Invariant — where it appears
Named by 87 essays across 9 fields — each of them below, with the objects they name alongside it.
The theorem that needed none of the theory
Bouton solved Nim completely in 1901, with an argument that mentions no value, no sum of games and no Grundy number, because none of the three existed. The argument is two closure properties and it is airtight — and run on any other game it fails at the step that does the work.
Two people, four years apart, one theorem
Roland Sprague proved it in 1935 and Patrick Michael Grundy proved it in 1939, neither knowing of the other. That looks like coincidence until the alternatives are examined — and the rule they both reached turns out to be the only one that can work at all.
"Left wins" has no short proof
A complete solution of Nim on heaps of 7, 11 and 13 is 480 table entries. A winning strategy for the same position — one move of the winner's at each of their turns, and an answer to every reply — has 56,167,022 nodes in it. The answer is smaller than the proof by a factor of a hundred thousand.
The strategy that is a symmetry
A pairing strategy is a symmetry of the board that turns one player's moves into the other's, and it wins without computing anything. Tested by playing it out rather than argued, it wins one of seven candidate symmetries across four games — exactly the Cram boards with both sides even, which is exactly where no domino is its own image.
When the regions add
The rung below described the NoGo boards whose regions add as the ones with symmetric walls, and said the description was a guess made from six examples. It is wrong: fourteen symmetric boards do not add and sixteen that add are not symmetric. What replaces it is a criterion about liberties — sound on all 117 boards, provable in a line, and complete on only nine of the twenty-four.
The reading that survives too much
Counting the empty squares in front of each coin gets a Push position right half the time, and the rung below said the failures were exactly the positions with two coins of opposite colour side by side. Sixty-six of the 1,072 failures have no such pair, the smallest is five squares long, and the condition that does decide it is not about the board at all — it is about every position the board can reach.
Where the nimbers run out
A single End-Nim heap is a Nim heap and every palindromic row is worth a nimber, so the impartial theory looks as though it might get a long way into a partizan game. It gets one row in thirteen. Five nimbers occur in five and a half thousand rows, the palindromes account for two fifths of them, and the rows worth something else run to 2,693 distinct values.
A compound of two different games
Every rule the compound theory has survives mixing exactly — the minimum-remoteness rule is right on all 5,184 mixed pairs and all 7,560 triples — and the reason is not that the rules are strong. It is that each of them reads one number per component, and a number does not remember which ruleset produced it. The thing mixing damages is the shortcut a reader carries instead.
The thirty that cancel themselves
Thirty values born by day three are equal to their own negatives, and every one of them has a mean of exactly nought and two stops that are exact opposites. Neither property comes close to picking them out — 496 values of the day have a mean of nought — and half of the thirty are hot, one of them the hottest value the day produces.
Looking for the symmetry
Answering every move with its mirror image wins Cram on a board with both sides even, which is the argument everybody meets. Asked of every connected shape of at most eight squares instead of of thirteen rectangles, it wins twelve — and accounts for a sixth of the second-player wins there are, because 852 of the 1,042 shapes have no symmetry to answer with in the first place.
The patch that generalised
Misère Nim takes a one-line patch: play the normal-play strategy until every heap holds a single counter, then invert. Moore's Nim, where a move may take from up to k heaps at once, takes exactly the same patch with exactly the same modulus — and the two rules disagree on six positions out of 923.
What restores the theorem
Fibonacci Nim breaks the recipe every impartial game is supposed to obey: one number per heap, exclusive-ored, gets a quarter of two-heap sums wrong. Index the recursion on the pair of heap size and cap instead and the recipe is exact on every pair and every triple — and the number a heap of nine carries turns out to be five rather than one.
The rule the symbols follow
Two genus symbols make a third by three lines and no lookup table: the base exclusive-ors, the sum is fickle only when every component is, and the symbol follows. Checked on 252 pairs across nine games it is right on 238 — and the fourteen failures are exactly the fourteen pairs with a wild heap in them, which is the boundary the genus is defined up to arriving as a measurement.
The rows that are their own mirror
Four hundred and ten End-Nim rows are worth nimbers and 168 of them are palindromes, so a condition covering the other 242 was outstanding. It is that the row is equal to its own negative — and on this game that condition is not merely sufficient but exact, which is more than the group law promises and is a fact about End-Nim rather than about games.
