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 five squares joined edge to edge. There are 5 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. If you find a smaller solution or solve an unsolved case, please let me know.

For pentomino compatibility with or without holes, see Pentomino Compatibility. For tetromino-pentomino compatibility, see Tetromino-Pentomino Compatibility.

Tetromino Names

These are Livio Zucca's names for the tetrominoes:

Hexomino Numbers

These are my numbers for the hexominoes:

Table

This table shows the smallest number of tiles known to suffice to construct a holeless figure tilable by the tetromino and the pentomino. Shaded cells indicate solutions that are minimal even if holes are allowed.

 ILNQT
12 34 62 32 34 6
22 32 32 32 34 6
34 64 62 316 244 6
42 34 62 3?4 6
52 32 32 34 64 6
64 62 38 12?8 12
72 32 34 62 34 6
84 62 32 3?8 12
94 62 32 3?4 6
102 32 38 12?8 12
114 64 64 6?4 6
124 62 32 34 64 6
132 34 64 64 64 6
142 34 62 34 64 6
154 64 62 3?4 6
162 34 64 64 64 6
178 122 32 3?8 12
188 122 32 34 64 6
194 64 62 3?4 6
202 32 32 3?4 6
212 32 32 34 64 6
228 124 64 6?4 6
2320 302 34 6?8 12
242 32 32 32 34 6
258 124 616 24?4 6
264 62 32 3?8 12
278 124 64 6?4 6
28?2 32 3?8 12
292 32 38 124 64 6
304 62 34 68 1216 24
312 32 34 62 34 6
328 124 62 3?4 6
332 34 62 3?4 6
34?4 68 12?4 6
35?4 64 6?4 6

Solutions

So far as I know, these solutions are minimal. They are not necessarily uniquely minimal.

3 Tetrominoes, 2 Hexominoes

6 Tetrominoes, 4 Hexominoes

12 Tetrominoes, 8 Hexominoes

More Tetrominoes and Hexominoes

Last revised 2015-11-07.

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Col. George Sicherman [ HOME | MAIL ]