<p>You are given an integer array <code>matchsticks</code> where <code>matchsticks[i]</code> is the length of the <code>i<sup>th</sup></code> matchstick. You want to use <strong>all the matchsticks</strong> to make one square. You <strong>should not break</strong> any stick, but you can link them up, and each matchstick must be used <strong>exactly one time</strong>.</p>

<p>Return <code>true</code> if you can make this square and <code>false</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/09/matchsticks1-grid.jpg" style="width: 253px; height: 253px;" />
<pre>
<strong>Input:</strong> matchsticks = [1,1,2,2,2]
<strong>Output:</strong> true
<strong>Explanation:</strong> You can form a square with length 2, one side of the square came two sticks with length 1.
</pre>

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

<pre>
<strong>Input:</strong> matchsticks = [3,3,3,3,4]
<strong>Output:</strong> false
<strong>Explanation:</strong> You cannot find a way to form a square with all the matchsticks.
</pre>

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

<ul>
	<li><code>1 &lt;= matchsticks.length &lt;= 15</code></li>
	<li><code>1 &lt;= matchsticks[i] &lt;= 10<sup>8</sup></code></li>
</ul>

class Solution:
    def makesquare(self, matchsticks: List[int]) -> bool:
        totalSum = sum(matchsticks)
        if totalSum % 4 != 0:
            return False
        self.matchsticks, self.subSum = matchsticks, totalSum//4
        self.memo = {}
        return self.recurse(0, 0, self.subSum, (1<<len(matchsticks))-1)
    
    def recurse(self, index: int, iteration: int, curSum: int, bitMask:int) -> bool:
        if iteration == 3:
            return True
        if index == len(self.matchsticks) or curSum < 0:
            return False
        if bitMask in self.memo:
            return self.memo[bitMask]

        if self.recurse(index+1, iteration, curSum, bitMask):
            self.memo[bitMask] = True
            return True
        
        isTaken = bitMask & (1<<index) == 0
        result = False
        if not isTaken and curSum >= self.matchsticks[index]:
            newMask = bitMask ^ (1<<index)
            if curSum - self.matchsticks[index] == 0:
                result |= self.recurse(0, iteration+1, self.subSum, newMask)
            else:
                result |= self.recurse(index+1, iteration, curSum-self.matchsticks[index], newMask)
        self.memo[bitMask] = result
        return result
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