Combinatorial Game Theory
who wins, and by how much
Generator

Where the count and the value part company

Where the count and the value part company
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.

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.

2 essays call territory-vs-value. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

The positions it draws

6 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
..LRxx.x. 0 | −2 N When a real board falls apart
..Lxx..xR {2 | −1, {1 | −1}} N When a real board falls apart
..LxxR.xx 1 | −1 N Amazons, and when a position becomes a sum · When a real board falls apart
..RLxx.x. 2 | 0 N When a real board falls apart
..RxxL.xx 1 | −1 N Amazons, and when a position becomes a sum · When a real board falls apart
.x..xxLRx 1 | −1 N When a real board falls apart

Where it is called

Changing this generator changes every one of these figures.

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. Particular games

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.

 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. Out in the world

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.

The whole library · The position index · The figures that play back