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59 lines
2.0 KiB
Python
59 lines
2.0 KiB
Python
"""
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Project Euler Problem 301: https://projecteuler.net/problem=301
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Problem Statement:
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Nim is a game played with heaps of stones, where two players take
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it in turn to remove any number of stones from any heap until no stones remain.
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We'll consider the three-heap normal-play version of
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Nim, which works as follows:
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- At the start of the game there are three heaps of stones.
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- On each player's turn, the player may remove any positive
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number of stones from any single heap.
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- The first player unable to move (because no stones remain) loses.
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If (n1, n2, n3) indicates a Nim position consisting of heaps of size
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n1, n2, and n3, then there is a simple function, which you may look up
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or attempt to deduce for yourself, X(n1, n2, n3) that returns:
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- zero if, with perfect strategy, the player about to
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move will eventually lose; or
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- non-zero if, with perfect strategy, the player about
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to move will eventually win.
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For example X(1,2,3) = 0 because, no matter what the current player does,
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the opponent can respond with a move that leaves two heaps of equal size,
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at which point every move by the current player can be mirrored by the
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opponent until no stones remain; so the current player loses. To illustrate:
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- current player moves to (1,2,1)
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- opponent moves to (1,0,1)
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- current player moves to (0,0,1)
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- opponent moves to (0,0,0), and so wins.
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For how many positive integers n <= 2^30 does X(n,2n,3n) = 0?
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"""
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def solution(exponent: int = 30) -> int:
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"""
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For any given exponent x >= 0, 1 <= n <= 2^x.
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This function returns how many Nim games are lost given that
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each Nim game has three heaps of the form (n, 2*n, 3*n).
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>>> solution(0)
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1
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>>> solution(2)
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3
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>>> solution(10)
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144
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"""
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# To find how many total games were lost for a given exponent x,
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# we need to find the Fibonacci number F(x+2).
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fibonacci_index = exponent + 2
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phi = (1 + 5**0.5) / 2
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fibonacci = (phi**fibonacci_index - (phi - 1) ** fibonacci_index) / 5**0.5
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return int(fibonacci)
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if __name__ == "__main__":
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print(f"{solution() = }")
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