Tic-Tac-Toe-like games
Most of us are familiar with Tic-Tac-Toe: players take turns placing X’s and O’s in a 3x3 square and the winner is the person who gets 3-in-a-row. Of course, it is possible that there is no winner!
Here are two variations on Tic-Tac-Toe:
Three-Dimensional Tic-Tac-Toe:
Think of three Tic-Tac-Toe boards stacked one on top of another. (You can draw this by placing the boards side by side on a piece of paper and thinking of them as top, middle and bottom.) The goal is still to be the first to get 3-in-a-row, but now the 3-in-a-row can be on any of the three levels, or between levels. For example, the two pictures below show 3-in-a-rows that use all three levels.
Top Middle Bottom Top Middle Bottom
How many different 3-in-a-rows are there?
Achi:
This is a game played by the Asante people of Ghana, West Africa. It is played on a board like the one to the right. Each player starts with four counters (like X’s and O’s) and takes turns placing them on the board as in Tic-Tac-Toe, with the goal of getting a 3-in-a-row. However, if the game is a draw after each has played their four counters, they take turns sliding a counter along the lines into the space left empty. The winner is the first player to get 3-in-a-row.
For example, if the players have played their counters as at the left – first player in gray and second in black – and the game is a draw, the first player then must slide the counter in the bottom right corner into the open (white) space, and then the second player slides one of his/her pieces, etc.
Go ahead and play a few games of either or both. As you figure out how you can win, you might consider the questions below:
1) Can the game end with no winner? (For Achi, if no one won after placing the counters, can the players always slide a counter until someone wins?)
2) Does either player have an advantage? If so, why? Do they have a strategy that guarantees they can win, no matter what the other player does? What does such a strategy look like?