<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:
        total = sum(matchsticks)

        target = total //4
        if total %4 !=0:
            return False
        self.memo = {}
        buckets = [0] *4
        used = 0
        def backtrack(used, ib ):

            #used = 0
            if ib == 3:
                return True
            if str(used) in self.memo:
                return self.memo[str(used)]

            if buckets[ib] == target:
                res =  backtrack(used, ib+1)
                self.memo[str(used)] = res
                return res

            for i in range(len(matchsticks)):

                if (used >> i) & 1:
                    continue
                if buckets[ib] + matchsticks[i] > target:
                    continue
                buckets[ib] += matchsticks[i]
                
                used = used | (1 << i)

                if backtrack(used, ib):
                    self.memo[str(used)] = True
                    return True
                buckets[ib] -= matchsticks[i]
                used = used ^ (1 << i)
            self.memo[str(used)]  = False
            return False
                

        ans = backtrack(used, 0)
        return ans


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