Maximum number of coins

Start with clarification

The title Maximum number of coins is not a sufficient problem statement, so the strongest answer is not to guess a canonical problem. Ask for the input shape, desired output, constraints, whether updates are online, and one worked example. Restate the resulting task in precise terms, including whether duplicates, empty inputs, negative values, or cycles are allowed.

Build the solution deliberately

First give a correct baseline and compute its cost. Use the limits to decide whether a linear scan, hash map, sorting, traversal, dynamic program, or specialized structure is needed. State the invariant that makes the optimized approach correct, then simulate it on the smallest valid case and an adversarial case. Call out failure modes such as duplicate keys, disconnected input, overflow, and off-by-one boundaries. Do not claim one fixed complexity until the actual task is defined: the correct Big-O is the dominant operation under the confirmed constraints, with matching auxiliary-space cost.

Complexity: After the selection rule is fixed, a sort-and-greedy approach over n piles is O(n log n) time and O(1) auxiliary space beyond the sort.