Combinatorial Game Theory
who wins, and by how much
Concept

Binary band — where it appears

A run of sizes sharing a power of two, which is the classification Cutcake counts by and which Maundy Cake does not use. The classification is the game's own rather than one imposed on it, which is why the same idea fails for a game with a different cutting rule.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Cutcake: every value an integer. The value of an m by n cake, for every small m and n. Left cuts down, Right cuts across, and neither player ever gains by moving — so nothing is ever at stake, every value is a whole number, and the number says exactly how many spare moves one player has.

Cutcake, where every value is a whole number

A partizan game in which no position is ever worth a fraction, a star or a fight. Every value is an integer, the integer is a count of spare moves, and the pattern it follows is decided by binary digits.

positions · Cutcake Maundy Cake: the pieces must be equal. The same cake as Cutcake, cut by the same two players, with one extra rule: a cut must divide the cake into equal pieces, and every piece stays in play. The values are still whole numbers, but the arithmetic that decides them is not Cutcake's — it counts prime factors rather than binary digits.

Maundy Cake

Cutcake with one word added: a cut must divide the piece into equal parts. The values are still whole numbers, and the rule this site has been repeating about them is false — over all 1,296 cakes to 36×36 the largest-odd-divisor account has 946 counterexamples. What survives is a count of prime factors, and it says who wins without saying by how much.

positions · Cutcake

Named alongside it

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

Cold gameCutcakeInteger valuedPartizanSpare movesBinaryCounterexampleDisjunctive sumExhaustive searchLargest odd divisorMaundy cakePrime factors

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