mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-23 21:11:08 +00:00
24d3cf8244
* The black formatter is no longer beta
* pre-commit autoupdate
* pre-commit autoupdate
* Remove project_euler/problem_145 which is killing our CI tests
* updating DIRECTORY.md
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
59 lines
2.0 KiB
Python
59 lines
2.0 KiB
Python
"""
|
|
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() = }")
|