The change-making problem for six coin values and beyond

TL;DR

This work analyzes the change-making problem by examining when the greedy algorithm is always optimal, formalized as $grd_{\mathcal{C}}(v)=opt_{\mathcal{C}}(v)$ for all $v>0$. Building on the prefix-pattern framework and key tools such as the Pearson theorem and the One Point Theorem, it first delivers a complete characterization of orderly coin systems with six values by enumerating four admissible prefix patterns and giving explicit forms (with precise inequalities) for the two patterns identified. It then extends the methodology to general $n$ via prefix-pattern analysis, highlighting the remaining hard case $+++-\cdots -+$ and introducing three infinite fixed-gap families $\mathcal{D},\mathcal{E},\mathcal{F}$ that realize this pattern and are proved orderly. The paper closes with computational observations and conjectures about broader structures (and a potential additional family $\mathcal{G}$), outlining directions for future theoretical work beyond the six-value case.

Abstract

The change-making problem asks: given a positive integer $v$ and a collection $C$ 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 $C$? For some coin systems $C$, 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.

Theorems & Definitions (40)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7: One Point Theorem