<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>

class Solution {
public:
    // Helper function to check if the current player can win
    bool sol(int maxChoosableInteger, int desiredTotal, int usedNumbers, vector<int>& dp) {
        if (desiredTotal <= 0) {
            return false; // If the current desiredTotal is <= 0, the current player loses
        }
        if (dp[usedNumbers] != -1) {
            return dp[usedNumbers]; // If this state has been computed before, return the result
        }
        
        // Try picking each number
        for (int i = 1; i <= maxChoosableInteger; i++) {
            int currentBit = (1 << (i - 1)); // Use bitmask to check if number 'i' is picked
            if ((usedNumbers & currentBit) == 0) { // If number 'i' hasn't been picked
                // If the opponent loses after we pick 'i', we win
                if (!sol(maxChoosableInteger, desiredTotal - i, usedNumbers | currentBit, dp)) {
                    dp[usedNumbers] = 1;
                    return true; // We found a way to win
                }
            }
        }
        
        dp[usedNumbers] = 0; // No winning strategy found
        return false;
    }
    
    bool canIWin(int maxChoosableInteger, int desiredTotal) {
        // Edge cases
        if (desiredTotal <= 0) {
            return true;
        }
        int totalSum = (maxChoosableInteger * (maxChoosableInteger + 1)) / 2;
        if (totalSum < desiredTotal) {
            return false; // If the sum of all numbers is less than desiredTotal, no one can win
        }
        
        vector<int> dp(1 << maxChoosableInteger, -1); // Use a bitmask to represent used numbers
        return sol(maxChoosableInteger, desiredTotal, 0, dp);
    }
};

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