Surfaces - Torus Tic-Tac-Toe

Think you've got Tic-Tac-Toe all figured out? ... Try Torus-Tic-Tac-Toe!

Glue each pair of opposite sides together, as shown to the left. Now there are more ways to get three in a row! (and more ways to lose -- keep your eyes open!) X wins in this torus tic-tac-toe game.

If you have a Web browser that runs Java applications (e.g. Netscape 3.0 or later), you can play Tic-Tac-Toe and other games on a torus or a Klein bottle at http://www.geometrygames.org/TorusGames/

Play a few games of tic-tac-toe on a torus:

Now try it on a Klein bottle. This is a bit trickier because of the flip.

Questions:

1. Does it matter which player goes first? Why or why not?
2. Is there a "best place" for the first marker? Where or why not?
3. For each possible first move, how many essentially different second moves are there?
4. Can you come up with a strategy so that you will always win?
5. Can there ever be a draw in this game?