Combinatorial Game Theory
who wins, and by how much
Concept

Infinitesimals — where it appears

The values smaller in size than every positive number, which decide a position once the points have all cancelled. They form an ordered family rather than a single size, so one of them can be enormous relative to another.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

One position, three ways of writing it, and only one of them adds. The same positions as a sentence about who wins, as a description of the position itself, and in the notation Winning Ways introduced. The first two columns carry identical information and support no operation whatever. The third column can be added — and the sums below it are values that no manipulation of the first two columns could reach, because two of these pairs start from the same two outcomes and finish differently.

The notation was the argument

Up, star and the brace form are not abbreviations for case analyses. They are the claim that these objects add — and the arithmetic they support is arithmetic that no table of outcomes could ever produce, because two positions with identical outcomes can have different sums.

history · Notation Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

A game where nobody can be ahead in moves

A blue stone beside a red one is a move for both players at once. So neither player can run out while the other still has something to do — and every value the game produces is smaller than every positive number, by the shape of the rule rather than by inspection.

positions · Clobber Three partizan positions against every nimber, and not one match. Sprague and Grundy give every impartial position a single number that is complete: two positions with the same value are interchangeable everywhere. The three positions here are partizan — the two players have different moves — and each is compared against every nimber up to eight. Nothing is equal to anything. The magenta cells are worse than inequality: a position confused with a nimber is not above it or below it either, so no ordering could rescue the substitution.

Where the impartial theory stops

Sprague–Grundy gives every impartial position one number, and the number is complete. The moment the two players have different moves no number works at all — not a harder one to compute, none — and three positions here are compared against every nimber to show it.

sums · Sprague–Grundy The days this site can compute, and the ones it cannot. Zero on the first day, ±1 on the second, and thereafter the simplest number in every remaining gap — the construction run by the game recursion, which produces only fractions with a power of two underneath however long it goes on. Below it, three objects the same recursion reaches when the stopping rule is removed, each written with its option set and the exact reason this site's machinery cannot hold it. They are named rather than drawn, which is the honest half of a figure-first collection.

The recursion this site cannot run

Remove the stopping condition from the construction and it reaches ω, its reciprocal, and one third — none of which this site's evaluator can represent, because it interns a position from a finite list of options. The figure draws what it computes and names what it cannot, which is where the boundary belongs.

values · Numbers A vocabulary that is not closed under its own arithmetic. Every pair of named values added together, with the answer sorted by whether it has a name. The named vocabulary covers every game born by day two and one in twenty-three born by day three, and coverage is the wrong measurement: the notation exists so that positions can be added. A sixth of the sums of two named values at day three cannot be written without opening a brace, and the first one to escape is a sum of two of the symbols anybody learns first.

Two names that add to nothing nameable

The special symbols reach one game in twenty-three at day three. Coverage is the wrong measurement. The notation exists so that positions can be added, and a sixth of the sums of two named values at day three cannot be written without opening a brace — starting with a sum of two of the six symbols anybody learns first.

history · Notation

Named alongside it

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

Canonical formDisjunctive sumNimberStar (∗)Up (↑)AdditivityComparisonNotationOutcome classAll-smallAtomic weightBirthday

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