<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 __init__(self):
        self.used = None
        self.memo = None
        self.target = 0

    def makesquare(self, matchsticks: List[int]) -> bool:
        if not matchsticks: return False

        if 4 > len(matchsticks): return False

        sum_ = sum(matchsticks)

        if sum_ % 4 != 0: return False

        self.target = sum_ // 4
        self.used = [0] * len(matchsticks)
        self.memo = {}

        return self.backtrack(0, 0, 0, matchsticks)

    def backtrack(self, count:int, sidelen:int, start:int, matchsticks:List[int]) -> bool:
        if count == 4:
            return True
        
        state = ''.join(map(str, self.used))

        if state in self.memo:
            return self.memo[state]
        
        found = False

        for i in range(start, len(matchsticks)):
            if found:
                break

            if self.used[i]:
                continue
            
            if matchsticks[i] + sidelen > self.target:
                continue
            
            self.used[i] = 1
            sidelen += matchsticks[i]

            if sidelen == self.target:
                # one side ready
                found |= self.backtrack(count + 1, 0, 0, matchsticks)
            else:
                found |= self.backtrack(count, sidelen, i + 1, matchsticks)
        
            self.used[i] = 0
            sidelen -= matchsticks[i]

        # state = ''.join(map(str, self.used))
        self.memo[state] = found

        return found

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