The family the Fibonacci numbers belong to
Fibonacci Nim lets a player take at most twice what the last one took, and the heaps the opener loses are the Fibonacci numbers. Two is an arbitrary number. At one the losing heaps are the powers of two, at three and four and five they are four more sequences, each with a linear recurrence whose lag is twice one less than the factor — until the factor is six, where the pattern stops.
What the numerals knew
Every factor in the Fibonacci Nim family gives a numeral system, and the rung below predicted its separation condition would be the lag of the recurrence the losing heaps satisfy. It is not. The gap is the factor — one at c = 1, Zeckendorf's two at c = 2, and c at every factor to eight — while the lag goes 1, 2, 4, 6, 8, 11, 14, 17 and leaves its own pattern at six. The numerals then solve every one of 194,480 states, cap and all.
The rule a smaller move breaks
Moore's Nim lets a player take from at most k heaps, and its winning condition is the binary columns summed modulo k + 1. Cap the amount as well and the obvious repair — reduce each heap modulo the cap plus one, then read the columns — is exact at every cap when k is one and wrong at every cap when k is two or three. The reason is stronger than a broken rule: at k ≥ 2 the residues do not determine the outcome at all, so nothing of that shape can work.
A game with nothing at stake
The rung below explained a coincidence with a claim it did not compute: that End-Nim carries no hot self-negative values. The census says something stronger. Not one of 10,919 rows across three shapes of board is hot, every one of them is worth an infinitesimal, and the 361 rows the earlier census called numbers are 361 rows worth nought. The reason is one line of the rule, and 2,693 distinct values sit underneath it.
The condition that survived the wider sweep
Which pairs of subtraction lists have a value sequence that repeats? Over the 49 pairs drawn from one, two and three, two conditions answer it identically — a translation and all-odd — and both are exactly right. Over the 961 pairs drawn from one to five, all 83 translations still repeat with no exception and four all-odd pairs do not, at heap ninety with a period as long as forty-two. Neither condition is necessary: 104 pairs repeat that satisfy neither.
The criterion that cannot exist
The rung below asked for a quantitative version of its condition — turn 'the reading survives mixing three quarters of the time' into a statement about the strip. Three strips of four squares settle it. `.LLR`, `.LRL` and `.RLL` have the same length, the same reading, the same coins and the same single run, and their readings are wrong by 1¼, ¼ and ½. The error is a fact about the order of the colours, and 207 of 805 statistical classes carry more than one of them.
The birthday is a floor
The rung below measured a correlation of 0.73 between a value's birthday and the size of its cheapest exhibit, and asked which values are dearer than the birthday suggests. The relation is not a trend. Over all 728 non-number values the exhibit is never smaller than the birthday and is exactly the birthday on 476 of them, and the excess on the other 252 belongs to the game rather than to the value — the ruleset accounts for 40 per cent of its variance.
The bend is the condition
The rung below offered a description of the class its stop reading is exact on — neither wall bends below the meeting point — and a route to proving it: that a bend happens precisely when some option is neither a number nor an infinitesimal. The first is exact on all 138 values, both directions, no exception. The second is half right: every bent value has such an option and 49 unbent ones do too. And the eight apparent exceptions to the first turn out to be a bookkeeping convention.
How big the answer is
The rung below found every early departure from a coupon stack caused by a position with a follow-up, and could not say more: its follow-ups were all of a similar size, so the class it measured was one bit. A pool graded by follow-up size answers it. With the position's own temperature held at one, the departure runs from coupon 1 to coupon 3.5 as the follow-up's temperature runs from 1 to 4 — and over the whole grid the players leave at the larger of the two temperatures.
A self-negative value costs a day
The rung below placed the thirty values equal to their own negatives on the temperature scale and asked whether being self-negative forces anything about when a value can be born. It does, exactly: the earliest self-negative value of temperature t is born the day after t itself, which accounts for the four temperatures that carry one and the four that carry none. The guess it offered — that the first value of each temperature is a self-negative one — holds at four temperatures out of five and is not the shape of the answer.
The closure that picks the nimbers
Closure under addition lets a sum be rewritten and turned out to admit companies that are not nimbers at all. Closure under forming options lets a subposition be rewritten, and it pulls the other way: every company this site computes in has it and none has the first, and among the seven finite addition-closed companies, keeping every option is exactly being a group of nimbers — four of seven, both directions, no exception. Demand both at once and nineteen of twenty-two day-two values generate nothing finite.
