<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:
    bool canIWin(int maxChoosableInteger, int desiredTotal) {
        int sum = 0;
        for (int i = 0; i < maxChoosableInteger; i++) sum += (i+1);
        if (desiredTotal <= 0) return true;
        if (desiredTotal > sum) return false;

        map<int, bool> cache;

        return canWin(desiredTotal, 0, maxChoosableInteger, cache);
    }

    bool canWin(int desiredTotal, int pool, int poolSize, map<int, bool>& cache) {
        if (desiredTotal <= 0) return false; 

        for (int i = 0; i < poolSize; i++) {
            if ((pool & (1 << i)) == 0) {
                int nextPool = (pool | (1 << i));
                int nextDesiredTotal = desiredTotal - (i+1);
                int key = nextPool;
                if (cache.find(key) == cache.end()) 
                    cache[key] = canWin(nextDesiredTotal, nextPool, poolSize, cache);
                
                if (cache[key] == false) return true;
            }
        }

        return false;
    }
};
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