### Sproutology

Sprouts is a simple and delightfully subtle pencil-and-paper game for two players. The game is set up by marking a number of spots on a page. Each player makes a move by drawing a curve that joins two spots, or that loops from a spot back to itself, without crossing any lines drawn earlier, and then marking a new spot on the curve.

A simple 2-spot game of Sprouts. The first player unable to draw a connecting line is the loser. (Image from Wikimedia Commons)

A maximum of three lines may link to a spot, and any spot with three lines is considered as dead, since it plays no further role in the game. Sooner or later, no further moves are possible and the player who draws the last line wins the game.

Sprouts was devised by two Cambridge mathematicians, John Horton Conway and Michael Stewart Paterson on Tuesday 21 February 1967. It has an addictive appeal, and it immediately became a craze, being played in mathematics departments around the world.

Despite the simple rules, the analysis of the game presents some challenges, and no general winning strategy is known. It is fairly easy to show that if there are n spots to start, the game will have at least 2n moves, and must end in at most 3n-1. Thus, with 8 spots to start, there will be between 16 and 23 moves.

The mathematics of Sprouts, which we might call sproutology, involves topology, a form of geometry that considers continuity and connectedness but disregards distances and shapes. Topology is often called rubber-sheet geometry since a figure drawn on an elastic sheet retains its topological properties when the sheet is stretched but not torn.

Sprouts is topological, since the precise positions of the spots is unimportant; it is only the pattern of connections between them that counts. The game exploits the Jordan curve theorem, which states that simple closed curves divide the plane into two regions. This apparently obvious result is actually quite difficult to prove.

The one-spot game of Sprouts is trivial: the first player must join the spot to itself and draw another spot; the second player then joins the two spots, winning the game. Games with a small number of starting spots have been fully investigated, and a pattern is evident: if the remainder when n is divided by 6 is 3, 4 or 5, the first player can force a win (assuming perfect play); otherwise, the second player has a winning strategy. This “Sprouts conjecture” remains unproven.

For up to seven spots to start, Sprouts can be checked by hand. But for larger numbers of spots it rapidly becomes too complex and a computer analysis is required. Recently, Julien Lemoine and Simon Viennot analysed games with up to 47 spots, and their findings support the Sprouts conjecture. Of course, the existence of a winning strategy does not guarantee a win.

Despite its elementary rules, Sprouts is surprisingly sophisticated, and prowess comes only with practice. You should start with a small number of spots, between 5 and 10, and gradually build up skill. But beware the addictive appeal of the game: you may well become a sproutaholic.