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18 September, 16:56

Is 4-heap Nim with heaps of sizes 22, 19, 14, and 11 balanced or unbalanced? Player I's first move is to remove 6 coins from the heap of size 19. What should player II's first move be?

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  1. 18 September, 17:21
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    Player II should remove 14 coins from the heap of size 22.

    Step-by-step explanation:

    To properly answer this this question, we need to understand the principle and what it is exactly is being asked.

    This question revolves round a game of Nim

    What is a game of Nim: This is a strategic mathematical game whereby, two opposing sides or opponent take turns taking away objects from a load or piles. On each turn, a player remove at least an object and may remove any number of objects provided they all come from the same heap/pile.

    Now, referring back to the question, we should first understand that:

    22₂ = 1 0 1 1 0

    19₂ = 1 0 0 1 1

    14₂ = 0 1 1 1 0

    11₂ = 0 1 0 1 1

    and also that the "bit sums" are all even, so this is a balanced game.

    However, after Player I removes 6 coins from the heap of size 19, Player II should remove 14 coins from the heap of size 22.
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