Normal play — where it appears
Named by 107 essays across 9 fields — each of them below, with the objects they name alongside it.
Misère play
Change one word — the player who cannot move wins — and the games are identical, the strategies are not, and almost every theorem of the normal-play theory stops being true. It is the cheapest possible modification and the most expensive.
Nim, and the nim-sum
Three heaps of counters, take as many as you like from one of them, and the player who takes the last counter wins. The winning condition is not a search, not a table, and not a heuristic — it is the bitwise exclusive-or of the heap sizes, and it was found in 1901.
The game in every exercise book
Dots and Boxes is played by more people than every game in this collection put together, and everybody is taught the same rule — take every box available. The rule is wrong. Establishing that takes a solver rather than an opinion, and the solver says how wrong, on which boards, and by how many boxes.
Who moves last
The player who cannot move loses. That single convention generates the whole theory — and it produces four outcomes rather than three, because a position can be confused with zero rather than greater, smaller or equal to it.
Loopy games
The whole theory assumes play stops. Allow a position to recur and the induction that every value rests on has nothing to stand on — and a fifth outcome appears that normal-play theory has no name for.
The chains decide it before the boxes do
Under every game of Dots and Boxes there is an impartial game with no score in it, and it settles the question the scoring game keeps asking — who ends up having to open. The rule players learn as folklore falls out of it, and so do the exceptions nobody mentions.
What survives misère play
Misère play destroys the value theory, and something much smaller grows back. Fix one game, look only at sums of its own positions, and the classes that behave alike form a monoid — computed here, and larger than the normal-play answer every time.
The rule that makes Go a finite game
A ko is a point in Go where a capture can be recaptured for ever, and every set of rules forbids it. That prohibition is not etiquette or tidiness — it is the hypothesis that puts Go inside the class of games every theorem on this site is about, and removing it removes the values.
Start at the end and work backwards
When play can return to where it started there is no bottom for the recursion to stand on. What replaces it begins at the positions where somebody has already lost and propagates outwards — and the positions it never reaches are exactly the draws. There is no test for a draw, and there does not need to be.
A game older than the theory
Kōnane was played on carved lava boards in Hawai‘i long before anybody wrote a brace notation, and its rule for losing is the normal-play convention arrived at some centuries early. Evaluate a row of it and the answers are halves, quarters, stars and infinitesimals — the theory's whole vocabulary, out of a game that was not built to display any of it.
Sprouts, and the game that is not one
Two games played with dots and curves, invented in the same room, all but indistinguishable on paper. One is unsolved past forty spots. The other has no decisions in it at all — the winner is fixed before the first curve is drawn.
The theorem that names a winner and no move
Strategy stealing proves that the first player wins Hex and wins Chomp, on every board, in about four lines. It exhibits no move, contains nothing a move could be extracted from, and is not going to. The moves have to come from somewhere else, and where they come from runs out almost immediately.
A winning strategy that is a spanning tree
The Shannon switching game was sold in a box in 1960 and solved in 1964, and the solution is not an assertion that somebody wins. It is a property of the graph anybody can check, and the strategy falls straight out of it — whichever link the opponent cuts, take its partner in the other tree.
Turn the board through a right angle
A two-by-four Domineering board is worth something no number can express, and Right is ahead on it. Turn a second board through a right angle, put the two side by side, and the total is exactly zero. Every position has an exact opposite, and that single fact is what makes subtraction — and therefore comparison — possible at all.
The move that gives counters back
Poker Nim adds one rule to Nim — a player may put counters back onto a heap from a private reserve. It looks as though a losing player could stall for ever. The winner is decided by exactly the same nim-sum, and the reason is the single most useful idea in the whole reduction apparatus.
The game that is a number system
In Sylver Coinage two players name integers and nobody may name a sum of what has already been named. Its positions are not boards — they are numerical semigroups, its termination is a theorem of Sylvester's from 1884, and the question of who wins after the opening move 16 has been worth a thousand dollars since 2017.
Counting at the end changes everything
Go is scored. So are Dots and Boxes, chess and almost everything anybody plays for money — and none of them is the kind of game this site's whole apparatus is built for. The simplest scoring game there is shows what that costs — the normal-play theory gives every position of it the same answer, and the answer is useless.
