Divisor Game

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

Initially, there is a number n on the chalkboard. On each player's turn, that player makes a move consisting of:

• Choosing any x with 0 < x < n and n % x == 0.
• Replacing the number n on the chalkboard with n - x.

Also, if a player cannot make a move, they lose the game.

Return true if and only if Alice wins the game, assuming both players play optimally.

Example 1:

Input: n = 2
Output: true
Explanation: Alice chooses 1, and Bob has no more moves.

Example 2:

Input: n = 3
Output: false
Explanation: Alice chooses 1, Bob chooses 1, and Alice has no more moves.

Constraints:

• 1 <= n <= 1000

SOLUTION:

class Solution:
    def DG(self, n: int) -> bool:
        if n in self.cache:
            return self.cache[n]
        for x in range(1, n):
            if n % x == 0 and not self.DG(n - x):
                self.cache[n] = True
                return True
        self.cache[n] = False
        return False

    def divisorGame(self, n: int) -> bool:
        self.cache = {}
        return self.DG(n)