README.md

1510. Stone Game IV

Alice and Bob take turns playing a game, with Alice starting first.

Initially, there are n stones in a pile. On each player's turn, that player makes a move consisting of removing any non-zero square number of stones in the pile.

Also, if a player cannot make a move, he/she loses the game.

Given a positive integer n, return true if and only if Alice wins the game otherwise return false, assuming both players play optimally.

Example 1:

Input: n = 1
Output: true
Explanation: Alice can remove 1 stone winning the game because Bob doesn't have any moves.

Example 2:

Input: n = 2
Output: false
Explanation: Alice can only remove 1 stone, after that Bob removes the last one winning the game (2 -> 1 -> 0).

Example 3:

Input: n = 4
Output: true
Explanation: n is already a perfect square, Alice can win with one move, removing 4 stones (4 -> 0).

Constraints:

• 1 <= n <= 105

Solutions (Python)

1. Solution

from functools import cache


class Solution:
    squares = [1]

    @cache
    def winnerSquareGame(self, n: int) -> bool:
        while n > self.squares[-1]:
            self.squares.append((len(self.squares) + 1) ** 2)

        return any(not self.winnerSquareGame(n - x) for x in self.squares[::-1] if x <= n)