛2²
• ∛2 × ∛2² lands on a multiple of 1
• ∛2 × 1 lands on a multiple of ∛2
• ∛2² × ∛2 lands on a multiple of 1
• ∛2² × ∛2² lands on a multiple of ∛2
• ∛2² × 1 lands on a multiple of ∛2²
• 1 × ∛2 lands on a multiple of ∛2
• 1 × ∛2² lands on a multiple of ∛2²
• 1 × 1 lands on a multiple of 1
• a bigger subgroup names a smaller field, never the other way round
• a rational is a whole numerator over a non-zero whole denominator
• a subgroup of order two fixes exactly one root
• and grow without bound
• and the product's at √2 · φ
• and the sign alternates
• both are monic
• divisors are taken of a positive whole number
• each ±√2 + ω^j∛3 is a root of the polynomial
• 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 trace step divides exactly, as it must for an integer matrix
• every coefficient is a whole number under 400
• every degree divides the degree of the field, six
• every product of basis elements is a multiple of a basis element
• every rational root found by sweeping is one the theorem listed
• no integer polynomial in the searched range vanishes at π
• no power of two is a multiple of three
• no whole number the rational-root theorem allows is a root of x³ − m
• 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 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 control polynomial really vanishes at its algebraic number
• the cube root really cubes to it
• 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 earlier powers are independent, so the solve is unique
• the exact value and the decimal one agree
• 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 leading coefficient is not zero
• 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 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 same search does find a polynomial for an algebraic number
• the searched number is a root of one of them
• the size of the subgroup times the degree of its field is the degree of the whole
• the tower has one basis element per subset of its roots
• the tower is built from one, two or three square roots
• the view is one the family draws
• 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
• 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
• 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 rectangle: and vertically
• a rectangle: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a rectangle: the outward normals weighted by edge length cancel horizontally
• a regular hexagon: and vertically
• a regular hexagon: it has a centre of symmetry, so its edges pair off and the quantity vanishes
• a regular hexagon: the outward normals weighted by edge length cancel horizontally
• a trapezium: and vertically
• a trapezium: the outward normals weighted by edge length cancel horizontally
• a triangle: and vertically
• a triangle: the outward normals weighted by edge length cancel horizontally
• a triangle's quantity does not vanish, so no number of cuts slides it into a rectangle
• 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 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 the arc from C
• and the ledger balances: 4 × 6 − 12 = 12
• and the other side at its midpoint
• and the same two tile the square
• 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
• 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 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
• every angle of each piece is a rational part of a half turn, so its Dehn invariant is nought
• 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 labelled point is on the side of the sphere facing the reader
• M is the midpoint of AB
• N is the midpoint of AC
• no edge has zero length
• no multiple of the tetrahedron's angle is a whole number of half turns
• no numerator in the sequence is a multiple of three
• P is the midpoint of A′′B
• P is the midpoint of A′B
• P is the midpoint of AB
• so does the arc from B
• 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 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 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 fan of triangles tiles the polygon
• the four corners and the octahedron fill the large tetrahedron's volume
• 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 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 octahedron has edge 1
• the octahedron has the volume of four small tetrahedra
• the octahedron's is −12 ⊗ θ
• 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 same three pieces tile the rectangle
• 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 solid has Euler characteristic two, so its faces close up
• the square's side is divisible by n(n+1) so every corner lands on a whole number
• the three pieces tile the triangle
• the triangle 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' areas add up to the polygon's
• the two pieces tile the rectangle
• the view is one of steps, polygon, dehn, hinge, translate, count, sydler, orthoscheme, tetocta, saccheri, lambert, lexell
• 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

• 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
• 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
• 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
• every 5 residues mod 3 contain 3 with a sum divisible by 3 ×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 conic y² = x² + b misses p by exactly one
• a ring of composite size has a pair of non-zero elements multiplying to zero
• addition and multiplication are commutative
• an irreducible polynomial of the right degree was found
• and it is exactly the composite ones that leave an element without a reciprocal
• and no curve y² = x⁵ + ax + b by more than 4√p
• and so is the mean fourth power, an eighth
• and that variable's power sums to 0 over GF(3)
• and the fewest is the smallest
• between 3 and 8 variables
• each walk ends within 2√p of zero
• every monomial has a variable whose power is not a positive multiple of p − 1
• every non-zero element has a reciprocal
• multiplication distributes over addition
• multiplication is associative
• no two non-zero elements multiply to zero
• odd primes up to 11
• one to three prime-power fields from 3 to 16
• so the number of zeros is 0 mod 3
• so, since (0, 0, 0) is one, there is another
