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.
Input: n = 1 Output: true Explanation: Alice can remove 1 stone winning the game because Bob doesn't have any moves.
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).
Input: n = 4 Output: true Explanation: n is already a perfect square, Alice can win with one move, removing 4 stones (4 -> 0).
1 <= n <= 105
fromfunctoolsimportcacheclassSolution: squares= [1] @cachedefwinnerSquareGame(self, n: int) ->bool: whilen>self.squares[-1]: self.squares.append((len(self.squares) +1) **2) returnany(notself.winnerSquareGame(n-x) forxinself.squares[::-1] ifx<=n)