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 integermaxChoosableIntegerand another integerdesiredTotal, determine if the first player to move can force a win, assuming both players play optimally.

You can always assume thatmaxChoosableIntegerwill not be larger than 20 anddesiredTotalwill 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.

S: 记忆化搜索 O(n!)

状态：当前还剩哪些数字可以选择，用一个int来表示，每个bit表示该数是否还可以选择

    bool canIWin(int maxChoosableInteger, int desiredTotal) {
        if ((1 + maxChoosableInteger) / 2 * maxChoosableInteger < desiredTotal) {
            return false;
        }
        if (desiredTotal <= 0) return true;
        int num = ~0;
        bool reach = false;
        return helper(num, maxChoosableInteger, desiredTotal, reach) && reach;
    }

    bool helper(int nums, int n, int target, bool& reach) {
        if (canWin.find(nums) != canWin.end()) return canWin[nums];
        for (int i = 1; i <= n; ++i) {
            if (nums & (1 << i)) {
                if (i >= target) {
                    canWin[nums] = true;
                    reach = true;
                    return true;
                } else {
                    int next = nums & ~(1 << i);
                    if (!helper(next, n, target - i, reach) && reach) {
                        return true;
                    }
                }
            }
        }
        canWin[nums] = false;
        return false;
    }
    unordered_map<int, bool> canWin;