Combinatorial Game Theory
who wins, and by how much
Generator

Wythoff from (5, 8) — and who wins

Wythoff from (5, 8) — and who wins
Wythoff from (5, 8) — and who winsA Wythoff position with the outcome named before anybody moves. The queen slides left, down or diagonally down-left, and whoever cannot move loses. Every reply was computed in advance from the table of values, so the machine is reading a theorem rather than searching, and a reader who wins has found a mistake in the table.shaded squares are worth nothing to the player who must move from themthe player to move winsstated before a move is madeyou move firstwith the script running, the squares the queen can reach become clickable and this claim can be tested

A Wythoff position with the outcome named before anybody moves. The queen slides left, down or diagonally down-left, and whoever cannot move loses. Every reply was computed in advance from the table of values, so the machine is reading a theorem rather than searching, and a reader who wins has found a mistake in the table.

2 essays call play-wythoff. 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
(5, 8) ∗2 N The digits say which move wins · Wythoff's game, and the ratio nobody put there

It plays back

The winner is named before a move is made, and every reply was worked out in advance.

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.

 The cold positions, written in Fibonacci base. The first several cold pairs of Wythoff's game with both heap sizes written in Fibonacci base — as sums of non-consecutive Fibonacci numbers, which every integer has exactly one of. Blue is the smaller heap and red the larger. Read as digits, the pair is a shift: the larger numeral is the smaller one with a zero appended, and the smaller one always ends in an even number of zeros. Out in the world

The digits say which move wins

Wythoff's cold positions are usually given as a pair of golden-ratio formulas. Written in Fibonacci base they are a statement about digits instead — the smaller heap ends in an even number of zeros and the larger is the same numeral shifted up a place — and a rule about digits answers a question about a heap of a trillion.

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