Game-it-Yourself! -- Sprouts

Sprouts

Designers: John Conway, Michael Paterson
Type: Abstract, Pen and Paper
Status: Public domain
BGG entry: http://www.boardgamegeek.com/game/7378
Available from: http://orion.math.iastate.edu/mathnight/activities/other/ppg/sprouts.shtml

From the BGG entry:

Sprouts is a two-player game, invented by John Conway (the creator of the game of Life) and Michael Paterson, while they were at the University of Cambridge (United Kingdom).

The rules are:

In the beginning - a few spots are drawn on the paper;

On every move, the player must connect two spots (or one spot to itself) with curve, which doesn't intersect other curves. After that, the player must set a new spot on just-drawn curve. To every spot can be attached maximum 3 curves.

The player who on his/her turn doesn't have a valid move, loses the game.

An animated example is available here: http://orion.math.iastate.edu/mathnight/activities/other/ppg/sprouts.shtml