<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>

from math import gcd
from collections import defaultdict

class Solution:
    def maxScore(self, nums: List[int]) -> int:
        cache = defaultdict(int)

        def dfs(mask, op):
            if mask in cache:
                return cache[mask]
            
            for i in range(len(nums)):
                for j in range(i + 1, len(nums)):
                    if (1 << i) & mask or (1 << j) & mask:
                        continue

                    newMask = mask | (1 << i) | (1 << j)
                    score = op * gcd(nums[i], nums[j])
                    cache[mask] = max(
                        cache[mask],
                        score + dfs(newMask, op + 1)
                    )
            return cache[mask]
        return dfs(0, 1)

        # my sol
        n = len(nums) // 2
        cache = {}

        def dfs(ops, chosen):
            if chosen in cache:
                return cache[chosen]
            if ops == n:
                return 0

            res = 0
            for i in range(len(nums)):
                for j in range(i+1, len(nums)):
                    if i == j or (chosen & (1 << i)) or (chosen & (1 << j)):
                        continue

                    # chosen[i] = True
                    # chosen[j] = True
                    new_mask = chosen + (1 << i)
                    new_mask += (1 << j)
                    res = max(
                        res,
                        (ops + 1) * gcd(nums[i], nums[j]) + dfs(ops + 1, new_mask)
                    )
                    # chosen[i] = False
                    # chosen[j] = False
                    # chosen -= (1 << i)
                    # chosen -= (1 << j)

            cache[chosen] = res
            return cache[chosen]

        return dfs(0, 0)

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