<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:
        n = len(matchsticks)

        total_sum = sum(matchsticks)

        if total_sum % 4 != 0:
            return False
        
        target_sum = total_sum //4
        matchsticks.sort(reverse=True)

        used = ['0'] * n 
        memo = {}

        def backtrack(start,count,curr_sum):
            if count == 3:
                return True
            if  curr_sum > target_sum:
                return False

            used_str = "".join(used)
            if used_str in memo:
                return memo[used_str]

            if curr_sum == target_sum:
                memo[used_str] = backtrack(0,count+1, 0)
                return memo[used_str]

            for j in range(start,n):
                if used[j] == '0':
                    used[j] = '1'

                    if backtrack(j+1,count,curr_sum+matchsticks[j]):
                        return True
                    
                    used[j] = '0'

            memo[used_str] = False
            return memo[used_str]

        return backtrack(0,0,0)    
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