Combinatorial Game Theory
who wins, and by how much
Generator

The cold squares, placed without a table

The cold squares, placed without a table
The cold squares, placed without a table. Wythoff's game as a square of positions, with the cold ones marked — and marked from the Fibonacci-base digits of the two heap sizes alone, without consulting a Grundy table. The two methods agree on every position checked. Beside it, what each costs as the heaps grow: a table needs an entry per position and a numeral has about as many digits as the logarithm of the number.

Wythoff's game as a square of positions, with the cold ones marked — and marked from the Fibonacci-base digits of the two heap sizes alone, without consulting a Grundy table. The two methods agree on every position checked. Beside it, what each costs as the heaps grow: a table needs an entry per position and a numeral has about as many digits as the logarithm of the number.

2 essays call digit-vs-table. 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.

Where it is called

Changing this generator changes every one of these figures.

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.

 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