Professor Preece's tredoku tilings

TL;DR

This work surveys Donald Preece's tredoku tilings, detailing their definition via $\tau$ and $\rho$, multiple equivalence notions, and a complete existence framework recently established by Simon Blackburn. It introduces run graphs as a key tool to study tilings, irreducibility vs reducibility, and to provide alternate nonexistence proofs; it also reports computer enumerations of tilings up to $\tau\le 16$, including irreducible/reducible classifications. The paper extends the study to quadridoku tilings and general $\kappa$-doku tilings, deriving bounds, constructing minimal exemplars, and discussing geometric and combinatorial constraints such as Turán triples and perimeter minimization. It further investigates tilings with holes, adapting leaf concepts and flip-equivalence, and presents Donald’s hole-containing tilings as a basis for future exploration. Overall, it builds a cohesive framework linking tiling theory and run-graph methods to classify existence, structure, and equivalence of tredoku-like tilings and their generalizations, while providing an archival record of Donald Preece’s contributions and guiding directions for future work.

Abstract

Shortly before he died in 2014, Donald Preece gave two talks about what he called tredoku tilings, inspired by the puzzle of the same name. In these talks he presented a conjecture about the existence of these tilings that has been proved recently by Simon Blackburn. This paper provides an overview of Donald's work in this area, including his work on a natural generalisation of a tredoku tiling that he called a quadridoku tiling. Additionally, the paper gives alternative proofs of some parts of the existence theorem for tredoku tilings, presents a computer enumeration of the isomorphism classes of tredoku tilings with up to 16 tiles and provides a brief introduction to tilings with holes.

Figures (44)

• Figure 2.1: The tredoku tiling dap7.4a, which has $\tau=7$ tiles and $\rho=4$ runs, drawn in vertical format (left panels) and horizontal format (right panels). The three possible tile types are shown in each case.
• Figure 2.2: The four runs of the tiling dap7.4a, highlighted in green.
• Figure 2.3: The dual and n-dual graphs of the tiling dap7.4a. The left panel shows the tiling, the middle panel the dual graph and the right panel the n-dual graph; for the n-dual graph, the colours of the edges indicate their weights (red=1, blue=2).
• Figure 2.4: Examples of runs in each of the three possible directions.
• Figure 2.5: The four 7.4 tilings with Donald's annotations. Green tiles are 4--tiles and blue dots and red filled squares indicate vertices where five or six tiling edges meet, respectively. A small dot indicates a leaf tile.

Theorems & Definitions (54)

• Theorem 3.1: Preece and Blackburn
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof
• Lemma 4.5