Consider a coin system where the greedy approach fails to produce the optimal solution, such as one with denominations like [1, 3, 4] and a target amount of 6.

Construct a scenario where the greedy algorithm picks the largest possible denomination first, but this leads to a suboptimal solution, requiring more coins than an alternative combination.

Design a coin system where the greedy choice (largest denomination ≤ remaining amount) does not align with the optimal solution, proving that greedy fails for this system.