What a fight does to a fight
A number added to a position shifts both its stops by itself; an infinitesimal moves neither. A hot game does neither: over 720 sums the two stops add on 330 and are wrong on the rest. What survives is the mean, which adds on every one of the 720 — and the failure has a bound, since no stop is ever out by more than twice the smaller of the two temperatures, a bound 222 of the sums attain exactly.
No fifth value
The colon reads a form rather than a value, and the rung below found the forms of a value disagreeing at exactly four of them — the values born by day one. It could only check forms whose options came from day two. Built one day deeper, by adding day-three gift horses to day-three values, eighteen thousand forms give no disagreement at all, while the same treatment still splits nought four ways. The class is about the width of the base's form and not the depth of its options.
What a game actually produces
Fifty-three per cent of the Domineering regions of at most eight squares are hot. Of the components eleven hundred random games actually produce, sixteen per cent are — and ten per cent once single squares are counted. The figure is the same on three sizes of board, so it is a property of play rather than of the board, and it says that every temperature census this site has taken over a catalogue overstates how hot the game is by a factor of three.
A subtraction, not a factor
The crossover factor was a half on fights whose answer starts another fight, measured on a pool with two three-deep positions in it. A pool built to be deep gives twenty, and the factor does not survive them: the crossover is the follow-up's temperature less a half on eighteen of the twenty, and a factor of a half agrees with that only where the temperature is one — which nearly every position in the earlier pool had.
What the class does not buy
Dead-ending is the hypothesis several modern misère results are stated under, and the rung below sorted this site's games into it without running the comparison those results are about. Running it: a quarter to a half of ordered pairs compare inside a ruleset's own universe, which is a great deal — and the ruleset that is not dead-ending sits in the middle of that range. Ten comparisons are lost when a universe is enlarged, and every one is lost to a dead-ending company.
Half a follow-up out
The rung below settled which values the stop reading is wrong about — the ones whose walls bend — and left the size of the error unmeasured. It is not bounded by anything readable off the diagram; it equals something readable off the diagram. On all thirty-two, the mean and the temperature are each out by exactly half the follow-up's temperature, and the temperature is always read too low.
The wider move is the easier game
An earlier essay ruled out every rule that reduces the heaps and reads the residues, and asked for a two-part statistic: the residues plus one more count. Four second parts are tested here and none of them decides. What turns up instead contradicts the premise the request was made under — a move that may reach three heaps is more predictable than one that may reach two, on every cap, every candidate rule, and after the change in the base rate is taken out.
Half of the smaller temperature
The correction to the sente crossover has been priced twice — first as a factor of a half, then as a subtraction of a half — each time on a pool whose answers were all about the same size. Over 128 fights with answers from a number up to a temperature of six, the correction is half the answer's temperature, saturating at half the fight's own. Both earlier readings are regions of that one law.
A bound with one number too many
The rung below bounded how far a hot addend can drag a stop — twice the smaller of the two temperatures — over sums whose addends were all plain switches, and conjectured that an addend with a follow-up would need twice the smaller of three numbers. Over 1,440 sums with bent addends the two-number bound holds everywhere and is attained 358 times, and the three-number version fails on 66.
The third digit
The rung below found 71 of the 255 two-digit hexadecimal codes repeating with a constant added rather than exactly, and asked whether the same share holds one digit wider. It rises. Of the 4,095 three-digit codes, 1,433 climb and 617 repeat exactly — seven in ten of the settled ones — so a saltus is the ordinary way a hexadecimal game settles and the exact repetition the octal survey was built to find is the special case.
At least five hundred and seventy-one
The rung below dated the self-negative values — the earliest of temperature t is born the day after t — and left the count to a day-four census nobody can run. The construction settles it instead: a value is its own negative exactly when its form is a mirror, so the subgroup can be built from subsets of the day below rather than sifted out of the day above. Day four supplies at least 571 against day three's 26, and the share of a day that is self-negative keeps falling.
The size of a cake
Ω gives the sign of a Maundy Cake and says nothing about the size, and the rung below left four values — 7, 10, 13 and 16 — unaccounted for. For a one-row cake they are a formula: write the prime factors largest first and add up their running products. The rule behind it is greedy — cut by the largest prime — and it is exact on every one-row cake to two hundred and wrong on a fifth of the two-sided ones.
