Combinatorial Game Theory
who wins, and by how much
Concept

Col — where it appears

The painting game forbidding a vertex next to one of the mover's own colour, whose every value is a number or a number plus star. Painting is a liability under its rule, which is why its positions never become worth fighting over.

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

Also named here as colouring game, snort — the same set of essays touches all of them, so they are one junction rather than several.

Col and Snort on a path of four. One graph, two games, and one word of difference between the rules. Col forbids painting next to your own colour, which makes every move a small self-harm and drives the values towards numbers. Snort forbids painting next to your opponent's, which makes every move a land grab and drives them towards fights. Both values are computed from the same recursion.

One board, two rules

Col forbids painting next to your own colour. Snort forbids painting next to your opponent's. One word differs, the boards are identical, and the values that come out are not the same kind of object.

positions · Colouring Every position of both games, and where the fights are. Every colouring of every graph in the library, under both rules, with the temperature of each computed. A position with a positive temperature is one both players want to move in. Col has none anywhere; Snort has them on every graph, and the hottest grows with the graph.

One rule makes it cold, the other hot

Col forbids painting next to your own colour and Snort forbids painting next to your opponent's. Over all 540 colourings of six graphs, not one Col position has anything at stake and 61 Snort positions do — the difference between a game that is a count and a game that is a series of fights, produced by one word.

positions · Colouring

Named alongside it

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

Cold gameColouring gameHot gamePartizanSnortStar (∗)SwitchTemperatureExhaustive searchNumbersThermograph

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