The odd triangle ring puzzle problem

TL;DR

This paper solves the odd ring puzzle problem related to Euclidean tessellations and CAT(0) complexes, identifying all possible puzzle families using Sidon sequences, including uncountable and finite sets.

Contribution

It provides a complete classification of odd ring puzzles associated with the odd Moebius–Kantor CAT(0) complex, introducing a novel method using Sidon sequences.

Findings

Exactly three families of puzzles identified

Two uncountable families and one finite family of twelve puzzles

Complete classification of odd ring puzzles in this context

Abstract

Ring puzzles are tessellations of the Euclidean plane respecting local constraints around vertices. Such puzzles may arise in geometric group theory, for example, as embedded flat planes in certain CAT(0) complexes of dimension 2. In the present paper, we solve the odd ring puzzle problem, which is associated with the unique odd Moebius--Kantor CAT(0) complex by the method of Sidon sequences. We prove that there are precisely three families of such puzzles, two uncountable families, and a finite family of twelve exceptional puzzles.