Combinatorial Game Theory
who wins, and by how much
State the winner, then lose

The figures that play back

The positions on this site small enough to be solved completely. Each one names its winner before a move is made, and every reply it can give was worked out in advance — so a reader who beats one has found a mistake in a theorem, not in a search.

Most game programs are heuristic: they look ahead as far as they can afford and guess about the rest. Nothing here does that. Each of these positions is small enough to solve completely, so the value is exact, optimal play falls straight out of the value, and the guarantee is a theorem rather than a statistic.

There are 20 such figures, across 20 essays, and between them 645 positions were solved before the page was served. Every one of those answers is in the payload the browser receives, which is what the claim rests on: the figure is following the analysis, not running a search.

Play corroborates and never carries. The position, its value and its outcome are in the server-rendered picture, so a reader with scripting off loses only the chance to test the claim. Nothing plays itself, nothing animates, and the winner is always named first — a figure that let a reader play and then announced the result would be a game rather than an argument.

EssayGameWorth OutcomePositions solved
"Left wins" has no short proof Domineering on 3×4 — and who wins −3/2 R 20
A game older than the theory Kōnane from oxo.oxo.oxo, and who wins 0 P 14
A game where nobody can be ahead in moves Clobber on 2×3 — and who wins 0 P 15
A position reached eleven ways is one position Domineering on 3×4 — and who wins −3/2 R 20
A row of coins is already a sum Nim from 3, 5, 6 — and who wins 0 P 70
Domineering Domineering on 3×4 — and who wins −3/2 R 20
Hackenbush is a numeral Hackenbush from LLRL and RRRL — and who wins −3/4 R 10
Nim, and the nim-sum Nim from 1, 2, 3 — and who wins 0 P 14
One row of Clobber Clobber on 1×6 — and who wins 0 P 9
Taking from several heaps at once Nim from 3, 5, 6 — and who wins 0 P 70
The digits say which move wins Wythoff from (5, 8) — and who wins ∗2 N 121
The move that gives counters back Nim from 1, 2, 3 — and who wins 0 P 14
The same strip without the jump The same strip with no jumping, T.FT.F — and who wins 0 P 2
The strip nobody has a formula for Toads and Frogs on T.FT.F — and who wins 0 P 7
The strip where every number is a whole one The same strip with no jumping, TT..FF — and who wins 0 P 2
The theorem that needed none of the theory Nim from 3, 5, 7 — and who wins ∗1 N 88
Three different claims are all called solved Clobber on 2×3 — and who wins 0 P 15
Toads and Frogs Toads and Frogs on T.FT.F — and who wins 0 P 7
Who moves last Nim from 2, 2 — and who wins 0 P 6
Wythoff's game, and the ratio nobody put there Wythoff from (5, 8) — and who wins ∗2 N 121

What playing costs a reader who does not

Nothing. That is the rule these figures are built to, and it is the same bargain the rest of the site makes with its pictures: the argument is in the SVG, and the interaction is a way of testing it rather than a way of receiving it. The script that drives all of these is requested only by pages that carry a playable figure — six essays of the 466 here — because a landing page full of thumbnails has no business paying for it.

Only games small enough to solve completely appear. A figure that played well without being provably optimal would be making a much weaker claim, and this site does not make it.

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