  The change-making problem asks: given a positive integer $v$ and a collection
$\mathcal{C}$ of integer coin values $c_1=1<c_2< c_3< \cdots< c_n$, what is the
minimum number of coins needed to represent $v$ with coin values from
$\mathcal{C}$? For some coin systems $\mathcal{C}$, the greedy algorithm finds
a representation with a minimum number of coins for all $v$. We call such coin
systems orderly. However, there are coin systems where the greedy algorithm
fails to always produce a minimal representation. Over the past fifty years,
progress has been made on the change-making problem, including finding a
characterization of all orderly coin systems with 3, 4, and 5 coin values. We
characterize orderly coin systems with 6 coin values, and we make
generalizations to orderly coin systems with $n$ coin values.