• some row is divisible by exactly the higher power Ax and Katz promise, and no more
• the drawn points and the count agree
• 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 largest power summed is a whole number between 8 and 20
• the mean square of the normalised error is the semicircle's, a quarter
• 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 powers of a primitive element close back onto one
• the size of the ring drawn beside it is a whole number between 0 and 12
• 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-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
• x² + y² over GF(7) has only the zero at the origin, since −1 is not a square

• class 000 has as many words as the code has ×8
• class 000 has exactly one lightest word ×8
• a row called exact really divides to a power of two
• above capacity it rises
• above it they take over
• 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 span is a distance between three and five and a longest length up to eight
• every codeword passes all three parity checks
• every pair of erasures is recoverable, since the distance is 3
• every rate is between zero and one
• every two words of the code found are at least the distance apart
• every word in the space is repaired to a codeword by its own syndrome
• four data bits give sixteen codewords
• more repeats, fewer errors
• most of the probability lies within two spreads of np
• nor Plotkin's, where it applies
• nor the Singleton bound
• one rate below 1 − ε and one above
• some length meets the sphere-packing bound exactly
• 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 exact answer never beats the sphere-packing bound
• the flip probability is between 0 and a half
• the Hamming code's rate is above capacity at p = 0.1
• the highlighted codeword is a whole number between -1 and 15
• the lightest non-zero codeword weighs what the closest pair are apart
• the longest word length in the table is a whole number between 5 and 23
• the longest word length searched is a whole number between 5 and 8
• the 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 rate at the longest length beats the rate at the shortest, at every distance drawn
• the received word is seven bits
• the repetition code and the Hamming code both meet the bound exactly
• the repetition code meets the bound at n = d
• the search finished rather than running out of budget
• the seven columns of the parity-check matrix are distinct
• the seven single errors give seven different syndromes
• the sum settles on the constant
• the syndrome of a single error is the column of the position it hit
• 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
• 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
• 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
• 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
• 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 projective plane over GF(q) has q² + q + 1 points
• 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 therefore there are at least as many blocks as points
• any two lines meet in exactly one point
• 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
• enough configurations were drawn
• every block's size times the block count is the point count times r
• every line carries q + 1 points
• every member of the base block is a residue modulo n
• every other point off the oval lies on exactly one
• every oval of PG(2, 5) lies on a conic
• every pair of points appears
• every point lies on q + 1 lines
• every tangent passes through the horizontal direction
• 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
• no point off the oval lies on exactly one tangent
• 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
• 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 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 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 forced corners are new points off the axis
• 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 line at infinity is the last line built
• 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 order is 3, 4, 5 or 7
• the order is 3, 4, 5, 7 or 8
• 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 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 points are labelled or not
• the residue rule and the divisibility rule agree at every size
• 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
• 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
• the count at order 1 is a positive number ×8
• a symbol was found for column 1 ×6
• and once in each column of the square for 1 ×6
• column 1 repeats no symbol ×6
• column 1 was given a symbol it could take ×6
• every symbol appears once in each row of the square for 1 ×6
• relabelling left the pair 1, 2 orthogonal ×6
• symbol 0 is missing from exactly n−k of the columns ×6
• a field of order 4 has 3 non-zero multipliers ×5
• the cyclic square of order 3 has 3 transversals ×5
• the relabelled square for 1 starts 0…3 along its first row ×5
• there is a field with 4 elements only if 4 is a prime power ×5
• row 1 uses every symbol once ×4
• every point lies on 4 lines, one from each class ×3
• the number of rows already placed is a whole number between 1 and 4 ×3
• the 3 available symbols are all used, so an extra square would repeat one ×2
• the order enumerated in full is a whole number between 1 and 4 ×2
• the order of the squares is a whole number between 3 and 8 ×2
• a square laid over itself produces only its own diagonal of pairs
• 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 9,408 of order 6
• and every table whose elements are all self-inverse has one
• and fifty-six of order five
• and four have every element its own inverse
• and 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 α
• and once in each column of the square for α+1
• and sixteen of them associate
• and three of them associate
• and twelve have an element of order four
• 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 square either associates or does not
• every line holds n of the points
• every non-zero element has a reciprocal
• every ordered pair of symbols occurs exactly once
• every pair of points is met by some line
