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