Combinatorial Game Theory
who wins, and by how much
Concept

Cold position — where it appears

A position neither player gains by moving in, whose value is a number or a number with an infinitesimal attached. Adding a number to it moves both stops and adding an infinitesimal moves neither, which is the whole difference between the two theories.

Named by 6 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

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.

impartial · Wythoff's game Why nobody moves in the number. A hot position added to a number. Left wins the sum whoever moves — but only by moving in the fight. Spending the move on the number instead hands the position back as a first-player win, with Right to move, which throws the win away. The theorem says this is always so, and here it is happening.

Numbers avoid numbers

In a position with a number in it and anything else, the number is never the right move. That is a theorem rather than a heuristic, and it is the closest this subject comes to advice a player can carry into a real game.

values · Numbers 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.

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.

applied · Wythoff's game The same population counted twice. The temperature scale over the positions this site has enumerated, once with every position counted and once with every distinct value counted. The two disagree about how much of the subject is hot, about what the commonest hot temperature is, and about whether a number is the usual thing for a position to be worth.

How hot a real position is

Counted one value at a time, a tenth of the subject is hot. Counted one position at a time — every board this site has enumerated, all 11,397 of them — it is a twentieth, two thirds of the positions are worth numbers outright, and ten of the seventeen rulesets never produce a hot position at all.

temperature · Cold 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.

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.

applied · Wythoff's game What the flip is worth, and what is at stake. Left's chances if Left moves and if Right moves, with the gap between them beside the position's temperature. The two are answers to the same question computed by different routes, and they do not order the positions the same way.

The coldest position has the biggest swing

How much a flip is worth is the gap between the coin's two branches, and it is a rival to the temperature — both answer how much is at stake. They disagree at once: the empty position has the lowest temperature there is and a swing of one, twice the hottest thing born on day two. On the small pool the two quantities look like a perfect three-way correspondence, and 255 of day three's values break it.

limits · Bidding

Named alongside it

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

Exhaustive searchBeatty sequenceGolden ratioHot positionMexTemperatureWythoff's gameClosed formComparisonDay threeGrundy valueHot game

All concepts