The game of sprouts is a two-person game invented by John Conway and Michael Paterson in 1967 (for some historical comments visit the encyclopedia). You just need pen and paper to play it. Here are the rules : Two players, Left and Right, alternate moves until no more moves are possible. In the normal game, the last person to move is the winner. In misere play, the last person to move is the loser. The starting position is some number of small circles called “spots”. A move consists of drawing a new spot g and then drawing two lines, in the loose sense, each terminating at one end at spot g and at the other end at some other spot. (The two lines can go to different spots or the same spot, subject to the following conditions.) The lines drawn cannot touch or cross any line or spot along the way. Also, no more than three lines can terminate at any spot. A spot with three lines attached is said to be “dead”, since it cannot facilitate any further action.

You can play sprouts online using this Java applet. There is also an ongoing discussion about sprouts on the geometry math forum. Probably the most complete information can be found at the world game of sprouts association. The analysis of the game involves some nice topology (the Euler number) and as the options for Left and Right are the same at each position it is an impartial game and the outcome depends on counting arguments. There is also a (joke) variation on the game called Brussels sprouts (although some people seem to miss the point entirely).

Some years ago I invented some variations on sprouts making it into a partizan game (that is, at a given position, Left and Right have different legal moves). Here are the rules :

Cold Antwerp Sprouts : We start with n White dots. Left is allowed to connect two White dots or a White and bLue dot or two bLue dots and must draw an additional Red dot on the connecting line. Right is allowed to connect two White dots, a Red and a White dot or two Red dots and must draw an additional bLue dot on the connecting line.

Hot Antwerp Sprouts : We start with n White dots. Left is allowed to connect two White dots or a White and bLue dot or two bLue dots and must draw an additional bLue dot on the connecting line. Right is allowed to connect two White dots, a Red and a White dot or two Red dots and must draw an additional Red dot on the connecting line.

Although the rules look pretty similar, the analysis of these two games in entirely different. On february 11th I’ll give a talk on this as an example in Combinatorial Game Theory. I will show that Cold Antwerp Sprouts is very similar to the game of COL, whereas Hot Antwerp Sprouts resembles SNORT.

apple, Conway, games, geometry, topology