First, clarify that the goal is to maximize the sum of medians by strategically forming groups of three. Then, sort the array and pair the largest elements with the smallest possible medians to maximize each median. Finally, derive the optimal grouping pattern and compute the sum.
Pro tip: Mention that this problem is equivalent to maximizing the sum of the second-largest elements in each group, which leads to a greedy solution. Also, note that the optimal grouping is to sort the array and take every second element from the upper half, ignoring the largest element.
Confirm that groups must have exactly three elements, and any leftover elements are ignored. The median of a group is the middle value when the group is sorted.
Sort the array in ascending order to easily identify the smallest and largest elements. This helps in forming groups that maximize the medians.
To maximize each median, pair the largest possible element as the maximum in a group, and the smallest possible element as the minimum, leaving the median to be as large as possible. The optimal pattern is to take the largest element as the maximum, the smallest as the minimum, and the second largest as the median for the first group, then repeat with the remaining elements.
After sorting, the medians will be the elements at indices n-2, n-4, n-6, ... down to the middle of the array (specifically, for an array of length n, the medians are at indices n-2, n-4, ..., n-2k where k = floor(n/3)). Sum these elements to get the maximum total.
Test the strategy with small examples to ensure correctness, and consider edge cases like arrays with length less than 3 or not divisible by 3.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.