Chocolate Distribution Problem

When numbers are sorted, any subset with minimal spread between its extreme values must occupy contiguous positions. Scrambled choices can only introduce wider gaps, so sorting converts a combinatorial search over combinations into a simple sliding window of fixed width.

Full solution

Sort, then slide a fixed window of m over the sorted packets: the tightest group of m values must be contiguous once sorted, so the answer is the smallest arr[i + m - 1] - arr[i]. O(n log n) from the sort, O(1) extra - trying every combination is O(n choose m).

def find_min_diff(arr: list[int], m: int) -> int:
    if m <= 1 or not arr:
        return 0
    sizes = sorted(arr)
    best = sizes[m - 1] - sizes[0]
    # windows start at 0..n-m; the last valid right edge is index n-1
    for i in range(1, len(sizes) - m + 1):
        best = min(best, sizes[i + m - 1] - sizes[i])
    return best

function findMinDiff(arr, m) {
  if (m <= 1 || arr.length === 0) return 0;
  const sizes = [...arr].sort((a, b) => a - b);
  let best = sizes[m - 1] - sizes[0];
  // windows start at 0..n-m; the last valid right edge is index n-1
  for (let i = 1; i <= sizes.length - m; i++) {
    best = Math.min(best, sizes[i + m - 1] - sizes[i]);
  }
  return best;
}

The theory behind it

Arrays — the ground this problem stands on. All Arrays problems

What Arrays is

An array is a row of fixed boxes laid side by side in computer memory, like numbered lockers in a hallway. Because each box occupies identical space and sits directly next to its neighbors, jumping to locker zero or locker ten thousand takes the exact same tiny fraction of time. Every box holds an item of the same type, addressed by an offset number called an index.

When to reach for it

Reach for an array when items arrive in a known sequence and need immediate retrieval by position number. Problems asking for running totals, prefix accumulations, cyclic rotations, or in-place rearrangements signal array mechanics. Whenever constraints require constant-time random lookups or contiguous cache scans across fixed collections, a flat sequence is the default container.

How the pattern works

Visualize a tape with zero-indexed slots stretching from start to end. Keep track of write and read cursors when modifying contents without allocating helper buffers. For running computations, maintain an invariant such as having processed all elements left of the current index while pending elements wait to the right. When modifying entries in place, consider scanning backwards from the end so unread data is not overwritten.