A math game of dodgeball

Let’s analyze a math game called dodgeball that’s a sort of twist on tic-tac-toe.

Here is how the game works. It’s a two player game with the following set-up.

Player 1 gets a 6×6 grid of squares as follows:

Player 2 gets a 6×1 grid of squares:

Here are the rules:

1. Player 1 begins the game by filling out the entire first ROW of his 6×6 grid, marking each square with either an X or an O.

2. Player 2 then goes by marking the just first SQUARE in his 6×1 grid, with either an X or an O.

3. On each subsequent turn, player 1 fills out the entire next row of his 6×6 grid with any combination of X’s and O’s. In turn, player 2 marks the next square of his 6×1 grid.

4. The game ends on the sixth round when both players have filled out their grids.

At this point, notice player 2′s grid has six squares filled with X’s and O’s. Player 1 has six such rows in his grid.

The winner is decided as follows: if player 2′s grid exactly matches one of the six rows in player 1′s grid, then player 1 wins. Otherwise, player 2 wins the game.

If you were given the choice, would you rather be player 1 or player 2? What is your strategy to win the game? Is the strategy foolproof meaning it will guarantee a victory?

The answer

It turns out player 2 can always win the game because he goes second and has an advantage.

This is not a hard game, but I will explain how it is interesting mathematically.

How can player 2 guarantee he is making a sequence that is not the same as any of player 1′s rows?

Player 2 does the following: on turn n, he looks at what player 1 writes in for the n^th square of the current row. Then player 2 marks exactly the opposite. For example, if player 1 begins the game by writing X in the first square, then player 2 should write O for the first square, and vice versa.

Here is an example for the 6×6 grid. After player 1 writes out a row, player 2 looks at the appropriate square and marks in the opposite. Here is what the two grids look like when the game is complete.

We can readily see that player 2′s row does not match any of the rows in player 1′s grid.

The reason is that player 2′s sequence differs from row n on the n^th spot, and hence the sequence must be different from any of the rows that player 1 created.

The same argument can be used to show player 2 can win for a row of any size. It even works for an infinite size grid! So even when player 1 writes an infinite number of sequences, player 2 can still make a unique sequence.

If you understood the strategy in the game, then you might know it is actually based on a more interesting mathematical proof. This strategy is the same idea used in Cantor’s diagonalization argument. This is a math proof that demonstrates the set of irrational numbers is larger than the set of rationals, and hence “uncountably” large.

As you can see, game theory can make set theory a bit more interesting. If you like this post, you will also like a previous one I wrote about a game that proves the real numbers are uncountably large.