Add, then reduce again
The homomorphism promises that a sum's reduced form can be computed from its parts', and says nothing about what the operation is. It is addition followed by a second reduction — needed on 431 of 3,600 pairs of day-three values, and on not one of the 1,751 pairs with a cold part. What the second pass removes is an option that only becomes dominated once the two fights are side by side.
A function with no formula
The rung below's composition rule is exact on tame pairs and wrong on all fourteen wild ones, which looked like an exact boundary. Two heaps further it is wrong on 34 of 35 and right on one — Kayles' five and nine — so the boundary was a boundary of the pool. What survives is stronger and stranger: the pair of symbols still determines the sum on the wild side, and no rule of that shape describes it.
A numeral in the empty squares
The rung below ruled out a quantitative criterion for Push and asked for a numeral over the coins combined with a count over the gaps. The two ingredients are the right way round: the colours pick a fraction — −1, −1/3, −1/7, −1/15 — and the empty squares give the binary precision, so a run of k coins before one of the other colour with g gaps is worth exactly (1 − 2^(−kg)) ÷ (2^k − 1). And it does not compose: a strip of two runs is not the sum of them, on any pair tried.
A region one player owns
On a strip, a region containing only one player's amazons is worth exactly its free-square count, on all 45,057 positions of the rung below's sweep — and it predicted the exactness would fail in two dimensions, where an amazon can be short of room in one direction and not another. It does not fail. Over 2,412 two-dimensional regions there is no exception, and the reason is one clause: an amazon may shoot back at the square it has just left.
Half the difference in odd runs
The rung below asked what the regions the packing reading fails on have in common, and whether it is something a player could see. It is: the reading itself. The count has a closed form — half the difference between the region's odd horizontal runs and its odd vertical runs — and it is exact seven times in ten when it claims one move of advantage, on none of the largest regions where it claims two, and it exaggerates four times in five when it is wrong at all.
The cliff a cut invents
The rung below asked for a correction term in the gap between two Push runs. There is none, because the gap's contribution vanishes: widen it and the strip's value converges geometrically, at a rate set by the back run's length alone, to a limit that is not the sum. And Shove — whose reading is exact everywhere — fails at the same cut, which says the broken thing is the cut and not the game.
Cut small unless you are behind
The rung below found the greedy rule — cut at the largest prime — wrong on 104 of 552 Maundy Cakes and asked for a description of them. On all 104 the best cut is at the smallest prime, the exact opposite. A middle divisor is never needed on any cake in a sixty by sixty grid, and which of the two extremes wins is decided by Ω alone: cut small when Ω(m) + 1 ≥ Ω(n), large otherwise, and that is exact on all 3,540.
The fractions that were not there
The rung below counted 1,452 fractions among the shared Amazons regions and asked which fractions they are. Fifty-six of them are fractions. The other 1,396 are hot positions with a fraction somewhere inside their options, counted by a regular expression looking for a slash — and the quantity the separation of the two amazons actually sets is not a denominator but a temperature.
The obstacle was the catalogue
The rung below could not measure the early game because its regions were too large for the catalogue, and asked for a bracket rather than a value. No bracket is needed: a twelve-square region evaluates in five milliseconds and an eighteen-square one in under a second. What was expensive was cataloguing every shape rather than sweeping the positions a board actually reaches — and the sweep says a board is hot four times in five three moves in, and cools when it breaks up.
Which end a sum lands at
The rung below found the errors in a translated stop clustered at the two ends of the range its bound allows — 660 at nought and 358 exactly on the bound — and asked for a rule saying which end a given pair lands at. There is one, in four lines, exact on all 1,440 sums. Three of the four cases are decided by the value being translated alone, and the property that decides them is the bend the switches ladder found for a different question entirely.
The option nothing names
The rung below found the arithmetic on reduced forms to be add and reduce again, needing the second pass on 431 of its sums, and asked whether the option that pass deletes can be named from the parts. Eight rules were scored and the best reaches four in five — and on a pool closed under negation it falls to under half, which says the near-miss is a property of the population. What the second pass does have is a shape and a cheap test that rules it out.
