<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 sum(matchsticks) % 4: return False 

        
        quarter = sum(matchsticks)//4
        
        dp = {}

        matchsticks.sort(reverse=True)


        def dfs(index,sides):
            if index == len(matchsticks): 
                return all(side==quarter for side in sides)
            
            state = (index,tuple(sides))
            
            if state in dp: 
                return dp[state]
            
            for j in range(4):

                if sides[j] + matchsticks[index] <= quarter: 
                    sides[j] += matchsticks[index]
                    if dfs(index+1,sides): 
                        return True 
                    sides[j] -= matchsticks[index]

                # if index == 0:
                #     break

            dp[state] = False
        
            return False 
        
        return dfs(0,[0]*4)

      
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