Tame and wild
The genus is a Grundy value with a tail — the misère values of the position with 0, 1, 2, … heaps of ∗2 added — and a game is tame when its symbols are the ones Nim heaps have. Computed here for seven games over heaps 1 to 14: Kayles goes wild at heap 5, Dawson's chess at heap 9, the octal game ·6 at heap 7, and heaps 3 and 11 of Dawson's chess are both worth ∗2 under normal play with only one of them tame.
Two misère outcomes are not enough
Knowing who wins each part does not say who wins the sum. Over 676 sums built from a pool of twenty-six positions, nine of the sixteen pairs of outcome classes settle the answer under normal play and not one of the sixteen settles it under misère — and the nine that work are theorems about a value being zero, which is exactly the thing misère play does not have.
A move that must be answered
Every argument on this site about sums assumes the parts are independent: a move in one leaves the others alone, and the reply may go anywhere. Top Entails denies it — take the top coin of a heap and the opponent must answer in that heap. The nim-sum then misreads 9 of 36 two-heap positions, and two heaps of two coins are a first-player win, which no impartial game the theory covers can be.
An environment made of coupons
Beside the game sits a stack of coupons worth 4, 3, 2, 1, 0, and a player may always take the top one instead of moving. Play the whole thing out and two quantities the theory computes are measured instead: {4 | 0} comes out worth exactly 2, its mean value, and the coupons stop at 2, its temperature. For {10 | {9 | 1}}, whose temperature is 1, they stop at 7/2 — because what the stopping coupon measures is the hottest temperature anywhere in the tree.
When never ending is a win
Retrograde analysis labels a position a win when somebody can force the opponent to be stuck, and leaves everything else blank. Calling the blanks draws is a rule from outside the game — and two other rules are available. The labelling does not change under any of them; only the residue does, and on a three-cycle that residue is every position on the board.
Equal in every company
Two games are equal when no third game can tell them apart — a quantifier over every position there is, discharged by one finite test. A search over 184 contexts separates all 5,790 unequal pairs it is handed and still calls two different games the same, which is exactly why G − H = 0 is a theorem and an exhaustive search is not.
Taking from several heaps at once
Moore's Nim lets a move take from as many as k heaps at a time, and the losing positions are still read off the binary columns — divisible by k + 1 rather than by two. The rule agrees with exhaustive search over 54,264 positions and never disagrees, and it decides every outcome while supplying no value at all: reading the same columns as a base-3 number gets the Grundy value right on 42 of 330 positions.
Cram
Domineering with one word of the rule changed: both players may place a domino either way up. That makes the game impartial, and the entire partizan apparatus collapses into a single Grundy value — on the 4 × 4 board, Domineering's canonical form runs to 114 characters of nested braces and Cram's answer is the one character 0.
The one outcome that adds
Finite outcomes do not add: two first-player wins can sum to anything. Loopy play has seven outcome classes instead of four and adds even less — of the 28 cells in the table, eleven hold several answers. Two do not, and they are the two worth having: a second-player win added to anything leaves the outcome alone, and a draw added to a draw is a draw. A draw added to anything else is not.
The values of every small board
Thirty Domineering rectangles, every value computed from the moves rather than looked up. The 1×n row obeys a formula and the 2×n row does not: its outcomes run L N N R three times over and then 2×13 comes out worth exactly 0, and its temperatures climb to 19/16 and fall back without settling.
A token on a graph
Geography is an impartial game whose position is a vertex and a history, so a ten-vertex graph has ten thousand states rather than ten. Take the arrows off and the same game is decided by a maximum matching — first player wins exactly when every maximum matching covers the start, verified on 41 vertices across eight graphs. One word in the rules separates a polynomial criterion from a PSPACE-complete problem.
Three players and no answer
Every theorem here is about two players, and the reason is not convenience. With two players the game is zero-sum, so 'play well' needs no further explanation. Add a third and the winner of a Nim position becomes a fact about the convention: two reasonable ones disagree on 56 of the 71 positions swept. The one question no convention touches — can a player force a win against the other two together — is answered 'nobody' in 65 of the 71.
A pass is not a move
Put a single pass token on a Nim board and one clause decides everything. If it may be taken at any time — including as the move that ends the game — the value of the whole is the nim-sum with a one added, in all 120 positions swept: the pass is a heap of one. Forbid it as the final move and the value stops being a function of the nim-sum at all, and 3 and 1 + 2 come apart.
