  The Change-Making Problem is to represent a given value with the fewest coins
under a given coin system. As a variation of the knapsack problem, it is known
to be NP-hard. Nevertheless, in most real money systems, the greedy algorithm
yields optimal solutions. In this paper, we study what type of coin systems
that guarantee the optimality of the greedy algorithm. We provide new proofs
for a sufficient and necessary condition for the so-called \emph{canonical}
coin systems with four or five types of coins, and a sufficient condition for
non-canonical coin systems, respectively. Moreover, we present an $O(m^2)$
algorithm that decides whether a tight coin system is canonical.
