<p>You are given <code>nums</code>, an array of positive integers of size <code>2 * n</code>. You must perform <code>n</code> operations on this array.</p>

<p>In the <code>i<sup>th</sup></code> operation <strong>(1-indexed)</strong>, you will:</p>

<ul>
	<li>Choose two elements, <code>x</code> and <code>y</code>.</li>
	<li>Receive a score of <code>i * gcd(x, y)</code>.</li>
	<li>Remove <code>x</code> and <code>y</code> from <code>nums</code>.</li>
</ul>

<p>Return <em>the maximum score you can receive after performing </em><code>n</code><em> operations.</em></p>

<p>The function <code>gcd(x, y)</code> is the greatest common divisor of <code>x</code> and <code>y</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2]
<strong>Output:</strong> 1
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 2)) = 1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [3,4,6,8]
<strong>Output:</strong> 11
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5,6]
<strong>Output:</strong> 14
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 7</code></li>
	<li><code>nums.length == 2 * n</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution:
    def maxScore(self, nums: List[int]) -> int:
        def backtrack(idx):
            # If we've performed N operations, return 0
            if idx == n // 2: return 0            
            # Use a tuple of used indices as a key for memoization
            key = tuple(used)
            if key in memo: return memo[key]   
            max_result = 0            
            # Iterate through all pairs
            for i in range(n):
                if used[i]: continue
                for j in range(i + 1, n):
                    if used[j]: continue
                    
                    # Mark these indices as used
                    used[i] = used[j] = True                    
                    # Calculate score for this operation
                    current_score = math.gcd(nums[i], nums[j]) * (idx + 1)                    
                    # Recur for the next operation
                    max_result = max(max_result, current_score + backtrack(idx + 1))                    
                    # Backtrack (unmark the indices)
                    used[i] = used[j] = False            
            memo[key] = max_result
            return max_result

        n = len(nums)  # Number of operations
        max_score = 0
        memo = {}
        # Start the backtracking with all indices unused and operation index 0
        used = [False] * len(nums)
        max_score = backtrack(0)
        
        return max_score
1587