The code names the move
If the lost rows of a coin-turning game are a linear code, then a won row is a codeword with errors in it and the winning move is whatever turns the errors off. Over all 256 rows of Mock Turtles on eight coins: 16 codewords, 240 won rows, none more than two coins from a lost one — and 64 of them whose cheapest winning move has to turn three coins anyway.
The same strip without the jump
Delete one clause from Toads and Frogs — the hop over an opponent — and the game is Elephants and Rhinos. Over the same 3,279 strips the values do not become simpler in the way a reader would guess: every value that is a number becomes an integer, against 172 fractions with the jump in, and the count of positions worth fighting over nearly doubles. Removing a move made the game hotter.
Taking from the ends
End-Nim is Nim's board with a player at each end, and it takes one sentence to state. Not one of its 5,460 small positions is worth a non-zero number — the game is all-small, so zero is the only number any of them can reach — and there are 2,693 distinct values between them. The outcome says a great deal more: 4,738 of those positions are won by the same player whoever moves, and on two heaps the rule is that the larger end wins.
No two heaps alike
Welter's game is Nim with one extra clause — no two heaps may be the same size — and the clause is fatal to the nim-sum, which gives the right answer in none of the 120 three-coin positions. What replaces it is a function of pairs: ⟨a | b⟩ = (a ⊕ b) − 1, exact on all 55 two-coin positions, and nim-added over every pair it is exact on the whole board provided the number of coins is even.
Nobody has to move
Every convention here rests on one sentence nobody examines — the players move alternately. Replace it with an auction and a position stops having an outcome class and starts having a number: the share of the money the second player needs. The 22 values born by day two collapse to seven of those numbers, eight of them landing on exactly a half; the new number respects the game order on all 179 comparable pairs, and is not determined by the parts under addition on 14 of 49.
It ends, and nothing says when
The recursion this site runs needs every line of play to reach a position with no moves, and the condition is usually met by an obvious decreasing quantity. The hydra meets it with no such quantity anywhere: the tree grows at nearly every step and the fight ends regardless, because the only thing that decreases is an ordinal. A four-node hydra dies in twenty chops; one level deeper and 279 chops reach forty thousand nodes with no end in sight.
One row of Clobber
Every string of blue, red and empty squares up to eight long — 9,840 rows — carries one of only 111 values, and every one of them is infinitesimal. A third of the rows are worth exactly zero. Six alternating stones are worth zero and eight are worth a form that takes four lines to print, so the values do not simplify as the row grows: they explode, while the row stays trivial to describe.
The fight never runs backwards
Left's stop is never below Right's — in every one of 1,780 distinct values, computed twice by two independently written routes, with nothing that disagreed anywhere. The inequality is what makes a mean value well defined and a fight a fight; and where it collapses to equality, 433 of the 460 cold positions turn out not to be numbers at all.
An option nobody would take
Every reduction of a form deletes. The gift horse principle adds: a move may be handed to a player for nothing, provided it is one they would never choose. Over all 484 additions to the values born by day two, 283 leave the value exactly where it was and the 201 that move it are precisely the ones the condition forbids — with the boundary at *not better*, which is a weaker demand than *worse*.
Nothing worth fighting over
Shove is a strip of coins beside a cliff, and both players have completely different moves. Every one of its 728 positions is worth a number, so nobody ever wants to move; the winner is the owner of the coin furthest from the cliff, in all 728; and the number the board is worth is not the sum of its coins — that reading is exact on 126 strips and wrong on 588 of the other 602.
The heap is not the position
Fibonacci Nim bounds a move by twice the previous move, which puts the state outside the board: a heap of six with a cap of two and a heap of six with a cap of five are different games. So there is nothing to add and no Grundy value to compute — and the game is completely solved anyway. The opener loses on exactly the nine Fibonacci numbers up to 120, and the smallest term of the Zeckendorf numeral is a winning move in all 110 winnable heaps.
Two players, two lists
Give each player their own list of how many counters they may take and the impartial theory stops applying. What survives is the outcome: it settles into a repeat, for every pair of lists, and that is a theorem. What does not survive is the value — on four of six pairs swept it has no repeat inside sixty heaps, and the birthdays are still climbing at the edge of the window.
