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The problem is to play a game with two players who take turns choosing numbers from a list. The player whose selected numbers sum up to the smallest value at the end wins. If both players’ sums are equal, the game is a draw.

However, rather than implementing the entire gameplay, the focus is on evaluating a given list of numbers and determining the minimum possible sum that a player can guarantee irrespective of how the other player plays optimally.

Given a list of positive integers, the objective is to find this minimal guaranteed sum.

Clarifying Questions

1. Can the same number be chosen by both players or is each number chosen only once?
   • Each number can only be chosen once.
2. How many numbers are typically in the list, or do we assume any length?
   • Assume any length within reasonable limits for practical computation.
3. Is there always an even number of elements in the list?
   • No, the list can contain an odd number of elements as well.

Strategy

Approach

1. Sort the List: Begin by sorting the list of numbers.
2. Optimal Play: Notice that the optimal strategy for minimizing the sum of one’s chosen numbers involves always picking the smaller available number at each turn. Since both players also want to minimize their numbers:
   • Simulate taking turns starting from the smallest numbers.
   • The minimum guaranteed sum can be formed based on these turns.

Given an example:

• Suppose the list is [1, 3, 2, 7, 5].
• Sorted list: [1, 2, 3, 5, 7].

Player One makes a choice and Player Two optimally counters.

Implementation

Write a function that performs these steps to compute the minimum guaranteed sum.

def minimum_number_game_out(nums):
    # Sort the list of numbers
    nums.sort()
    
    # Minimum Sum - we will alternate taking minimums
    mini_sum = 0
    
    # Since the problem guarantees optimal selection and turn-based approach,
    # we consider taking every second number starting from the smallest.
    # Essentially, we sum the smallest half of the sorted list split by turns
    for i in range(len(nums)//2 + 1):
        mini_sum += nums[i]
    
    return mini_sum

# Example test case
nums = [1, 3, 2, 7, 5]
print(minimum_number_game_out(nums))  # Output will be the minimum guaranteed sum

Time Complexity

• Sorting: O(n log n).
• Summing: O(n), but we only sum half the elements.

Overall time complexity: O(n log n).

Explanation

• Sorting efficiently orders the list.
• We then simulate taking the smallest available numbers first, guaranteeing the minimized sum by summing the first half of the sorted list based on turn-based strategy.

This approach ensures an optimal selection is made even if the other player plays optimally on their turns.

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