1510-stone-game-iv

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Description

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

Solution(Python)

class Solution:
    #     @cache
    #     def winnerSquareGame(self, n: int) -> bool:
    #         if n == 0:
    #             return False

    #         for i in range(1,int(pow(n,0.5)+1)):
    #             if not self.winnerSquareGame(n-i*i):
    #                 return True
    #         return False
    # Time Complexity : N^(1+1/2)
    # Space Complexity: N
    def winnerSquareGame(self, n: int) -> bool:
        dp = [False] * (n + 1)

        for i in range(1, n + 1):
            for j in range(1, int(pow(i, 0.5) + 1)):
                if not dp[i - j * j]:
                    dp[i] = True
                    break
        return dp[n]