A tree is still a number
A Hackenbush string spells its own value in binary. Put a fork in it and the numeral has nothing to read — there is no leftmost anything. The value is still a number, in all 10,066 forests up to six edges; it is still computable, by the ordinal sum, in all 3,238 single-trunk trees; and the reading is right on 762 of them, of which 126 are the strings it was written for.
Nobody wants to move here
A position is a number exactly when every move loses ground for the player making it. The test never mentions numbers, it disagrees with the ordinary one on none of the 1,474 values born by day three — and the reason a position fails it is not that somebody wants to move. It is that somebody cannot afford to wait.
Topple it from either end
A row of blue and red is the picture this site opens with, and under Hackenbush's rules it is always a number. Knock the pieces over instead of cutting them — everything on the chosen side falls — and 480 of the 510 rows up to eight pieces stop being numbers. The two games agree on sixteen rows, every one of a single colour, and the temperature of the hottest row climbs by exactly a half for each domino added.
The reduction that puts options back
Canonical form is presented as simplification, and half of it is. Deleting a dominated option takes one away. Bypassing a reversible one substitutes the answer's whole option list, so it can leave the form wider than it started — and 60 of 32,428 forms end up with a canonical form wider than they are.
The other way to move a row
Shove has a cliff and Push has a wall, and that is the whole of the difference. Both games make every one of the 728 strips up to six squares a number, so neither ever has anything worth fighting over — and the two rules do not agree on the value of a single position. The obvious board-reading is exact on 446 strips under the wall and on 140 under the cliff, and 486 strips contain a coin its owner cannot move at all.
The reduction that always shrinks
Canonical form is two reductions and they are not the same kind of operation. Deleting a dominated option removes one option and can do nothing else; bypassing a reversible one substitutes a whole option list. Over the 256 forms born by day two, deleting alone finishes 225 of them and accounts for 480 of the 520 options that come off — and the 31 it cannot finish are almost all the ones with a star in them.
Every group must keep breathing
NoGo is Go with no captures at all: a stone may be placed only if, afterwards, every group on the board still has a liberty. That makes a move's legality a fact about the whole board rather than about the squares it occupies — and a board therefore almost never breaks into independent parts. Of 117 boards here whose empty points fall into two regions, 24 are the sum of their regions and 93 are not.
The values nobody's game produces
The construction hands down 1,474 values by day three. Seventeen rulesets on this site, swept to eleven thousand positions, produce 1,193 — and only 116 of those are on the construction's list. Two of the twenty-two values born by day two are produced by no position of any game here, and 1,077 of the values that are produced are born later than day three. A value's birthday and a value's reachability have almost nothing to do with each other.
Nobody comes back
There is a class of games in which running out of moves is permanent, and it is the setting almost every modern misère result is stated in. Nine of this site's eleven rulesets belong to it across 5,334 positions; the two that do not are Toads and Frogs and Amazons, and Toads and Frogs loses the property to a single clause — delete the hop and it joins the list.
What a wider pool rescues
The misère outcome table has sixteen cells, and over a pool of ten positions fifteen of them hold fewer than four outcomes — which looks like structure and might be a shortage of positions. Thirteen values further on there is nothing left: every pair of outcome classes takes every outcome, so the near-misses were the pool, and the prediction the rung below made was right.
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.
The weight that blunts the count
The rung below found that counting the opponent's replies gets the direction of a comparison right once the gap reaches three, and proposed a repair: weigh each reply by whether it leaves the opponent anything. Weighing it makes the count worse. The threshold goes from three to four, the failures from 1,124 to 1,320, and all seventy-two of the pairs the repair was written for come through it unchanged.
The threshold was a fact about the census
Two rungs failed to account for the seventy-two pairs where a mobility count gets the direction of a comparison wrong, and the third looks at them one at a time. They are not a class of shapes. All seventy-two are on the largest board in the census, at two depths, and sixteen positions up to symmetry — and one board larger the count fails at a margin of three, which the ladder has been quoting as the point at which it never does.
Three rules and a tie-break
An exhaustive table of what a Domineering strategy has to remember is 3,308 lines. Three rules applied in order answer 94.5 per cent of it — leave the opponent fewest replies, then keep the region whole, then take whichever placement comes first — and the fourth and fifth rules answer not one more. The residue is 181 decisions in which every rule scores the candidates the same and one of them is worse.
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 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.
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 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.
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 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.
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.
