Combinatorial Game Theory
who wins, and by how much
Generator

The first move, and who knows what about it

The first move, and who knows what about it
The first move, and who knows what about it. Every opening move in Sylver Coinage up to sixteen, with what is known about it and where that knowledge comes from. Blue wins, red loses, magenta is unknown; underneath each is whether this site computed the answer, cited it, or has none. The site settles exactly one of them by its own search, because after a single number is named the position still has infinitely many moves.

Every opening move in Sylver Coinage up to sixteen, with what is known about it and where that knowledge comes from. Blue wins, red loses, magenta is unknown; underneath each is whether this site computed the answer, cited it, or has none. The site settles exactly one of them by its own search, because after a single number is named the position still has infinitely many moves.

2 essays call sylver-openings. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

The gaps of ⟨5, 7⟩, which are the moves. A Sylver Coinage position drawn as the numerical semigroup it is. Gold squares are the numbers already named; plain squares are sums of them, and so cannot be named again; magenta squares are the gaps, which are exactly the legal moves. The largest gap is the Frobenius number, marked F — past it every integer is reachable, which is why the game has finitely many moves left and must end. Out in the world

The game that is a number system

In Sylver Coinage two players name integers and nobody may name a sum of what has already been named. Its positions are not boards — they are numerical semigroups, its termination is a theorem of Sylvester's from 1884, and the question of who wins after the opening move 16 has been worth a thousand dollars since 2017.

 Hydras, and how long each takes to kill. Six small hydras with the ordinal the termination proof assigns to each and the exact number of chops it takes to finish it. Two of them are not finished here: the fight is guaranteed to end and the machine runs out of memory long before it does, which is the gap between a termination proof and a bound. Where it stops

It ends, and nothing says when

The recursion this site runs needs every line of play to reach a position with no moves, and the condition is usually met by an obvious decreasing quantity. The hydra meets it with no such quantity anywhere: the tree grows at nearly every step and the fight ends regardless, because the only thing that decreases is an ordinal. A four-node hydra dies in twenty chops; one level deeper and 279 chops reach forty thousand nodes with no end in sight.

The whole library · The position index · The figures that play back