<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 not matchsticks:
            return False
        
        if sum(matchsticks) % 4 !=0:
            return False
        
        each_side = sum(matchsticks) // 4

        matchsticks.sort(reverse=True)
        
        memo = {}

        def backtrack (side1,side2,side3,side4,index):
            if index == len(matchsticks):
                if side1 == side2 == side3 == side4 == each_side:
                    return True
            
            key = (index,side1,side2,side3,side4)
            if key in memo:
                return memo[key]
            
            result = False

            if matchsticks[index] + side1 <= each_side:
                result = result or backtrack(matchsticks[index] + side1,side2,side3,side4,index+1)
            
            if not result and  matchsticks[index] + side2 <= each_side:
                result = result or backtrack( side1,matchsticks[index] +side2,side3,side4,index+1)
            
            if not result and matchsticks[index] + side3 <= each_side:
                result = result or backtrack( side1,side2,matchsticks[index] + side3,side4,index+1)
            
            if not result and matchsticks[index] + side4 <= each_side:
                result = result or backtrack( side1,side2,side3, matchsticks[index] + side4,index+1)
            
            memo[key] = result
            return memo[key]
        
        return backtrack(0,0,0,0,0)

    
      
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