The clause that turns the class off
Three rungs failed to find the dead-ending class doing measurable work, and each time the population was blamed. Toads and Frogs with and without the jump is the matched pair the anchor wanted — the same board with the class switched on and off — and on it the test the class licenses gains less from the class than a control that has never heard of it.
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 pairing that is not a symmetry
Every pairing strategy this ladder has found is a rigid motion of the square, and the requirement mentions no geometry at all. Searching all 8.8 million fixed-point-free involutions instead of the eight maps more than doubles what a pairing explains — and the smallest new one turns out to be a reflection with its two fixed squares swapped.
The quantity that carried nothing
The rung below proposed predicting a pre-period's length from the saltus and the period. The saltus correlates with it at −0.03, which is nothing; the period correlates at 0.77 with a coefficient of one, so a pre-period is about one period long. The digit that predicts whether there is one predicts nothing at all about how long.
The table that changes its mind
The advice that ten entries chosen by use serve nine lookups in ten was untested: it describes a table sorted after the fact rather than a solver that only ever held ten. A solver that only ever held ten gets 94.2 per cent — beating the best ten chosen with the whole run in view, because there is no best ten.
A description, and not a detector
The rung below noticed that the count of shapes attaining the hottest temperature grew across a plateau and collapsed at the step, and proposed it as a way to read a plateau off a single size. The growth is exact — five plateaus, no exception — and the rule is impossible: five orbits precede a rise at seven squares and no rise at eight.
The easy case was not the reason
The rung below found the mobility rule reaching a failure rate of exactly nought near the endgame and named what a proof would need: that a decomposed board's comparable options are ordered by reply count. That statement is false on all five boards, at margins up to two — and split positions go exact two squares of depth before whole ones, so decomposition is the easy case rather than the cause.
The price of asking what the parts are
The third licence lets a solver look up a region rather than a position, and the rung below priced it by the entries it stores. Priced by the work it costs, it saves between a third and two thirds of the expansions and pays for them with a flood fill at every node — six times the total. A square would have to be ten times cheaper than a table probe before it broke even.
One expression proved, and one withdrawn
The census closed with three expressions exact on 1,440 pairs, and the rung above asked for derivations. The cold one has a four-line proof. The straight one has a threshold the census cannot determine — any constant between 4/3 and 3/2 fits it — and eight more addends of the same family break it on 38 pairs while leaving the bound above it untouched.
The four paragraphs prove something else
Three rungs earned the right to write the crossover law's proof as two straight walls meeting where the law says. The walls are straight — one of them everywhere, for a trivial reason, and the other only above the answer's own temperature. The crossover sits below that height, so the geometry holds nowhere the law is about, and where it does hold it proves the temperature instead.
The dual was the value table
A coin-turning game's losing rows form a linear code, and a code has a dual that nothing in the game appeared to read. It reads it constantly: the dual is spanned by the bit-planes of the Grundy values — the parity checks are the value table stood on end — and on Mock Turtles over eight coins the losing rows are exactly the span of the table that decides them.
One proof, and one wrong lemma
Two measured identities were left for a proof: the move rule by induction, the gap condition from the reply bound. The induction is exact on 31,731 heaps at eight factors. The reply bound holds at c = 2 and on one index pair in twenty-seven at c = 3 — and the inequality that does the work is a third one nobody proposed.
The pairing the formula hides
Welter's closed form sums a function over every pair of coins and needs an extra term when the count is odd, which the rung below called a surprise. Read as a matching it is not: an odd number of coins cannot be paired, the left-over coin contributes its own square, and some matching gives the value on every position measured.
A cross in the table
Which pairs need a second reduction had never been asked. Sorted by the two temperatures the answer is a cross — the whole gap-of-a-quarter column and the whole colder-is-three-quarters row — and it is exactly necessary on all 431 with no exception, and wrong 477 times in the other direction.
The proof needs both reductions
The gift-horse theorem was to be proved by showing the added option dominated. It is, on 97.8 per cent — and the other 232 are reversible instead, with nothing left over. The case the proposal missed is almost entirely one follower: none under a positive number, 190 under a negative one.
Two measures bounded, and one not
A sum is born no later than its parts' birthdays together, and it has no more options than they have between them — a bound nobody had checked, and it is attained. What runs away is the length of the written form: 27 pairs of 231 exceed it, the worst by 29 characters, on a sum with exactly as many options as it was entitled to.
