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The odd triangle ring puzzle problem

Sylvain Barré, Othmane Oukrid, Mikaël Pichot

TL;DR

The paper solves the odd triangle ring puzzle problem associated with the unique odd Moebius–Kantor complex using the method of Sidon sequences, identifying a complete classification into three families: two uncountable families formed by unions of height-1 or height-2 flat strips of period $6$, and a finite family of twelve exceptional puzzles. It advances the study by introducing root distributions and the mesoscopic-rank framework within a CAT(0) 2-complex setting to translate local constraints into global tessellation structure. The main result demonstrates that puzzle solutions organize into the three families and provides explicit constructions and counts for each, notably the twelve exceptional cases arising from the special distribution $D_0$. This approach deepens the connection between geometric group theory, CAT(0) geometry, and combinatorial tiling problems, with potential implications for understanding Sidon-type puzzle problems in related complexes.

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.

The odd triangle ring puzzle problem

TL;DR

The paper solves the odd triangle ring puzzle problem associated with the unique odd Moebius–Kantor complex using the method of Sidon sequences, identifying a complete classification into three families: two uncountable families formed by unions of height-1 or height-2 flat strips of period , and a finite family of twelve exceptional puzzles. It advances the study by introducing root distributions and the mesoscopic-rank framework within a CAT(0) 2-complex setting to translate local constraints into global tessellation structure. The main result demonstrates that puzzle solutions organize into the three families and provides explicit constructions and counts for each, notably the twelve exceptional cases arising from the special distribution . This approach deepens the connection between geometric group theory, CAT(0) geometry, and combinatorial tiling problems, with potential implications for understanding Sidon-type puzzle problems in related complexes.

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.
Paper Structure (5 sections, 10 theorems, 4 equations, 27 figures)

This paper contains 5 sections, 10 theorems, 4 equations, 27 figures.

Table of Contents

  1. Introduction
  2. The direct approach
  3. Roots in CAT(0) complexes and ring puzzles
  4. Classifying the odd root distributions
  5. Proof of Theorem \ref{['T - main theorem']}

Key Result

Lemma 1.1

The ring set in Fig. Fig 2 defines a unique edge marking (by elements in $S$) in every odd ring puzzle.

Figures (27)

  • Figure 1: The shape set of the odd ring puzzle problem
  • Figure 2: The set $\Theta_s$ of rings ($s\in S$) in the odd ring puzzle problem
  • Figure 3: The twelve special odd puzzles
  • Figure 4: Eight configurations around the initial triangle (here $u,v,w\in \{1,2\}$)
  • Figure 5: Four possible extensions
  • ...and 22 more figures

Theorems & Definitions (24)

  • Lemma 1.1
  • proof
  • Theorem 1.2: Classifying the odd ring puzzles
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 4.1: 1-periodic geodesics
  • proof
  • Lemma 4.2: 2-periodic geodesics
  • ...and 14 more