<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:
        # if total sum not evenly divisible by 4, no solution possible
        # because we must use all matchsticks
        if sum(matchsticks) % 4 != 0:
            return False
             
        matchsticks = sorted(matchsticks)[::-1]

        cache = {}
        requiredLength = sum(matchsticks) // 4
        def backtrack(l1, l2, l3, l4, index):
            if index == len(matchsticks):
                return l1 == l2 == l3 == l4
            elif l1 > requiredLength or l2 > requiredLength or l3 > requiredLength or l4 > requiredLength:
                return False
            elif tuple(sorted([l1, l2, l3, l4])) in cache:
                return cache[tuple(sorted([l1, l2, l3, l4]))]

            s1 = backtrack(l1+matchsticks[index], l2, l3, l4, index+1)
            s2 = backtrack(l1, l2+matchsticks[index], l3, l4, index+1)
            s3 = backtrack(l1, l2, l3+matchsticks[index], l4, index+1)
            s4 = backtrack(l1, l2, l3, l4+matchsticks[index], index+1)

            sorted_sides = sorted([l1, l2, l3, l4])
            cache[tuple(sorted_sides)] = s1 or s2 or s3 or s4
            return s1 or s2 or s3 or s4
        return backtrack(0,0,0,0,0)
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