# Solution to puzzle 16: Zero-sum game

Two players take turns choosing one number at a time (without replacement) from the set {−4, −3, −2, −1, 0, 1, 2, 3, 4}. The first player to obtain three numbers (out of three, four, or five) which sum to 0 wins.

Does either player have a forced win?

Consider a 3×3 magic square, wherein all of the rows, columns, and diagonals sum to 0; example below. It's not difficult to see that the aim of the game, as stated, can be satisfied if, and only if, the three integers fall in the same row, column, or diagonal.

Hence the game is equivalent to tic-tac-toe, or noughts and crosses, a game which, with best play, is well known to be a draw.

Therefore neither player has a forced win.

## Further reading

- How many games of Tic-Tac-Toe are there?

Source: Traditional

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