Sprouts

Sprouts is an easy-to-play combinatorial game of mathematical interest created by John Conway.

Sprouts can be played with pencil and paper as follows:

• Start with a few points on a piece of paper. For a simple game, you can start with two points. For a more complex game, use 16 points, arranged in a 4 × 4 grid.
• Players take turns. For each turn, a player draws a line from any point to any point, and marks a new point somewhere on it. Note that the lines do not have to be straight and that a line can be drawn from one point back to the same point.
• No line can cross itself or any other line, or pass through a point.
• No point can have more than 3 lines connected to it.
• The winner is the last person able to play.

Some things you might want to think about:

• How long can a game continue? Could a game go on forever, or does it have to end sooner or later?
• Since sprouts is a combinatorial game and either one player or the other must win, either one player or the other must have a winning strategy. Which player is it? Does this change if you add or subtract dots?