README.md

1799. Maximize Score After N Operations

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

In the ith operation (1-indexed), you will:

• Choose two elements, x and y.
• Receive a score of i * gcd(x, y).
• Remove x and y from nums.

Return the maximum score you can receive after performing n operations.

The function gcd(x, y) is the greatest common divisor of x and y.

Example 1:

Input: nums = [1,2]
Output: 1
Explanation: The optimal choice of operations is:
(1 * gcd(1, 2)) = 1

Example 2:

Input: nums = [3,4,6,8]
Output: 11
Explanation: The optimal choice of operations is:
(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11

Example 3:

Input: nums = [1,2,3,4,5,6]
Output: 14
Explanation: The optimal choice of operations is:
(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14

Constraints:

• 1 <= n <= 7
• nums.length == 2 * n
• 1 <= nums[i] <= 106

Solutions (Python)

1. Solution

import math
from functools import cache


class Solution:
    def maxScore(self, nums: List[int]) -> int:
        @cache
        def gcd(x: int, y: int) -> int:
            return math.gcd(x, y)

        n = len(nums) // 2
        pairmask = [[] for _ in range(n + 1)]
        dp = [0] * (1 << len(nums))

        for num in range(len(dp)):
            if bin(num).count('1') % 2 == 0:
                pairmask[bin(num).count('1') // 2].append(num)

        for i in range(1, n + 1):
            for j in range(len(nums)):
                x = nums[j]
                for k in range(j + 1, len(nums)):
                    y = nums[k]
                    for prevmask in pairmask[i - 1]:
                        mask = (1 << j) | (1 << k)
                        if prevmask & mask == 0:
                            dp[prevmask | mask] = max(
                                dp[prevmask | mask], dp[prevmask] + i * gcd(x, y))

        return dp[-1]