The thread: One clause decides it — page 5
Impartial games
Cram with the last move losing
Under misère play every Cram rectangle to twenty squares is tame at the start except 3 × 6 and 4 × 5, and twelve of the fifteen boards two rows deep behave like one counter or none. The pairing that settles even boards under normal play hands the misère game to the first player, who wins by any vertical domino. And the tame boards are not tame inside: wild positions appear on boards as small as 3 × 4, always at ten empty squares, the length at which the strip goes wild.
Where it stops
What survives a coalition
Three-player Nim has no winner without a convention and no sum theory at all. Fix one player against the other two and both come back: every position has a winner, the winner is read off the number of heaps of one modulo three and the number of larger heaps, and forty-five positions fall into six interchangeable classes — fewer than Nim's sixteen. Fix a scoring rule instead and the winner comes back, identical to one of the conventions, and nothing else does: no two positions are interchangeable.
Where it stops
One value more than Nim
Top Entails broke the nim-sum: nine of thirty-six two-heap positions came out wrong when a move could compel the reply. The repair is one extra value. Read each heap off the heap itself, and every heap of Top Entails and of a second entailing game is either a nimber or loony — a win for the mover in any company. One rule then decides every sum of those games and Nim together: 4,016 positions, and it is never wrong. A loony heap is a pass, and the compulsion is what buys it.
Where it stops
A coalition counts to three
Three-player Nim with one player against two collapses to six classes, because every heap of two or more is a store of tempo either side can draw on. Put the same coalition on games whose heaps are not all stores and it does not collapse: subtraction {1, 2} keeps five kinds of heap, Kayles six, subtraction {1, 3} ten and {2, 3} twelve, each count settling as the range grows. Past a threshold the large heaps are interchangeable again — and in every one of the five, unlike Nim, a single large heap hands the game to the coalition. The classes neither refine the two-player values nor coarsen them: {1, 3} has two Grundy values and ten coalition classes.
Impartial games
Two takes, one line
A subtraction game with two legal takes, a and b, has a Grundy sequence that fits in one sentence: cut the heap's remainder modulo a + b into blocks of a, alternate nought and one, and turn every even block at or past b into a two. The proof is three cases long and holds for every pair. Add a third take and the sentence has no successor — but it does not vanish. The period of a three-take game is a sum of two of its takes six times in seven, which sum is decided by the third take's remainder modulo the first two's period, and the long irregular periods cluster on the one arrangement where the third take is the other two added together.
Out in the world
The same rule on a star
Kōnane turned out to hold the theory's whole vocabulary, and one old game is an anecdote. Mū tōrere, a Māori game on an eight-pointed star, has the same rule for losing — a player who cannot move loses — and is small enough to solve whole. Of its 630 arrangements, not one is a win for whoever moves first or a loss for whoever must move, sixteen go to each side whoever starts, and most are drawn. A random player still loses it every time. The vocabulary was never in the rule for losing; it was in the captures.
Temperature
Why cooling has to add
Cooling came through a sweep of 1,768 sums untouched while heating failed 263 times, and the sweep's own account was cautious: cooling is not additive in general, and a richer pool would break it. Aimed exactly where that account said to look — infinitesimals hidden under low temperatures, at taxes of a quarter and a half — 107,816 more pairs break nothing, because there is nothing to break. Cooling distributes over every sum of short games. Its freezing clause fires on one side of the equation and not the other in 6,032 of those pairs, and costs nothing every time, because what it hands back is the mean, and means add.
Temperature
The bonus an integer is owed
Heating fails to distribute over a sum where cooling never does, and the repair proposed for it was to change what its exception returns: heat a number into a switch instead of leaving it alone. Run on the same 1,768 pairs, that repair fails on every one of them, beginning with nought plus nought. What adds is a heating that changes which numbers it exempts and pays the bonus on the moves an integer holds in reserve — Norton's product, and heating with no exception at all, which at a whole-number bonus is the same operator. Given heating's exception, cooling stops adding too.
Impartial games
The third take is the other two
A subtraction game whose third take is its first two added together was where the long irregular periods of three-take games clustered, and the question was whether that one arrangement has a formula or is where the formulas stop. It has one. Divide out any common factor and look at the heap one short of the middle take: if it would be a loss when only the smallest take is allowed, the period is the smallest take times twice the middle take plus a remainder, otherwise it is short and the whole sequence is two bands of blocks. The formula gives the exact period of every one of the 3,160 games to eighty.
Where it stops
A compulsion that points elsewhere
Top Entails and entailing Nim are decided by one value more than Nim because the compelled reply stays in the heap that compelled it. Point it at a different heap instead and the two games part company. Entailing Nim does not notice — its heaps are worth exactly what Nim's are, because Nim's winning reply was never in the heap just moved. Top Entails loses its loony heaps and then every value of the Nim kind: no assignment of nimbers explains its sums, because a heap of three beside two heaps of one is a win and beside two heaps of two is a loss.
How it was found
A rule that let the game run for ever
In 1929 German chess drew a game when the same sequence of moves came three times running, and the rule was meant to make chess finite. Max Euwe, six years from the world title and trained as a mathematician, showed that it does not, using a word that had already been found twice for other reasons. Two round trips are enough. The board returns to one position thousands of times, no block of play ever comes three times running, and the rule has nothing to say.
How it was found
How strict a rule can be
A repetition rule can be tightened one notch at a time, and each notch either leaves unending games alive or kills them outright. With two moves a block may not come twice, and every game is over by the fourth letter; relax that by a single letter of overlap and unending games appear. Three moves stop dead at 38 letters under a rule that a whisker's relaxation frees. Four stop at 121. Past each boundary the words go on for ever, and the boundary sits at a fraction nobody would have guessed.
How it was found
The opponent makes it repeat
Euwe's unending game needs both players to keep to it. Give the letters to two players in turn, with one of them wanting the draw that a repetition brings, and the rule of 1929 fires within seven letters on a plan of one sentence: always play A. The word that defeats the rule exists and nobody can be held to it. Add a third round trip and the same player can no longer force anything in twenty-six letters. Existence and force turn out to be separate questions, with separate boundaries.
What it costs
The number in the rule
Draw on a repeated position and no result moves. Refuse the move instead, as Go does, and nothing is left drawn: over every game on four positions, 10,324 of 16,384 starts are drawn with no rule and every one of them acquires a winner. Which winner depends on the number written into the rule. Bar the third arrival rather than the second and 1,392 of those starts change hands. A question with a linear answer has become one that needs the history, and the extra difficulty lies entirely in the positions the labelling called drawn.
How it was found
A rule that cannot see whose turn it is
The stable sets a game hands over work by telling the players apart. Forbid that, and read every board as mine and theirs, and the odd cycles come straight back. On Mū tōrere, with rotations and reflections identified too, the 892 drawn positions fall into 33 kinds, and among those there is exactly one stable set. It names sixteen shapes. A player who always hands the opponent one of them holds the draw against labelled play without knowing a colour, an orientation or an outcome, and the opening itself is one of the sixteen.
Temperature
An exemption that costs nothing below one
Heating that leaves integers alone fails to add at a bonus of one and adds on everything at a bonus of a half, and the natural guess was that a half is where it starts to go wrong, since that is where a fraction's moves become worth fighting over. It is not. Across twelve bonuses nothing fails at any value below one, and dozens of pairs fail at one exactly. The reason is the simplicity rule: below one, the recursion sends every integer to itself, so the exemption is not doing anything at all.