1025. Divisor Game

1025. Divisor Game (Easy)

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

Example 2:

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

 

Note:

1. 1 <= N <= 1000

Companies:
Visa

Related Topics:
Math, Dynamic Programming

Solution 1.

// OJ: https://leetcode.com/problems/divisor-game/
// Author: github.com/lzl124631x
// Time: O(N * sqrt(N))
// Space: O(N)
class Solution {
private:
    unordered_map<int, bool> m;
public:
    bool divisorGame(int N) {
        if (N == 1) return false;
        if (m.find(N) != m.end()) return m[N];
        bool ans = false;
        for (int x = sqrt(N); x >= 1 && !ans; --x) {
            if (N % x) continue;
            if (!divisorGame(N - x)) ans = true;
        }
        return m[N] = ans;
    }
};

Solution 2.

Denote F(N) as the result.

1. If F(N) = false, then F(N + 1) = true. Because for case N + 1, Alice can simply pick x = 1 to win.
2. If N is odd number, it only have odd factors. So after the first move, it will become even number.

• F(1) = false
• F(2) = true according to Rule #1.
• F(3) = false since 2 is the only possible situation for Bob and F(2) = true.
• F(4) = true according to Rule #1.
• F(5) = false since all even numbers (2 and 4) lead to true for Bob.
• ...

So we can see, even numbers always win; odd numbers lose.

// OJ: https://leetcode.com/problems/divisor-game/
// Author: github.com/lzl124631x
// Time: O(1)
// Space: O(1)
// Ref: https://leetcode.com/problems/divisor-game/discuss/274566/just-return-N-2-0-(proof)
class Solution {
public:
    bool divisorGame(int N) {
        return N % 2 == 0;
    }
};