In another, any three spaces constitute a winning position. One of those is equivalent to tic tac toe on a torus. You can also try and analyze tic tac toe on a cylinder, the real projective plane, and the Klein bottle. Having trouble visualizing this? See the links below. Shifting either of the two diagonal patterns in any of the four directions for either one or two squares will only yield four new patterns. Shifting a diagonal gives a new sort of pattern.Shifting a row just gives the same or another row, so there are no new winning patterns coming from rows.To find all winning patterns in torus tic tac toe, all you have to do is try shifting each of the ordinary winning patterns off each of the possible edges. In ordinary tic tac toe, there are three winning patterns: row, column, and diagonal. Instead of including an image, I think it’s more fun to leave exploration of this as an exercise. You can do the same thing with tic tac toe! In other words, in completing a line of three X’s (or O’s), you can walk off the edge of the board and finish on the other side. The game of Pac-Man is played on a (2-dimensional, real) torus. In Pac-Man, you’ve probably noticed that if you run off the edge of the screen, you reemerge on the opposite side.
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