<p>In the &quot;100 game&quot; two players take turns adding, to a running total, any integer from <code>1</code> to <code>10</code>. The player who first causes the running total to <strong>reach or exceed</strong> 100 wins.</p>

<p>What if we change the game so that players <strong>cannot</strong> re-use integers?</p>

<p>For example, two players might take turns drawing from a common pool of numbers from 1 to 15 without replacement until they reach a total &gt;= 100.</p>

<p>Given two integers <code>maxChoosableInteger</code> and <code>desiredTotal</code>, return <code>true</code> if the first player to move can force a win, otherwise, return <code>false</code>. Assume both players play <strong>optimally</strong>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> maxChoosableInteger = 10, desiredTotal = 11
<strong>Output:</strong> false
<strong>Explanation:</strong>
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 &gt;= desiredTotal.
Same with other integers chosen by the first player, the second player will always win.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> maxChoosableInteger = 10, desiredTotal = 0
<strong>Output:</strong> true
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> maxChoosableInteger = 10, desiredTotal = 1
<strong>Output:</strong> true
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= maxChoosableInteger &lt;= 20</code></li>
	<li><code>0 &lt;= desiredTotal &lt;= 300</code></li>
</ul>

typedef int state;
class Solution {
    vector<int> dp; // store the value of each state, -1 if not visited, 0 if can't win, 1 if can win
public:
    bool dfs(state k, int n, int t) {
        if (dp[k] != -1) return dp[k] == 1;
        
        // won by previous turn
        if (t <= 0) return false;
        
        for (int i = 0; i < n; i++) {
            if ((k & (1 << i))) continue;
            state k_prime = k | (1 << i);
            if (!dfs(k_prime, n, t-i-1)) {
                dp[k] = 1;
                return true;
            }
        }
                                        
        dp[k] = 0;
        return false;
    }
    
    bool canIWin(int maxChoosableInteger, int desiredTotal) {
        int M = maxChoosableInteger;
        int T = desiredTotal;
        int sum = M*(M+1)/2; // sum of entire choosable pool

        // I just pick 1 to win
        if (T < 2) return true;

        // Total is too large, nobody can win
        else if (sum < T) return false;

        // Total happens to match sum, whoever picks at odd times wins
        else if (sum == T) return M%2;
        
        dp = vector<int>(1 << M, -1);
        return dfs(0, M, T);
    }
};
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