Holeless Tetromino-Hexomino Compatibility

Introduction

A tetromino is a figure made of four squares joined edge to edge. A hexomino is a figure made of six squares joined edge to edge. There are five tetrominoes and 35 hexominoes, not distinguishing reflections and rotations. They were first enumerated and studied by Solomon Golomb.

The compatibility problem is to find a figure that can be tiled with each of a set of polyforms. Polyomino compatibility has been widely studied since the early 1990s, and two well-known websites, Poly²ominoes by Jorge Mireles and Polypolyominoes by Giovanni Resta, present the results of their authors' systematic searches for compatibility figures. The sites include solutions by other researchers, especially Mike Reid. So far as I know, polyomino compatibility has not been treated in print since Golomb first raised the issue in 1981, except in a series of articles called Polyomino Number Theory, written by Andris Cibulis, Andy Liu, Bob Wainwright, Uldis Barbans, and Gilbert Lee from 2002 to 2005.

The websites and the articles show only minimal solutions with no restriction. Here I show minimal known tetromino-hexomino compatibility figures without holes, only where the solutions differ from the minimal solutions with holes allowed. If you find a smaller solution or solve an unsolved case, please let me know.

For tetromino-hexomino compatibility allowing holes, see Resta's Tetrominoes. For pentomino compatibility with or without holes, see Pentomino Compatibility. For hexomino compatibility without holes, see Holeless Hexomino Compatibility. For pentomino-hexomino compatibility without holes, see Holeless Pentomino-Hexomino Compatibility.

Last revised 2011-02-10.

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Col. George Sicherman [ HOME | MAIL ]