A floor, and not a decline
Comparability fell eighteen points from day two to day three and the next day cannot be enumerated. It can be built — and the construction's bias measured one day lower, where the truth is known. Corrected, day four comes to 60.6 per cent against day three's 59.7: the fall was a one-day event.
One number, stated two ways
Twice the height of the cut held and was loose; the height alone failed. The smallest true constant is three halves — exact and attained as a bound on how far the value can fall, and an infimum attained nowhere as a bound on the value. The gap between the two is one move.
A factor, and not an overhead
Deciding who wins a form searches the form's own tree. Reducing it to canonical form searches a difference game for every comparison, and a difference game is a sum. Over 256 forms the reduction expands 5.46 times as many positions — and the ratio runs from 0.58 at one option to 9.80 at eight.
Close calls nothing resolves
The same value panel that settles 118 of the 202 silent decisions settles four of the seventy-two the rules get wrong. Every rule that helps at all must replace connectivity rather than follow it, and the cheapest one breaks twenty-six decisions for every one it saves.
Three groups, and three yields
The conjecture was that each ruleset's yield tends to the reciprocal of its symmetry group's order. Three rulesets have a trivial group and predicted yields of one; they measure 1.000, 0.531 and 0.204. And every colliding value in Push and Shove — all 175 of them — has two rows no symmetry relates.
The parts are worth nothing and the sum is not
Every chain and every loop in Nimstring, taken alone, has Grundy value nought. So the Sprague–Grundy theorem predicts that every position built from them is worth nought — and ninety-six of the two hundred and seven positions checked here are not. The theorem is not being misapplied; it does not apply, because a capture keeps the turn. What replaces it is smaller and sharper: count the short chains, and one long component of any kind reverses the parity.
A parity with a first exception
Sort every Sylver Coinage position by how many numbers are still unnameable and the game very nearly falls to parity: odd rows are between a fifth and a half positions the mover loses, and the first three even rows hold none at all. The rule has a first counterexample at genus eight, where it is a single position out of sixty-seven, and eleven more at genus ten. It is a tendency wearing away from both ends rather than a law with exceptions.
Two ways to count a finished board
Territory scoring and area scoring are both in daily use and they are not variants of one rule. Over seventy-nine small endgames they agree exactly on the thirty-nine with an even number of neutral points and on none of the forty with an odd number — sixteen name a different winner, and twelve are played differently, which is not something a convention is supposed to do.
The two moves that are not captures
Kōnane begins from a full board and the first two moves lift stones rather than take them, which is the only time in the whole game anybody does. Nothing on this site applies to them, and the choice is not free — on a 3 × 5 board two of Black's eight openings leave a position a whole move worse than the other six, and on a 3 × 3 board none of the five does.
What a pass is worth to a theory
The rung below finds fifteen of twenty-seven coin rows where having the move is a disadvantage, and those are exactly the rows Milnor's mean-value theory has to assume away. Allow a pass and the hypothesis stops being a hypothesis — nought violations, on every row in range. What it costs is the convention the rest of this site is built on.
Nothing to subtract with
Comparison is defined by contexts and computed by subtraction, and the equivalence between the two is a theorem about groups. A scoring game is not one — sixty-six of eighty-one coin rows do not cancel against their own negatives — and the difference test then fails on a row compared with itself, which every context accepts and nothing certifies.
The convention Dawson actually used
Dawson published his puzzle as a problem where running out of moves loses you the game, and every compact result about ·137 is about the other convention. Under his own, nine values become a classification that doubles the moment a wild heap enters the range, and a heap stops carrying a number at all.
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.
Using up the edges instead
Undirected geography is decided by a maximum matching when a move uses up the vertex it leaves. Use up the edge it crosses instead and the matching is exact on every tree — on a tree the two games are one game — and on nothing else. Over every connected graph on up to six vertices it names the winner at 480 of 745 starts once there is a cycle, it gets a four-cycle wrong from every start, and the more cycles a graph has, the more of its misses are wins that are really losses.
The capture that has to be made
Dawson's chess is quoted as the octal game ·137, and the step from a pawn diagram to a row of counters has been taken on trust. Searched as a chess position, the diagram agrees with ·137 on every board from one file to twelve, under both endings, and every exchange it can start is an odd number of moves that lands on one of ·137's options. The whole reduction rests on one rule of the diagram that the octal code never mentions: a capture, when one is available, must be made. Make it optional and the winner changes on two, three, six and seven files.
