Combinatorial Game Theory
who wins, and by how much
Generator

Wythoff's game, and the line the losing squares lie on

Wythoff's game, and the line the losing squares lie on
Wythoff's game, and the line the losing squares lie on. A queen moves left, down, or diagonally down-left any distance, and whoever cannot move loses. Every square carries the Grundy value the mex rule gives it. The squares worth nothing — the ones a player wants to hand over — lie along two lines whose slopes are the golden ratio and its reciprocal, in a game with no geometry and no continuous quantity in its rules.

A queen moves left, down, or diagonally down-left any distance, and whoever cannot move loses. Every square carries the Grundy value the mex rule gives it. The squares worth nothing — the ones a player wants to hand over — lie along two lines whose slopes are the golden ratio and its reciprocal, in a game with no geometry and no continuous quantity in its rules.

2 essays call wythoff-board. 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

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

PositionWorth OutcomeDrawn in
queen at (6, 10) 0 P A golden ratio thirty years early · Three complete solutions in nine years · Wythoff's game, and the ratio nobody put there

Where it is called

Changing this generator changes every one of these figures.

Wythoff's game, and the line the losing squares lie on. A queen moves left, down, or diagonally down-left any distance, and whoever cannot move loses. Every square carries the Grundy value the mex rule gives it. The squares worth nothing — the ones a player wants to hand over — lie along two lines whose slopes are the golden ratio and its reciprocal, in a game with no geometry and no continuous quantity in its rules. Impartial games

Wythoff's game, and the ratio nobody put there

Two heaps, three kinds of move, and losing positions that lie along a line of irrational slope. Nothing in the rules mentions a ratio, a length or a continuous quantity — and the golden ratio comes out anyway.

 One set, described three ways that share no arithmetic. Wythoff's cold positions can be stated as the Beatty pairs of the golden ratio, as a greedy construction over the integers that mentions no constant, and as a condition on Fibonacci numerals. None of the three consults the game. The fourth column is the game — a mex table over the moves — and all four name the same set of cold pairs over the whole square, which is what the figure counts. Out in the world

A set with three descriptions, and a function with none

Wythoff's cold positions can be written three ways that share no arithmetic — an irrational constant, a greedy rule, a condition on Fibonacci digits — and all three are exact. The same game's Grundy values have no closed form at all. Both facts are about one table, and the gap between them is the subject.

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