Every essay — page 12
How it was found
The theory looks inevitable in retrospect and the record says otherwise. The older arguments are run here rather than recounted — and one of them still answers a question nothing since has answered.
The solution that was a losing set
In 1944 von Neumann and Morgenstern defined the solution of a game by two conditions on a set of outcomes: none overrules another, and each outcome outside is overruled by one inside. They were writing about coalitions. The same two conditions, read with a move in place of an overruling, describe exactly the lost positions of a game that must end. Where play can loop, the same definition gives several answers or none, and three positions are enough to show both.
An odd cycle is the only way to have none
A cycle of two positions has two stable sets and a cycle of three has none. Moses Richardson proved in 1953 that this is the whole story of absence: a graph with no odd cycle always has a stable set. Run over all 9,608 games on five positions, the theorem holds without exception, sits inside a chain of five nested families, and leaves open the larger question its converse invites, since 7,452 of those games have an odd cycle and a stable set anyway.
What a stable set knows about a draw
On a graph with cycles, outcome labelling calls a position lost, won or drawn, and stops. The stable sets go further in exactly one place. On every game of up to five positions they hold every lost position, no won one, and among the drawn positions are exactly the stable sets of the drawn part taken alone. So a stable set is a certificate that the player who keeps returning to it never gets stuck. It can distinguish draws that labelling cannot, and two parts with no stable set can have one between them.
A stable marriage is a stable set
Draw a graph whose positions are pairs of people, with a move from a pair to any pair one of its members prefers. Its stable sets are exactly the stable marriages, on all 46,656 profiles of three men and three women. So Gale and Shapley's theorem of 1962 is an existence theorem of the kind Richardson's was, on graphs where 49 per cent have odd cycles and his hypothesis is useless. The reason it holds is that every clique in the graph is one person's list, and four roommates show what happens when it is not.
The list that cannot run out
Give each cell of an n × n square its own list of n colours, and ask for a colouring from the lists with no repeat in any row or column. Dinitz asked in 1979 whether that is always possible, and reading the cells in order fails about four times in ten. Galvin's proof of 1994 needs one page. Use an ordinary Latin square to turn each row and column into a preference, so that every cell has exactly n − 1 moves, and peel off a stable set for one colour at a time. A list of n cannot run out against n − 1 moves.
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 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.
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 counting positions buys
A rule that counts arrivals at a position ends every game, which the rule of 1929 could not do, and it needs no proof beyond arithmetic: with L arrivals barred on n positions, no game is longer than (L − 1)n − 1 moves. What it does not do is change a single result. Over every game on three and four positions and a sample on five, drawing on the second arrival or the third gives exactly the labels retrograde analysis had before any rule was written. The rule pays for the game's ending and for nothing else.
Taking turns removes the hard part
Whether a graph has a stable set is NP-complete, and every hard instance depends on odd cycles. A real game's position graph has none. A position records whose turn it is and every move changes that, so all cycles are even, Richardson's theorem applies, and the set of all drawn positions with one player to move is a stable set of the drawn part, always. Mū tōrere and Three Men's Morris both confirm it. The hard question never reaches a game with turns, and the sets it hands over carry no information at all.
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.
Impartial games
Both players have the same moves. Every such position is a Nim heap, and the theorem that says so is the field's first.
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.
Every impartial game is a Nim heap
Sprague and Grundy proved, independently and four years apart, that any position in any impartial game is equivalent to a single heap of counters. Not similar to one — equal to one, interchangeable with it inside any larger game.
Grundy sequences, and where they stop being predictable
Computing one Grundy value is a mex. Computing all of them produces a sequence, and the sequences do something nobody has fully explained — most of them eventually repeat, some of them take thousands of terms to start, and for a few nobody knows whether they ever do.
Naming a game with a number
An octal code is a rule table compressed into an integer. It turns "which game" into something that can be counted through, which is how the family was swept — and how the games nobody can solve were found.
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