# Three Glass Puzzle

### Graph Theoretical Approach

Oystein Ore gave a worldly twist to the Three Glass puzzle and solved it in the framework of the Graph Theory.

Every distribution of wine in the three jugs A, B, and C, can be described by the quantities b and c of wine in the jugs B and C, respectively. Thus every possible distribution of wine is described by a pair

puz(WaterPuzzle) in this case consists of all integer pairs (b,c) connected by edges wherever it's possible to move from one node to another by pouring wine between the jugs. Thus, from

On the diagram I only showed some of the edges. In particular, there is a walk from the starting node

(0,0) | (A->B) |

(5,0) | (B->C) |

(2,3) | (C->A) |

(2,0) | (B->C) |

(0,2) | (A->B) |

(5,2) | (B->C) |

(4,3) | (C->A) |

(4,0) |

Complete puz(WaterPuzzle), as it follows from the proof of one of the statements derived previously, consists of all possible horizontal, vertical, and diagonal (upper-left-to-bottom-right) edges that connect perimeter points.

## Remark

It's perhaps relevant to note that puz(WaterPuzzle) with nodes on the perimeter of the 6×4 rectangle resembles the diagram obtained in describing a slanted cut of the torus with a rational slope. In this case the cut here proceeds in the mirror direction of that on the torus. As I just noted all the diagonal lines connecting points on the perimeter are included. This shows that the serpentine band is indeed of constant width.

## Reference

- O. Ore (R. J. Wilson),
*Graphs And Their Uses*, MAA, New Math Library, 1990.

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