Combinatorial Game Theory
who wins, and by how much
Concept

Coin-turning — where it appears

A game on a row or grid of coins whose moves turn some over, with the last one turned going face down so that play must end. Its positions decompose inside a single picture, so a row of coins is already a sum rather than something to be split.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Turning Turtles: a row of 12 coins. A row of coins, some heads and some tails. A move turns some of them over, and the rightmost coin turned must go from heads to tails — which is what makes the game end. The number under each place is what a lone head there is worth, and the row is worth the exclusive or of the places showing heads.

A row of coins is already a sum

Everywhere else on this site a sum is several positions side by side. In a coin-turning game it is one row — each coin showing heads is a game in its own right, and the row is worth the exclusive or of them. The decomposition is inside a single picture.

impartial · Sprague–Grundy A grid of coins, and a multiplication table. The Grundy values of a two-dimensional coin-turning game, computed from its own move rules by a mex at every cell. Down the left and across the top are the one-coin values of the two one-dimensional games it is built from. Every cell is the nimber product of its two edge values — the multiplication defined for the nimber field on algebraic grounds — and beside the grid are the three combining rules a reader would try first, each killed on a named cell.

The tartan theorem

The nimbers are a field, with a multiplication defined by a mex-style rule that looks like an algebraist's amusement. Lay two coin-turning games on a grid and the Grundy value of each square is the nimber product of its two coordinates — which is the point at which the multiplication stops being a curiosity and starts computing answers.

impartial · Nim Mock Turtles on 8 coins: every lost position. The rows a player to move has already lost, drawn in full. A filled disc is a coin showing heads. The set is closed under turning over every coin two of its members disagree about, which is what makes it a linear code, and the count of heads in the sparsest of them is the fewest coin turns that separate two lost positions.

The losing positions are a code

Turn over one, two or three coins, and the rows a player has already lost turn out to be closed under adding two of them together. That makes them a linear code — and on eight coins it is the extended Hamming code exactly, sixteen words with a weight enumerator of 1 + 14x⁴ + x⁸, produced by a move rule that knows nothing about codes.

impartial · Codes Mock Turtles on 8 coins: finding the move is decoding. Every row of the game, sorted by what it takes to win from it. The lost rows are the codewords; a won row is a codeword with errors, and the winning move is the error pattern that turns them off. The distance column is a fact about the code and the coins column is a fact about the rules, and the two do not quite agree.

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.

impartial · Codes Welter positions and what they are worth. Coins on a strip, with the Grundy value the recursion returns and the nim-sum the squares would have if they were independent heaps. The two columns are the essay: they hardly ever agree.

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.

impartial · Welter The dual is the value table's span. The dual code against the span of the bit-planes of the one-coin Grundy values, on every game measured.

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.

impartial · Codes What the theorem replaces. Every grid a brute-force solve can reach, valued both ways, with the grid the theorem is normally drawn at underneath. Twelve squares is four thousand arrangements against twelve products; sixty-four squares is eighteen quintillion against sixty-four.

Four hundred and seventy steps

The tartan theorem replaces a search with a multiplication. Measured on every grid a brute-force solve can reach, the two agree on all of them — and the ratio doubles with every square added. On the 8 × 8 grid the theorem is normally drawn at, the search would have to value eighteen quintillion arrangements; the theorem needs twenty-six different nimber products, and computing all of them by the rule that defines them looks at four hundred and seventy pairs.

impartial · Nim

Named alongside it

The objects these essays reach for when they reach for this one.

Grundy valueImpartialNim-sumMexExhaustive searchXORBinaryDecompositionDisjunctive sumNormal playP-positionCertificate

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