What identifies two subsets
Every subset of a day gives a self-negative value by mirroring it, and 1,793 subsets of day two give thirty values. The collapse happens in two stages with different characters: domination takes the 1,793 to 96 antichains and is a theorem, and the rest of the reduction takes 96 to 30 and is concentrated almost entirely on two values — nought, which has an exact description, and star, which has none.
The count of odd heaps
The rung below refused a family of two-part rules for bounded Moore's Nim and asked what the 364 losing positions have in common as a set. They have an invariant, and it is a statistic of the whole position rather than of a heap: how many heaps hold an odd number. Every all-even position is lost, at every width of move, by a restoring strategy — and the count settles every position at one heap a move and at four, and a little over half at two.
The only way to split into three
Nineteen three-digit hexadecimal codes climb by three, and the rung below asked whether they share a form and what digits they have in common. The digits are exact: on eighteen of them the only way to split a heap into three is by taking exactly three counters, and taking three counters can do nothing else. The form is not shared — the eighteen carry four distinct sequences, and exactly one of the four counts in base three.
The two numbers at the top
The rung below found the crossover of a sente fight to be its temperature less half its answer's, and said a proof would settle the depth question with it. The depth question is settled without the proof, by construction: group the positions by their two top temperatures and the crossover is single-valued on every group, however far apart the third temperature is — and the formula survives a fourth level of fight, which the rung below never reached.
The check that was not a check
The rung below asked for a depth-conditioned solver and named the board to measure it on. Building it found two things first. The pairing check is unsound at interior positions — on a four by five Cram board it fires on 8,613 positions and 1,026 of them are losses — and the board it named has twenty-five squares, so the check can never fire there at all. Repaired, the check is right everywhere, and the policy that pays is the root alone.
The wild side does not close
The rung below asked for the wild composition table and for two things about it: whether the wild genus symbols form a small closed set, and whether that set is a misère quotient in disguise. Building the table needed a wider sweep — nine counters a heap gives a diagonal rather than a table — and both answers are no. Not one of the twelve entries is a symbol any wild heap carries, and two wild heaps added together are tame two thirds of the time.
A catalogue that knows what it will meet
The rung below priced a catalogue of regions by its reach and found the coverage saturating, and asked what a catalogue ordered by frequency would cost instead. Eight shapes answer half the components a played Domineering board produces; a catalogue by size needs fifteen for the same, and 1,042 for what 119 chosen by frequency reach. Three quarters of a size-ordered catalogue never turns up in play at all.
The short side only says how many
The rung below settled which cut to make in a Maundy Cake and left the value open. With the cut settled the recursion is a walk, the walk unrolls, and what it unrolls into is the running products of the long side's prime factors, largest first. The short side never enters the products at all — it decides how many of them there are and nothing else, so sixty-two different short sides give one value.
Read from the back forwards
The rung below found a two-run Push strip converging at a rate set by the back run and asked what a third run does — whether the rate is still the rearmost run's, or whether the rates compound. It is the rearmost run's, and for every gap: widen the front gap of a three-run strip, two whole runs away, and the value still dies at the last run's rate. Shove, the game one clause away, compounds.
One domino every three cells
The rung below gave the optimistic packing count as a formula in odd runs and asked for the other end of the interval, expecting a formula in the even ones. Parity is the wrong arithmetic: the smallest maximal packing is a sum of ⌈(len−1)/3⌉ over the runs, exact on all 1,042 shapes. That makes the whole interval readable off a drawing — and shows it can never reach the value, because regions with the same runs have different values.
Eight squares, and no hotter
The rung below found no Domineering position hotter than three halves on four boards and asked for the position that attains it. It is a region of eight squares, there are five of them up to symmetry, three are the hot core of an attaining board on every size swept — and the ceiling holds at nine and ten squares too, where the obvious extrapolation predicted seven quarters.
The parities, in size order
The rung below settled four of six parity classes in bounded Moore's Nim and asked whether the sizes pick out the losing positions in the two it could not. They do — but only through the order they put the parities in. Sort the heaps largest first, read off their parities, and that five-bit word settles the whole game at every width of move, with the losing words forming a subspace.
The short side is not in the lemma
The closed form for a two-sided Maundy Cake rested on one unproved statement: that no divisor beats the largest prime. Written out, that statement never mentions the short side — it is an inequality between a multiset of primes and a term count — and once it is stated that way it has a two-line proof, term by term. The ladder ends in a theorem rather than a grid.
