Combinatorial Game Theory
who wins, and by how much
Concept

Arrow — where it appears

The square an Amazon burns after moving, which is what makes the game shrink and what makes a strip of it partizan rather than a race. Without it the game is loopy and has no values at all, so the clause about burning a square is what makes the theory apply.

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

Amazons, after the arrows have cut the board in 2. An amazon moves like a queen and then shoots an arrow, also like a queen, which burns the square it lands on. Late in a game the burnt squares cut the board into regions no amazon can cross — and from that moment the position is a sum of independent games, which is the shape the whole theory was built for, arrived at by the play rather than assumed.

Amazons, and when a position becomes a sum

Every technique on this site starts from a position already broken into independent parts. Amazons does not begin that way — the board is one fight until the arrows cut it, and the moment of cutting is something the play produces rather than the analyst assumes.

positions · Amazons Amazons on one line. A one-dimensional Amazons board: an amazon slides along the row and shoots along the row, and the square the arrow lands on is burnt for the rest of the game. The whole board fits in a sentence, and the values it produces are already of several different kinds.

Amazons on one line

A board one square high is small enough to evaluate completely: every strip from two to ten squares with one amazon a side is 37,886 positions taking 81 distinct values, and every one of them is an integer, a switch, a number plus a star, or a bare star. Not one is a fraction — and forcing the arrow onto the square just vacated, which takes a freedom away rather than adding one, produces 1,196 that are.

positions · Amazons

Named alongside it

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

AmazonsDecompositionRegionBoard partitionCanonical formComparisonDisjunctive sumDynamic decompositionEndgameExact evaluationExhaustive searchLoopy

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