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Puzzles where both players play optimally

  1. Nov 7, 2012 #1
    Puzzles like the following(they are actually programming questions) :-
    How do you solve such puzzles?
    Since they say that both play optimally does that mean given the conditions in the game(here lets say number N is given) we already know who will be the winner?
     
  2. jcsd
  3. Dec 18, 2012 #2
    Playing optimally means that both players will have insight into all possible next moves and will play in such a way to maximize their chance of winning without making a mistake. However, if all moves lead to the other player winning, then this player still has to play, and any move would be equivalent.

    In the first puzzle, a player loses when their number is 1. This means that the player at 2 can force their opponent to lose by subtracting 1, and a player at 3 will lose since they can only subtract 1. Similarly, a player at 4 can win by subtracting 1, and a player at 5 will lose.

    Note that all odd numbers only have odd factors, so a player at an odd number can only get to an even number since he or she can only subtract an odd number from odd. A player at an even number can then subtract 1 to force the opponent to get to an odd number.

    Thus if the game begins on an even number, Alice wins. If the game begins on an odd number, Bob wins.
     
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