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.

## Comments