Container With Most Water

Water volume is width multiplied by the shorter of the two boundary heights. Starting at the widest possible separation puts width at its absolute maximum. Moving the taller line inward decreases width without any chance of increasing the bottleneck height; only moving the shorter line inward has any mathematical hope of finding a taller wall to compensate.

Full solution

Inward two-pointer shrinkage: width is maximal at the endpoints, and only the shorter wall has any hope of a taller replacement, so it never needs to check a pair twice. O(n) time, O(1) space.

def max_area(height: list[int]) -> int:
    left, right = 0, len(height) - 1
    best = 0
    while left < right:
        width = right - left
        best = max(best, width * min(height[left], height[right]))
        # the shorter wall is the bottleneck, so only moving it can ever help
        if height[left] < height[right]:
            left += 1
        else:
            right -= 1
    return best

function maxArea(height) {
  let left = 0;
  let right = height.length - 1;
  let best = 0;
  while (left < right) {
    const width = right - left;
    best = Math.max(best, width * Math.min(height[left], height[right]));
    if (height[left] < height[right]) left++;
    else right--;
  }
  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.