<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 {
public:
    void helper(vector<int>& matchsticks, int target, int start, int cur_mask, unordered_set<int>& valid) {
        if (target == 0) {
            valid.insert(cur_mask);
            if (cur_mask == 0)
                cout << target << " " << start << " " << cur_mask << endl;
            return;
        }
        if (start >= matchsticks.size() || matchsticks[start] > target)
            return;
        if (start == 0 || matchsticks[start] != matchsticks[start - 1] || (cur_mask | (1 << (start - 1))) != 0) {
            helper(matchsticks, target - matchsticks[start], start + 1, (cur_mask | (1 << start)), valid);
        }
        helper(matchsticks, target, start + 1, cur_mask, valid);
    }

    bool makesquare(vector<int>& matchsticks) {
        int sum = 0, len = matchsticks.size();
        sort(matchsticks.begin(), matchsticks.end());
        for (int m : matchsticks)
            sum += m;
        if (sum % 4 != 0) {
            return false;
        }
        unordered_set<int> valid;
        helper(matchsticks, sum / 4, 0, 0, valid);
        if (valid.size() < 4)
            return false;
        unordered_set<int> valid_2;
        for (int edge1 : valid) {
            for (int edge2 : valid) {
                if ((edge1 & edge2) == 0) {
                    valid_2.insert(edge1 | edge2);
                }
            }
        }
        for (int edge1 : valid_2) {
            for (int edge2 : valid_2) {
                if ((edge1 & edge2) == 0) {
                    return true;
                }
            }
        }
        return false;
    }
};
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