Combinatorial Game Theory
who wins, and by how much
Concept

Territory — where it appears

The squares a player will end up owning, counted rather than fought over. In Amazons a region containing only one player's pieces is worth exactly its free-square count, so the whole strategic content of the game is which player ends up owning which region.

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

Where the count and the value part company. Amazons endgames whose arrows have already cut the board into regions, with the territory count beside the computed value. Territory gives every empty square to whichever amazon can reach it in fewer moves, which is what Amazons programs compute. The positions drawn are the ones where that number gets the outcome wrong, and they have something in common: each is worth a switch, so there is no number for the count to have been right about.

When a real board falls apart

Amazons is played competitively, and late in a game the arrows have cut the board into regions no piece can cross. From that moment the position is a disjunctive sum — arrived at by the play rather than assumed — and the territory count every program uses can be measured against what the sum is actually worth.

applied · Amazons Still a count of squares. One-sided Amazons regions on two-dimensional boards, swept exhaustively. Every one is worth exactly the number of free squares in it.

A region one player owns

On a strip, a region containing only one player's amazons is worth exactly its free-square count, on all 45,057 positions of the rung below's sweep — and it predicted the exactness would fail in two dimensions, where an amazon can be short of room in one direction and not another. It does not fail. Over 2,412 two-dimensional regions there is no exception, and the reason is one clause: an amazon may shoot back at the square it has just left.

positions · Amazons

Named alongside it

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

AmazonsComponentDecompositionPartizanCounterexampleDisjunctive sumEnumerationExhaustive searchHeuristicIntegerInvariantNumber

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