<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:
        s = sum(matchsticks)
        target = s//4
        if s % 4 != 0 or any(match > target for match in matchsticks): return False
        n = len(matchsticks)
        dp = {}
        def rec(mask, fin):
            tot = 0
            for i in range(n-1, -1, -1):
                if not (mask & (1<<i)):
                    tot += matchsticks[n-1-i]
            if tot > 0 and tot % target == 0:
                fin += 1
            if fin == 3:
                return True
            if (mask, fin) in dp:
                return dp[(mask, fin)]
            ans = False
            rem = target*(tot//target + 1) - tot
            for i in range(n-1, -1, -1):
                if matchsticks[n-1-i] <= rem and mask & (1<<i):
                    if rec(mask ^ (1 << i), fin):
                        ans = True
                        break
            dp[(mask, fin)] = ans
            return ans
        return rec((1 << n) - 1, 0)

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