Start with two numbers; letís say 9 and 14. Each turn a player can subtract 1, 2 or 3 from one of the numbers. The one who makes the last move wins.
We find the winning strategy by finding the key positions. What are the key positions for a two numbers subtraction game ?
If one of the numbers reaches zero, the game would become a one number subtraction game. So the position (0, n) is a key position if and only if
††††††††††††††††††††††† n mod 4 = 0
What are the other key positions?
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Read on if you know what does n mod t = m mod t †mean.