Two strips that end the same way
If a Push strip's sensitivity is governed by its last run, then two strips agreeing at the far end should behave the same however different their fronts. On 78 of 80 tails they do, exactly. On two of them a single empty square in the prefix reaches across a gap that grows without bound and halves the rate — and the run reading turns out to be sound in one direction only.
A bend that never reaches the surface
How many levels of the recursion a thermograph needs before its stops suffice is a number attached to a position, and the rung below conjectured it was the depth of the deepest bend in the tree. It is not: on 124 values a bend one level down costs nothing at all. What the number counts is the longest unbroken chain of bends running down from the top, exact on 2,400 of 2,403.
The mirror was the floor
Toppling Dominoes' share of genuinely new values had fallen from one to a half over eight sizes, and the rung below could not tell a floor from a slow fall. Four more sizes settle it: the distance above a half halves every two sizes. And the half is not a shortage of values but a symmetry — a row played from the other end is the same game, and the ruleset is as injective as that allows.
A heuristic that becomes a theorem
The mobility rule's failure rate had been measured at two depths on each of five boards and found to fall. Swept at every depth it does not merely fall — it accelerates, and it reaches exactly nought before the endgame. From four to eight empty squares onwards the rule has no exceptions at all, which turns a rule of thumb into a guarantee for the last few moves.
Where the runs meet
A Domineering region's value interval is a function of its run lengths and its value is not — twenty-one groups of shapes share a multiset and disagree. The crossing count separates none of them, and neither do ten other local statistics, fifteen sets of which agree on everything and differ in value. What separates nineteen of the twenty-one is where along its runs each crossing sits.
An effect that changes sign
Which squares two Amazons share turns out to matter about as much as how many — three shared squares in a line run at 0.63 where three scattered run at 2.51. But the effect of clumping is hotter at one distance and colder at the next, so the arrangement predicts well and describes nothing, which is not what the four rungs below it produced.
The parameter was the difference
The losing words of bounded Moore's Nim form a linear subspace and no map was known whose kernel they are. The equations exist, four conditions cover all thirteen cases at three to six heaps, and they are indexed not by the heap count but by the heaps less the width of a move — which turns the failure at six heaps into a prediction about seven.
Where the value stops mattering
Fourteen straight-walled pairs missed the bound on a translated stop and had only a threshold to explain them. Their stops move by exactly the addend's temperature — a formula with the value nowhere in it — which turns the threshold into the boundary between two lines and closes a census of 1,440 pairs that has been open for four rungs.
A symmetry that is not a pairing
The quarter turn was the last symmetry a Cram pairing argument had not tried, and the one a square board seemed to offer. It fires on the empty four by four and it settles nothing the half turn misses — and the reason is a clause four rungs of this anchor never had to write down, because every map tried so far was its own inverse.
A pattern that has not started yet
A pre-period was supposed to be rarer in this family than a defect. Two hexadecimal codes in five have one, 321 have a pre-period longer than their own period, and the code the rung below found slow takes fifty-four heaps to settle rather than two blocks — which is also the account of three defects the rung below recorded and could not explain.
The licence that weighs nothing
The third substitution licence is constant in the number of components, exactly as predicted, and it saves under two times where the first saves seventy-six million. Priced by its table instead of by its saving it is the only one of the three whose cost does not run away — which reverses the order three rungs of this anchor have put them in.
Which top is the top
The crossover law's proof rests on the walls above the crossover being governed by the top two options, and the check was never run. Run on 23,586 heights it holds exactly — but only when the options are ranked by mean value. Ranked by the temperatures the law is stated in, it fails on a fifth of them.
The ceiling was a plateau
Three halves of a move looked like a ceiling on a Domineering region's temperature: it held at eight squares, at nine and at ten, and the rise that had been a quarter every two sizes stopped. At eleven squares four regions reach seven quarters — and they contain the hottest eight-square shapes and are hotter than them, so the extra material is not cold.
The family with two witnesses
Six predictions about seven heaps were written down and deliberately not run. Five of them held. The one that broke is the condition that had been checked against two cases when it was proposed — the fewest of the four — and at seven heaps it does not merely give the wrong answer, it asks a question the parity word has stopped being able to answer.
