Combinatorial Game Theory
who wins, and by how much
Concept

Brussels Sprouts — where it appears

The pencil game whose length is fixed by its starting position, so the winner is settled before anybody draws. Its length being fixed makes it a parity puzzle rather than a game, which is what separates it from Sprouts.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

One of these is a game. Every length that came out of 40 random games from each starting position. Brussels Sprouts always ends after exactly five crosses less two moves, so whoever is to move at that point was decided before the first curve was drawn. Sprouts ends at different lengths depending on how it is played, which is what makes it worth playing.

Sprouts, and the game that is not one

Two games played with dots and curves, invented in the same room, all but indistinguishable on paper. One is unsolved past forty spots. The other has no decisions in it at all — the winner is fixed before the first curve is drawn.

positions · Sprouts 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.

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.

limits · Termination 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.

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.

history · Sprouts How far a plain search gets. An exhaustive search of Sprouts and Brussels Sprouts, run on this site, with the number of positions each size costs. Sprouts settles at three spots and Brussels Sprouts at two crosses; the published results on Sprouts go to forty-seven.

What computing further has bought

Sprouts has been searched harder and longer than almost any game, and the period-six pattern has survived every extension. This site's own exhaustive search settles three spots; the published results reach forty-seven, and the gap is not a gap in hardware — the gentler of the two measured growth factors puts forty-seven spots at ten to the hundred and twenty-fifth positions. Beside it sits Brussels Sprouts, which has five million positions holding a choice and not one choice that changes who wins.

history · Sprouts

Named alongside it

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

Exhaustive searchOutcome classSproutsCertificateComplexityEuler formulaIntractableNimPeriodicityPlanar mapTerminationBirthday

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