Combinatorial Game Theory
who wins, and by how much
Concept

Encoding — where it appears

How a position is written down, which decides whether a game counts as easy and is not a property of the game. Nim is easy in binary and hard if the heaps are written in unary, which is a fact about the writing rather than about the game.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The same position, and two rulers to measure it with. Nim positions with the length of their input under two encodings — the heap sizes in binary, and the counters themselves — beside the work the nim-sum does. The work never changes. Which of the two lengths it is compared against decides whether the same algorithm reads as linear or as exponentially fast, and hardness claims are always made against one particular encoding.

Nim is easy, in binary

Three heaps of a thousand counters take thirty bits to write down and three thousand counters to lay out. The nim-sum does three exclusive-ors either way. Whether that counts as fast depends entirely on which of the two numbers the work is compared against.

complexity · Complexity 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 Every heap up to 40, won or lost. Heap sizes with the outcome for the player who moves first. The lost ones are shaded; they are exactly the Fibonacci numbers, which is a fact about a game with one heap, no board and no geometry in it anywhere.

The heap is not the position

Fibonacci Nim bounds a move by twice the previous move, which puts the state outside the board: a heap of six with a cap of two and a heap of six with a cap of five are different games. So there is nothing to add and no Grundy value to compute — and the game is completely solved anyway. The opener loses on exactly the nine Fibonacci numbers up to 120, and the smallest term of the Zeckendorf numeral is a winning move in all 110 winnable heaps.

impartial · Fibonacci nim A turn is not a bit. The number of turns a game lasts, beside the number of quantified bits those turns amount to. Each ply is measured over the positions actually reachable at it rather than along one line, and the bits are the logarithm of the branching, which is what a quantifier prefix would need one of.

A turn is not a bit

The prefix a game is read as gives each player one quantifier a turn, and a turn on a board is a choice among however many moves there are. Nim on heaps of 3, 4 and 5 lasts twelve moves and carries 23.6 bits of choice; a Toads and Frogs strip lasts eleven and carries two. Corrected for that, the model predicts a strategy 539 times too large on one board and 67 times too small on another, and the two failures have different causes.

complexity · Alternation

Named alongside it

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

Exhaustive searchGrundy valueBeatty sequenceClosed formComplexityGolden ratioZeckendorf representationAlternationBinaryCertificateCold positionComponent

All concepts