Nine Balls
Ann and Bob play a game that starts with the nine pool balls numbered 1 through 9. On each turn, a player takes one of the balls off of the pool table and keeps it. The first person who can make exactly three of their balls add up to 15 is the winner.
If Ann goes first, and both play optimally, who will win?
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1 solution
The balls can be arranged to form a magic square of side length 3, so its magic sum will be 15.
4 3 8 9 5 1 2 7 6
Every line in this square makes a sum of 15, and every possible sum of three numbers is represented by some line in the square, so there is a 1 to 1 correspondence between the two games.
This means that the described game is equivalent to a game of Tic-Tac-Toe on a standard 3 × 3 board. For this game, it has been proven that the game will always end in a draw .