Professor Preece's tredoku tilings

TL;DR

This paper reviews Donald Preece's work on tredoku tilings, discusses recent proofs of their existence, explores generalizations like quadridoku tilings, and presents enumeration results and related tiling concepts.

Contribution

It provides a comprehensive overview of Preece's work, including new proofs, enumeration data, and extensions to tilings with holes, advancing the understanding of tredoku tilings.

Findings

Recent proof of the existence of tredoku tilings

Enumeration of isomorphism classes up to 16 tiles

Introduction of tilings with holes

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.