Combinatorial Game Theory — who wins, and by how much An illustrated collection of essays about the theory of two-player games with no chance and no hidden information — values, sums, temperature and the games that are not numbers. Every value is computed, and the figures play back. https://www.combinatorial-game-theory.com/ 2026-08-04T00:30:24.932Z Who moves last https://www.combinatorial-game-theory.com/essays/who-moves-last/ 2026-08-04T00:30:24.932Z The player who cannot move loses. That single convention generates the whole theory — and it produces four outcomes rather than three, because a position can be confused with zero rather than greater, smaller or equal to it. Nim, and the nim-sum https://www.combinatorial-game-theory.com/essays/nim-and-the-nim-sum/ 2026-08-04T00:30:24.932Z Three heaps of counters, take as many as you like from one of them, and the player who takes the last counter wins. The winning condition is not a search, not a table, and not a heuristic — it is the bitwise exclusive-or of the heap sizes, and it was found in 1901. The sum is the object https://www.combinatorial-game-theory.com/essays/the-sum-is-the-object/ 2026-08-04T00:30:24.932Z Real positions come apart into independent regions, and a move happens in exactly one of them. That operation — the disjunctive sum — is what the whole theory is built to survive, and it is the reason values exist at all. What is at stake https://www.combinatorial-game-theory.com/essays/what-is-at-stake/ 2026-08-04T00:30:24.932Z Some positions both players are desperate to move in, and some neither player wants to touch. The difference is a number — how much the move is worth — and it turns out to be the most useful single quantity for deciding where to play. Hackenbush is a numeral https://www.combinatorial-game-theory.com/essays/hackenbush-is-a-numeral/ 2026-08-04T00:30:24.932Z Draw a stalk of coloured edges. Read it as a string, blue for one and red for zero, and the string is the binary expansion of what the position is worth. Not approximately — exactly, and the site computes it both ways and refuses to build if they disagree. Misère play https://www.combinatorial-game-theory.com/essays/misere-play/ 2026-08-04T00:30:24.932Z Change one word — the player who cannot move wins — and the games are identical, the strategies are not, and almost every theorem on this site stops being true. It is the cheapest possible modification and the most expensive. Every impartial game is a Nim heap https://www.combinatorial-game-theory.com/essays/sprague-grundy/ 2026-08-04T00:30:24.932Z Sprague and Grundy proved, independently and four years apart, that any position in any impartial game is equivalent to a single heap of counters. Not similar to one — equal to one, interchangeable with it inside any larger game. The simplicity rule https://www.combinatorial-game-theory.com/essays/the-simplicity-rule/ 2026-08-04T00:30:24.932Z When both players' options are numbers, the position is worth the simplest number strictly between them. Not the midpoint, not the average, and the difference between "simplest" and "middle" is the entire content of the rule. Comparing positions https://www.combinatorial-game-theory.com/essays/comparing-positions/ 2026-08-04T00:30:24.932Z One position is worth at least another when the second player wins their difference. That is the only definition there is, it is a computation rather than a judgement, and it produces an order in which some pairs are simply not comparable. Reading a thermograph https://www.combinatorial-game-theory.com/essays/reading-a-thermograph/ 2026-08-04T00:30:24.932Z A thermograph is two walls rising from a number line, closing in as the tax on moving increases, and meeting where the position stops being worth fighting over. Everything about a position's hotness is in the shape. Domineering https://www.combinatorial-game-theory.com/essays/domineering/ 2026-08-04T00:30:24.932Z One player places dominoes vertically, the other horizontally, on a shared grid. The rules take one line, the values are a mess, and that mess is the point — this is what the theory looks like applied to a game nobody designed for it. Loopy games https://www.combinatorial-game-theory.com/essays/loopy-games/ 2026-08-04T00:30:24.932Z The whole theory assumes play stops. Allow a position to recur and the induction that every value rests on has nothing to stand on — and a fifth outcome appears that normal-play theory has no name for. Grundy sequences, and where they stop being predictable https://www.combinatorial-game-theory.com/essays/grundy-sequences/ 2026-08-04T00:30:24.932Z Computing one Grundy value is a mex. Computing all of them produces a sequence, and the sequences do something nobody has fully explained — most of them eventually repeat, some of them take thousands of terms to start, and for a few nobody knows whether they ever do. Canonical form https://www.combinatorial-game-theory.com/essays/canonical-form/ 2026-08-04T00:30:24.932Z Two positions are worth the same when neither player can tell them apart inside any larger game. Deciding that could be an infinite search. Instead there is a normal form — delete what nobody would play, bypass what backfires — and equality becomes a comparison of two small trees. Outcomes do not add https://www.combinatorial-game-theory.com/essays/outcomes-do-not-add/ 2026-08-04T00:30:24.932Z Knowing who wins each part of a position tells almost nothing about who wins the whole. Two first-player wins can sum to a second-player win, or to another first-player win, and no rule distinguishes the cases from the outcomes alone. Playing the hottest https://www.combinatorial-game-theory.com/essays/playing-the-hottest/ 2026-08-04T00:30:24.932Z Given several independent fights, play in the one with most at stake. The rule is simple, it is what strong Go players do without being told, it is provably close to optimal — and it is provably not optimal, which is the interesting part. Toads and Frogs https://www.combinatorial-game-theory.com/essays/toads-and-frogs/ 2026-08-04T00:30:24.932Z Toads shuffle right, frogs shuffle left, and either may jump over one of the other. A strip six cells long is worth exactly up. Another six-cell strip is worth exactly down. Nobody has a formula for which. How hard is it https://www.combinatorial-game-theory.com/essays/how-hard-is-it/ 2026-08-04T00:30:24.932Z Every theorem on this site stays true at any size. The answers stop being reachable long before the games get interesting — deciding the winner of a generalised board game is PSPACE-complete, and an exact evaluator gives out after a few dozen moves. Infinitesimals https://www.combinatorial-game-theory.com/essays/infinitesimals/ 2026-08-04T00:30:24.932Z Some positions are positive — Left wins them whoever moves first — and smaller than every positive number, including a millionth and a millionth of that. They are the values that decide close games, and the smallest of them is a single move's worth of nothing.