From Video Games to Tic-Tac-Toe

34. Hyperspace Hangout

From Video Games to Tic-Tac-Toe

The cube of your space capsule (see "Hyperspace Hangout") is a weird geometric object in which opposite sides appear connected through a higher dimension. Going through one side brings you back through the opposite side.

Many video games have a similar feature in order to keep a figure on the screen. When a game figure moves off the right side of the screen, it reappears on the left. When it moves off the top edge, it reappears at the bottom.

It's as if the screen were bent around to form a doughnut or torus.

You're probably used to playing games like tic-tac-toe on a flat surface. Imagine what would happen if a 3-by-3 tic-tac-toe game board were on the surface of a torus instead.

In the game shown below, neither X nor O has three squares in a row, so neither wins. If the board were turned into a torus, however, there would be new ways to have three in a row.

What if side A were glued to side B? Would that make X or O win the game?

TRY IT!

Neither X nor O is the winner in the standard tic-tac-toe game shown above. Figure out whether X or O would win if the tic-tac-toe board were turned into a cylinder or torus, and you applied the same rule of winning.

You will need:

• pencil
• sheet of paper
• ruler or straight edge (optional)

What to do:

1. Turn your sheet of paper into a big tic-tac-toe board so the edges of the game board are the edges of the sheet of paper.
2. Copy the game above onto your sheet of paper and label the sides A, B, C, and D, as shown.
3. Bend your sheet of paper into a tube so that side A meets side B, with the game facing outward.
4. Notice that three of the X's now form a diagonal line, so X wins the game!
5. In a torus, sides C and D also would meet. To see how the squares would line up, make your sheet of paper flat again, then bend it so C meets D. Look for three X's in a row or three O's in a row to determine the winner.

A tic-tac-toe board rolled around the outside surface of a cylinder.

Answers:

When sides A and B come together, X wins. The bottom row is beside the top row, as shown in the dotted squares at the top of the diagram below. Also, the top row is next to the bottom row, as shown in the dotted squares at the bottom of the diagram.

This creates a diagonal row of three X's.

Bringing sides C and D together also produces a diagonal row of three X's, as shown below.

See also "More than Just a Plane Game."

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