Python/project_euler/problem_301/sol1.py

"""
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() = }")