<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:
        perimter = sum(matchsticks)
        max_side_length = perimter // 4
        if max_side_length * 4 != perimter:
            return False
        
        matchsticks.sort(reverse=True)
        @cache
        def dfs(idx: int, n1: int, n2: int, n3: int, n4: int) -> bool:
            if idx == len(matchsticks):
                if n1 == n2 == n3 == max_side_length:
                    return True
                return False
            if n1 + matchsticks[idx] <= max_side_length:
                if dfs(idx + 1, n1 + matchsticks[idx], n2, n3, n4):
                    return True
            if n2 + matchsticks[idx] <= max_side_length:
                if dfs(idx + 1, n1, n2 + matchsticks[idx], n3, n4):
                    return True
            if n3 + matchsticks[idx] <= max_side_length:
                if dfs(idx + 1, n1, n2, n3 + matchsticks[idx], n4):
                    return True
            if n4 + matchsticks[idx] <= max_side_length:
                if dfs(idx + 1, n1, n2, n3, n4 + matchsticks[idx]):
                    return True
        return dfs(0, 0, 0, 0, 0)
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