Question

A subtraction game Subtraction games are two-player games in which there is a pile of objects, say coins. There are two players, Alice and Bob, who alternate turns subtracting 4.9. A SUBTRACTION GAME 19 from the pile some number of coins belonging to a set S (the subtraction set). Alice goes first. The first player who is unable to make a legal move loses. For example, suppose the initial pile contains 5 coins, and each player can, on his turn, remove any number of coins belonging to the set S = {1, 2, 3}. Who wins? Alice goes first. On her turn she removes 1, 2, or 3 coins from the pile. If she removes 3, then the game reduces to a 2-coin game with Bob going first. Bob wins on his next move. Similarly, if she removes 2, then the game reduces to a 3-coin game with Bob going first, and Bob wins on his next move. But, if she removes 1, then the game reduces to a 4-coin game with Bob going first, and no matter what move Bob makes, Alice wins on her next move. In any subtraction game, the winner can be determined if we know how many coins are in the pile, and which player is next to play. Suppose there are n coins in the pile. If the player next to play take some coins and leave a position from which the opponent (who becomes the player next to play) has no winning strategy, then he can win. If he can not do this, then every legal move leaves a position from which the opponent has a winning strategy, and so the player whose turn it is cannot win.

Consider the following variation of the subtraction game. Each player can (only) remove 1 or 2 coins from the pile on their turn and, now, the last player to move loses (i.e., the player who takes the last coin loses). Suppose Alice and Bob play this game starting from a pile of 15 coins, and Alice makes the first move. Who wins?

Answer 1

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hank you