464. Can I Win

In the "100 game," two players take turns adding, to a running total, any integer from 1..10. The player who first causes the running total to reach or exceed 100 wins.

What if we change the game so that players cannot re-use integers?

For example, two players might take turns drawing from a common pool of numbers of 1..15 without replacement until they reach a total >= 100.

Given an integer maxChoosableInteger and another integer desiredTotal, determine if the first player to move can force a win, assuming both players play optimally.

You can always assume that maxChoosableInteger will not be larger than 20 and desiredTotal will not be larger than 300.

Example

Input:
maxChoosableInteger = 10
desiredTotal = 11

Output:
false

Explanation:
No matter which integer the first player choose, the first player will lose.
The first player can choose an integer from 1 up to 10.
If the first player choose 1, the second player can only choose integers from 2 up to 10.
The second player will win by choosing 10 and get a total = 11, which is >= desiredTotal.
Same with other integers chosen by the first player, the second player will always win.

Approach #1: DFS. [C++]

class Solution {
public:
    bool canIWin(int M, int T) {
        int S = M * (M + 1) / 2;
        return T < 2 ? true : S < T ? false : S == T ? M % 2 : dfs(M, T, 0);
    }
    
private:
    int m[1<<20] = {};
    
    bool dfs(int M, int T, int k) {
        if (T <= 0 || m[k]) return T > 0 && m[k]>0; 
        for (int i = 0; i < M; ++i) 
            if (!(k&1<<i) && !dfs(M, T-i-1, k|1<<i)) 
                return m[k] = 1;
        return !(m[k] = -1);
    }
};

　　

Analysis:

State of Game: initially, we have all M numbers [1, M] available in the pool. Each number may or may not be picked at a state of the game later on, so we have maximum 2^Mdifferent states. Note that M <= 20, so int range is enough to cover it. For memorization, we define int k as the key for a game state, where

• the i-th bit of k, i.e., k&(1<<i) represents the availability of number i+1 (1: picked; 0: not picked).

At state k, the current player could pick any unpicked number from the pool, so state k can only go to one of the valid next states k': 

• if i-th bit of k is 0, set it to be 1, i.e., next state k' = k|(1<<i).

Reference:

https://leetcode.com/problems/can-i-win/