# 1510. Stone Game IV

#### Hard

***

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`

```python
class Solution:
    def winnerSquareGame(self, n: int) -> bool:
        return self.recursion(n)
    
    @lru_cache(None)
    def recursion(self, n):
        if n == 0:
            return False
        for i in range(1, int(sqrt(n)) + 1):
            if not self.recursion(n-pow(i,2)): # Since We can remove sqaure of numbers
                return True
        return False
```