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

Code

關鍵在於奇數的因數不會有偶數！

class Solution {
public:
    bool divisorGame(int n) {
        return n % 2 == 0;
    }
};

證明：

If N is even.
We can choose x = 1.
The opponent will get N - 1, which is a odd.
Reduce to the case odd and he will lose.
 
If N is odd,
2.1 If N = 1, lose directly.
2.2 We have to choose an odd x.
The opponent will get N - x, which is a even.
Reduce to the case even and he will win.
 
So the N will change odd and even alternatively until N = 1.

reference

Link

• Divisor Game