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A Graph-Theoretic Model for a Generic Three-Jug Puzzle

Suresh Manjanath Hegde, Shashanka Kulamarva

TL;DR

This work addresses the generic three-jug puzzle by introducing a graph-theoretic framework that abstracts state transitions into a directed graph $G_Q$ built from an ordered quadruple $Q=(a,b,c,d)$ with $a>b>c$, $d=2k$, and $b\ge k$. A central result proves that an edge in $G_P$ corresponds exactly to a single measurable pour between distributions, and that a directed path from $s=(\tilde b,\tilde c)$ to $t=(\frac{d}{2},0)$ in $G_P$ is equivalent to the existence of a solution to the puzzle $\mathbb{P}^{\tilde a,\tilde b,\tilde c}_{a,b,c}$; this provides a concrete solvability criterion. The paper then sketches an algorithm: form $Q$, build $G_Q$, check for an $st$-path, and, if present, follow a shortest path to obtain a minimal-pours solution; otherwise, conclude unsolvability. The approach yields a puzzle-independent method with potential for efficient computation and extension to related liquid-transfer problems, offering a formal, graph-based lens on state-space exploration and optimal strategies.

Abstract

A classic three-jug puzzle asks, given three jugs $A$, $B$, and $C$ with fixed maximum capacities, with jug $A$ filled with wine to its maximum capacity, whether is it possible to divide the wine into two halves by pouring it from one jug to another without using any other measuring devices. However, we consider a generic version of the three-jug puzzle and present an independent graph-theoretic model to determine whether the puzzle has a solution at all. If it has a solution, then the same can be determined using this model. We also present the sketch of an algorithm to determine the solution of the puzzle.

A Graph-Theoretic Model for a Generic Three-Jug Puzzle

TL;DR

This work addresses the generic three-jug puzzle by introducing a graph-theoretic framework that abstracts state transitions into a directed graph built from an ordered quadruple with , , and . A central result proves that an edge in corresponds exactly to a single measurable pour between distributions, and that a directed path from to in is equivalent to the existence of a solution to the puzzle ; this provides a concrete solvability criterion. The paper then sketches an algorithm: form , build , check for an -path, and, if present, follow a shortest path to obtain a minimal-pours solution; otherwise, conclude unsolvability. The approach yields a puzzle-independent method with potential for efficient computation and extension to related liquid-transfer problems, offering a formal, graph-based lens on state-space exploration and optimal strategies.

Abstract

A classic three-jug puzzle asks, given three jugs , , and with fixed maximum capacities, with jug filled with wine to its maximum capacity, whether is it possible to divide the wine into two halves by pouring it from one jug to another without using any other measuring devices. However, we consider a generic version of the three-jug puzzle and present an independent graph-theoretic model to determine whether the puzzle has a solution at all. If it has a solution, then the same can be determined using this model. We also present the sketch of an algorithm to determine the solution of the puzzle.
Paper Structure (5 sections, 2 theorems, 6 equations, 3 figures)

This paper contains 5 sections, 2 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Graph Model
  3. Main Results
  4. Sketch of an Algorithm
  5. Conclusion

Key Result

Theorem 3.1

Let $\mathbb{P}^{\tilde{a},\tilde{b},\tilde{c}}_{a,b,c}$ be a generic three-jug puzzle and let $G_P$ be the graph obtained from the ordered quadruple $P=(a,b,c,\tilde{a}+\tilde{b}+\tilde{c})$ using the graph model in Definition def:Model. Then there exists a directed edge from a vertex $(i,j)$ to a

Figures (3)

  • Figure 1: Directed path indicating a solution to the three-jug puzzle $\mathbb{P}_{10,7,3}$
  • Figure 2: A generic three-jug puzzle $\mathbb{P}^{\tilde{a},\tilde{b},\tilde{c}}_{a,b,c}$
  • Figure 3: Graph $G_Q$ corresponding to the quadruple $Q=(7,4,2,6)$

Theorems & Definitions (6)

  • Definition 1.2: Measurable Pour
  • Definition 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof