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Math Help - Picking Stones

  1. #1
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    Picking Stones

    Consider the following game: a pile of n stones is placed in front of two players. The players
    take turns removing 1, 2 or 3 stones from the pile. The player who removes the last stone
    loses the game. Prove that if n gives a remainder of 1 when divided by 4 then the second
    player has a winning strategy, i.e. can always win the game no matter how the first player
    plays. Also prove that in all other cases the first player has a winning strategy.
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  2. #2
    Senior Member Shanks's Avatar
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    If n leaves a remainder of 1 divided by 4:
    then if the first player pick k stones, the second player can 4-k to make the number of rest stone always leaves a remainder of 1 divided by 4. the second player always win the game.
    In other case,
    if n leaves a remainder of r (r=2,or 3) divided by 4:
    the first player pick r-1 stones at the first step. and then If the second pick i stones, he will pick 4-i stone to make the number of rest stones always leaves a remainder of 1. the first player win the game!
    if n leaves a remainder of 0 divided by 4, then the first player will pick 3 stones, after that he adopt the similar strategy mentioned above, he always win the game!
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