For each player, the first move is to pick one of the corners of the triangles and draw a path along the triangle's edge, connecting that corner to an adjacent corner. Thereafter, each player moves by connecting the last corner they landed on to an adjac ent corner. The first player to land on a corner that is already occupied loses.

Ex: Each player has made a total of four moves. The S shows where each player started.

Variation: Glue one, two, or three pairs of opposite sides together (see game boards and Dots and Boxes). What new moves does this allow?

Game Board 1: Snakes on a 'torus'.

Game Board 2

: Snakes on a `projective plane'

Game Board 3: Snakes on a 'Klein bottle'

Game Board 4: ?????

Surprise!

If we can stretch, bend, and twist Board 4 any way we like, then we can make it look like one of the other boards. These two boards are called topologically equivalent. Can you tell which board is topologically equivalent to board 4? (Hint: Some traits about the boards won't change no matter how they are bent and stretched. Thus, boards which are equivalent will share these traits in common. Look for these traits in how the sides are glued and how the games are played.)

Further Variations: Fill some squares with isoceles right triangles and glue one or two pairs of opposite sides together. (Try to make all possible combinations.) Play on these squares. Are any of the squares topologically equivalent to the above game boards?