• every set of columns can still take at least as many symbols as it has columns
• every 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 α
• every symbol appears once in each row of the square for α+1
• four of the sixteen have every element its own inverse
• multiplication distributes over addition
• multiplication is associative
• no table with an element of order four has a transversal
• no two non-zero elements multiply to zero
• no two pieces share a wrapped diagonal
• no two standardised squares carry the same entry under the corner
• one shift per row already placed
• 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
• 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
• sixteen tables of order four associate
• some square of order 6 has none
• the cyclic square has transversals exactly at odd orders
• the cyclic square of even order 6 has no transversal
• the drawn order agrees with the table
• the field is closed
• the four self-inverse tables have the same number of transversals
• the largest order enumerated is a whole number between 3 and 6
• the Latin square view is one of transversal, mols, bound, net, count, extend, quasi, mates, 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 quasigroup view is one of counts, tables
• the quoted orders are orders between 7 and 11 and larger than the ones enumerated
• the radius of the points is a whole number between 3 and 8
• the relabelled square for α starts 0…3 along its first row
• the relabelled square for α+1 starts 0…3 along its first row
• the second square uses a multiplier that makes it a different square
• the size of the field is a whole number between 2 and 16
• the size of the permuted set is a whole number between 1 and 8
• the squares for 1 and α produce every ordered pair once
• the squares for 1 and α+1 produce every ordered pair once
• the squares for α and α+1 produce every ordered pair once
• the total at order four agrees with every square of order four, listed
• there are 56 reduced Latin squares of order 5
• there are 576 Latin squares of order four
• there are four reduced squares of order four
• there are twelve Latin squares of order three
• twelve of them have an element of order four
• two associative squares were found to draw
• 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

• 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
• all 4 windows of 2 bits occur somewhere in the period ×7
• the recovered rule matches output 0, which was given ×4
• the modulus is a whole number between 16 and 65536 ×3
• the register of 4 bits visits every nonzero state before repeating ×3
• a coin that can land either way
• a multiplier consistent with the outputs exists
• a multiplier is a whole number between 2 and 1048576
• a short vector the modulus annihilates exists
• 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 so is the second
• and the all-zero window never appears
• and they occur equally often, but for the one the all-zero state would have supplied
• and visits none of them twice
• applying the filter twice returns the word
• each extra level adds, and none passes the entropy
• each of them exactly once
• each window is between one bit and the register's width
• every generator's points fall on a genuine family of parallel lines
• every point of the sequence satisfies the relation exactly
• how many high bits are predicted is a whole number between 1 and 8
• no filter passes the ceiling
• no more rows than state bits
• no short whole-number relation holds between consecutive outputs
• 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
• 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 all-zero state is the one the register cannot leave
• the all-zero window is the one that is short, by exactly one
• the criterion and the measured period agree
• the dimension is a whole number between 2 and 3
• the example word is a whole number between 0 and 255
• the filter sends the 256 words to 256 different words
• the first prime is a whole number between 3 and 4000
• the first prime is three modulo four, which the construction requires
• the flips drawn is a whole number between 8 and 48
• the highest tap is the register's own width, or it is a shorter register
• the increment is a whole number between 0 and 1048576
• the linear solve, run on this generator, predicts the next output wrongly
• the modulus is small enough that the arithmetic stays exact
• the multiplier is a whole number
• the multiplier is a whole number between 2 and 1048576
• the multipliers really do differ, by more than a factor of two in spacing
• the number of outputs it then predicts is a whole number between 2 and 16
• the number of outputs the solver is given is a whole number between 3 and 8
• the number of points is a whole number between 16 and 4096
• the number of predictions is a whole number between 50 and 4000
• the number of triples is a whole number between 100 and 4000
• the occupied hyperplanes are consecutive, so the family has no gaps
• the output 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 predictor gets essentially every output right
• the register does not repeat before its full period
• the register visits every non-zero state before repeating
• the register width is a whole number between 5 and 14
• the relation involves the middle coordinate, so an edge-on view exists
• the relation involves the second coordinate
• the relation's normal lies in the screen, so the planes are seen edge on
• the second prime is a whole number between 3 and 4000
• the solver is given only a handful of outputs
• the taps are positions inside the register
• the triples lie on a small number of planes
• the twisted generator runs through every non-zero state before repeating
• 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
• the width of the register is a whole number between 3 and 6
• three to eight multiplier-and-increment pairs
• two multipliers below the modulus
• two taps
• which is enormously better than guessing the leading bits
• while the independent coin's output shows no correlation

• 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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