Combinatorial Game Theory
who wins, and by how much
Concept

Fixed length game — where it appears

A game whose every play takes the same number of moves, so that the winner is decided by parity before anybody sits down. Its winner is a parity rather than a strategy, which is what separates a puzzle from a game.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

One of these is a game. Every length that came out of 40 random games from each starting position. Brussels Sprouts always ends after exactly five crosses less two moves, so whoever is to move at that point was decided before the first curve was drawn. Sprouts ends at different lengths depending on how it is played, which is what makes it worth playing.

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.

positions · Sprouts Hydras, and how long each takes to kill. Six small hydras with the ordinal the termination proof assigns to each and the exact number of chops it takes to finish it. Two of them are not finished here: the fight is guaranteed to end and the machine runs out of memory long before it does, which is the gap between a termination proof and a bound.

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.

limits · Termination Two ways to be certain and ignorant at once. Ten positions from two games that both terminate for reasons no bound comes out of. Sylver Coinage's proof counts something that goes down and can be counted; the hydra's counts an ordinal, which cannot, and the last column shows what that difference is worth.

Two ways to end with no bound

Sylver Coinage and the hydra are both guaranteed to finish and neither will say when. The difference is that one of them carries its own bound: every move in Sylver removes at least one gap, the gaps can be counted in a moment, and over ten openings the longest play uses every single one. The hydra has no decreasing quantity a solver can hold — three hydras of five nodes each take seven chops, twenty-one, and a number past two hundred and seventy-nine that this machine never reaches.

limits · Termination

Named alongside it

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

Exhaustive searchInductionNormal playOrdinalSylver CoinageTerminationBackward inductionBoundBrussels SproutsCertificateComplexityDecreasing quantity

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