Combinatorial Game Theory
who wins, and by how much
Concept

Down (↓) — where it appears

A negative value smaller in size than every positive number, written ↓, and the mirror image of up. Adding two of them to a position Left wins by an infinitesimal margin hands the win to Right, without crossing any number.

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

Smaller than every positive number, and not zero. Values that sit between zero and every positive number, each compared with zero and with 1/1024. Every relation drawn was computed by playing the difference, and one of them is confusion — neither greater, smaller nor equal. None of these is a number, and in a close game they are the entire margin.

Infinitesimals

Some positions are positive — Left wins them whoever moves first — and smaller than every positive number, including a millionth and a millionth of that. They are the values that decide close games, and the smallest of them is a single move's worth of nothing.

values · Infinitesimals Toads and frogs. Toads move right and frogs move left, one square into a gap or hopping over exactly one opponent. A player unable to move loses. It can be played on squared paper by anybody, and its values are immediately stranger than the game looks.

The strip nobody has a formula for

Some toads, a gap, some frogs. Two counts and a spacing is the whole description, and the values that come out of it are integers, stars, switches with eighth-point options and a down — four classes inside one two-parameter family, which is why nobody has written the formula.

positions · Toads and Frogs

Named alongside it

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

InfinitesimalToads and FrogsAll-smallAtomic weightCanonical formExact evaluationExhaustive searchHackenbushNimberOutcome classPartizanPosition graph

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