Combinatorial Game Theory
who wins, and by how much
Generator

One Brussels game from 2 crosses, counted

One Brussels game from 2 crosses, counted
One Brussels game from 2 crosses, counted. One randomly played Brussels game, with the map counted after every move. Joining two arms uses two up and the new crossbar puts two back, so the number of free arms never moves. Every move either cuts a face in two or joins two separate pieces of the drawing, never both — so the faces and the piece count together rise by exactly one a move, and the faces alone do not. The game stops when every face holds one arm, and nothing a player does changes when that happens.

One randomly played Brussels game, with the map counted after every move. Joining two arms uses two up and the new crossbar puts two back, so the number of free arms never moves. Every move either cuts a face in two or joins two separate pieces of the drawing, never both — so the faces and the piece count together rise by exactly one a move, and the faces alone do not. The game stops when every face holds one arm, and nothing a player does changes when that happens.

2 essays call sprouts-bookkeeping. 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.

Poker Nim from 3, 5, 7, with reserves of 4 and 4. Nim with one extra kind of move: a player may put any number of counters back onto a heap from a private reserve. It looks as though a losing player could stall for ever. They cannot, and the winner is decided by exactly the same nim-sum as ordinary Nim — checked here over every position within a stated range rather than argued. Where it stops

The condition the recursion rests on

Not that the moves run out, and not that the options are few. Poker Nim's heaps can grow without bound and it ends; the game called `on` has one option and never does. What every value on this site needs is that no infinite run of moves exists — and there are three separate ways to fail it.

 One Sprouts game from 3 spots, counted. One randomly played Sprouts game, with the map counted after every move. A move spends two lives and the new spot brings one, so the lives fall by exactly one every time — and unlike the arms of a Brussels cross they are not replaced. Every move either cuts a face in two or joins two separate pieces of the drawing, and how many of each a game contains is up to the players, which is why the length is not fixed. How it was found

A conjecture from hand play

Sprouts was invented over tea and its outcome pattern was guessed from games played with a pencil. Computers have checked it far past where a person could go, and this site's own solver gives out at three spots — so the honest figure states the frontier it reaches rather than the number somebody else published.

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