A fraction does not reach
Two Push tails read their prefix when every other tail ignores it, and the previous rung guessed the deciding bit was a shape — whether the prefix's last coin stands alone. The full census says it is a number. The rate changes exactly when the prefix is worth a whole move, and three quarters of a move is not enough.
Three distances too many
The junction descriptor records how far a crossing sits from four ends, and the rung below asked what the value does when one crossing slides along its run. It reads one bit — the offset's parity — and only when the run has odd length. The other three distances reach the value not at all.
Two clauses and a third question
A component can carry its own rule when two things hold: its moves are a function of what it carries, and a move in it leaves every other component alone. Two rulesets built to fail one clause each are both caught on a named witness. The four real games sort exactly — every one the recipe gets right fails no clause, every one it gets wrong fails one — and the two clauses still miss something, because Fibonacci Nim and a held pass fail the same clause and only one of them can be repaired.
The third point on the curve
Four rungs below this one measure a shared Amazons region's temperature against how far apart its two amazons are, and every one does it on a 3 × 3 board, where the distance can only be 1 or 2. Two points make a direction, not a curve. A 4 × 4 board reaches distance 3 — and cannot be evaluated at all until the regions are cut to five free squares. Restricted that far, the answer is neither a sign that flips nor an oscillation: the rise continues and it is running out, the second step being 36 per cent of the first.
Three complete solutions in nine years
Bouton in 1901, Wythoff in 1907, Moore in 1910 — three airtight solutions of three games, all published before there was any theory of games at all. Asked about each other's games they all fail, and two of them fail by being wrong while one fails by having no form for the question. Only the last kind of failure decides anything.
A set with a short description
Bouton's argument is a closure argument about a set, and every impartial game has such a set — its own losing positions. So the method is complete and proves nothing. What made 1901 a theorem is that his set had a description shorter than the game, and swept over fifty-six subtraction games, exactly seven have one of his kind.
The sentence that solved the other convention
Bouton's paper solves misère Nim too, in one line, and it is the only misère result in the subject that fits on one. Transplanted the way the normal criterion is, it fails differently — the normal one calls losses wins and never the reverse, and this one errs in both directions on every game tried, because the clause it adds is about counters rather than about moves.
The first move is a link that is not there
Lehman's criterion answers one question about a switching game — who wins when Short moves second. The other question has the same answer asked of a different graph: add one link from A to B, and Cut is forced to spend its first move deleting it. The trees of that larger graph then name Short's opening, and on every subgraph of seven graphs they name a winner.
Every move closes the largest gap
A census of Sylver Coinage by genus finds a parity that nearly decides the game and asks whether any known property of a numerical semigroup predicts the outcome. One does, completely: a semigroup whose gaps pair off around the largest one is never lost for the player to move — none of 583 up to genus sixteen. The reason is strategy stealing, and it is the same reason the top-right square decides Chomp: every move from such a position closes the largest gap.
Two graphs a rule cannot tell apart
The repair proposed for the matching criterion was to read the cycle as well. Over every connected graph with exactly one cycle up to six vertices — 21 graphs, 114 starts — eleven such rules reach at most 91, and the winner is not a function of the matching, the cycle's length, the start's distance from it, its degree and the edge count together: six cells of that table hold both verdicts, the smallest a pair of five-edge graphs.
A key is a code, and two squares come free
The families of positions a Zobrist key confuses are the words of a binary linear code — the sets of squares whose words cancel — so choosing a key is choosing a code. On 4 × 5 Domineering the textbook choice, a code with the largest minimum distance, confuses more stored positions than a random key at fourteen, sixteen and nineteen bits. The choice that reads the board confuses none at eighteen: two squares of one shade, in rows of different parity, are decided by the other eighteen, and no seventeen-bit key is exact.
The picture Bouton's proof leaves behind
His argument is two closure properties of one set, and the Sprague–Grundy theorem is the same two sentences with nought replaced by a variable — checked here on five games and every value in range, with no move staying inside a class and no class failing to be reachable from above. What the argument also leaves behind is a picture in which the values descend, and that is false: 99 of 444 moves here raise a value, and none of them is in Nim.
Named alongside it
The objects these essays reach for when they reach for this one.
EnumerationCounterexampleValueExhaustive searchImpartialApproximationTemperatureDecompositionDisjunctive sumPartizanHeuristicGrundy value