Proving a loss means answering everything
A win is established by one move and a loss by every move, so the two verdicts are certified by objects of different shapes. Measured over every position of four games, a loss costs between 1.07 and 2.31 times a win — a small constant, never an exponential. The obvious explanation is the branching and it is wrong: Nim answers six options at a losing turn and pays 2.18, not six.
The auction never gets to the money
The critical fraction is computed and never played. Played out with a countable pool of chips — twelve positions, four pool sizes, every split of the chips, every bid answered — the verdict does not move with the money on a single one of the forty-eight sweeps, and the rule for equal bids settles all forty-eight. The reason is one line long: declining every auction wins, and bidding nothing declines.
A coin needs no tie-break
The same recursion has a second derivation: a fair coin decides who moves at each turn, a player whose turn it is with no move has lost, and both play to win. Written from those rules it comes out identical on every position — and it needs no rule for equal bids, because there are no bids. The number is a probability, it belongs to Left rather than Right, and the empty position is the one where the coin decides everything.
Left always wins, and loses more often than not
Alternating play answers with one of four classes and the coin answers with a chance, and the two do not have to agree. Over the twenty-two values born by day two they never disagree and the margin is exactly nothing — the lowest chance on a position Left wins whoever moves is a half. Over the 1,474 born by day three, seven of them sit at seven sixteenths, and seven mirror them on the other side.
The best chance is the wrong move
Maximising a probability and denying an opponent a reply are different objectives, and on 189 of the 904 day-three positions where Left has a choice and a winning move, the option the coin prefers is one that loses the alternating game outright. The smallest case is two options and one line of arithmetic: five eighths beats a half, and a half is the move that wins.
The endgame theory arrives late
Every component the chain-and-loop theory names has coins of degree two, so a position it can read is one where every surviving coin holds exactly two strings. Over a six-box board that is 1,033 of the 28,028 positions with no free box on the table — 3.7 per cent — and more than half of them only after twelve of the board's seventeen strings have been cut.
A thousand positions and no exception
The parity law was fitted to constructed bags of chains and loops inside a string budget. A board's positions are a different population — the sizes are what the geometry allows, the components come correlated, and a six-box board holds exactly one position that is a loop of six. Tested on all 1,032 of them and all 160 of the four-box board's, the law is right every time, against a verdict computed from the strings by a walk that has never heard of a component.
A coin with three strings is worth something
Every chain and every loop is worth nought on its own, whatever its size, and that is exactly what makes their nim-sum useless. A coin with three strings on it is worth nought, one, two or three depending on its arms — 31 of the 35 measured are not nought, and the four that are are the ones whose arms are all long. A coin with four strings is back to nought every time.
Three heaps and a pass
Nim with a single pass that may not end the game is easy on one heap and on two: a heap swaps each odd size with the even one above it, and two heaps lose exactly at (2k − 1, 2k). On three heaps the losses are known only as a list. Fix the smallest heap and each slice of the list settles into a pattern after an irregular start — period 4, 8, 10, then 160 at a smallest heap of ten, and nothing visible from eleven.
The follower does the reversing
The gift-horse theorem for the ordinal sum needs two cases, and the second — the added option is reversible — was counted and not described. Recorded move by move, the reversing answer is always Right's move inside the follower: on all 410 escapes under five followers, and on every one of the 2,628 gift horses under every follower that gives Right a move at all. The case split is by follower, not by horse.
The split slips one day deeper
The reversal case of the gift-horse theorem was described in one line — the follower's own move reverses every gift horse, whenever the follower has one — and tested only where it was found. In the mirror it holds exactly, with 1 and −1 trading places. One day deeper it fails: under ↑ and ½, three gift horses on built day-four bases are not reversed by the follower's move. All three are dominated, so the theorem stands; the clean split by follower does not.
Twenty draws and a second recipe
Day four's comparability was reported as 60.6 per cent against day three's 59.7, from one built sample and one calibration. Built twenty times with each of two recipes whose biases differ by six points, and calibrated against all 1,474 day-three values rather than a quarter of them, the corrected figure spreads over twelve points from seed to seed and the two recipes agree within one standard error. The floor survives; the decimal was one draw.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchOutcome classEnumerationImpartialCanonical formGrundy valueCounterexampleDecompositionDisjunctive sumComparisonNimTemperature