Tetromino-Pentomino Compatibility

Introduction

A tetromino is a figure made of four squares joined edge to edge. A pentomino is a figure made of five squares joined edge to edge. There are 5 tetrominoes and 12 pentominoes, 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.

Here I show minimal known tetromino-pentomino compatibility figures, allowing or disallowing holes. If you find a smaller solution or solve an unsolved case, please let me know.

The solution shown below for the square tetromino and the T pentomino is a variant of the minimal solution found in 2011 by deepgreen. His home page is here.

For compatibility of two pentominoes with or without holes, see Pentomino Compatibility. Resta's page of tetromino-pentomino compatibilities was originally on GeoCities, a defunct web host. It is now here. Mireles's site was also on GeoCities and has not been rebuilt. The link above is to the Internet Archive.

Tetromino Names

These are Livio Zucca's names for the tetrominoes:

Pentomino Names

These are Golomb's names for the pentominoes:

  • Solutions Allowing Holes
  • Variants Without Holes
  • Solutions With Diagonal Symmetry

    • Solutions Allowing Holes

    Table

    This table shows the smallest number of tiles known to suffice to construct a figure tilable by the tetromino and the pentomino.

     FILNPTUVWXYZ
    I4 54 54 54 54 58 108 108 104 5×4 58 10
    L4 54 54 54 54 54 54 54 54 58 104 54 5
    N4 54 54 54 54 58 104 54 54 58 104 54 5
    Q8 104 54 54 54 568 8564 804 516 20×8 104 5
    T4 58 104 58 104 54 58 108 108 104 54 58 10

    Solutions

    Variants Without Holes

    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.

     FILNPTUVWXYZ
    I4 54 54 54 54 58 108 108 104 5×4 58 10
    L4 54 54 54 54 54 54 54 54 58 104 54 5
    N4 54 54 54 54 58 104 54 54 512 154 54 5
    Q?4 54 58 104 5??4 5?×8 104 5
    T4 58 108 108 104 54 58 108 108 108 104 58 10

    Solutions

    Here I show only solutions that are larger than the corresponding minimal solutions allowing holes. So far as I know, these solutions are minimal. They are not necessarily uniquely minimal.

    Solutions With Diagonal Symmetry

    Solutions with diagonal symmetry are hard to find. For those that do not have full (square) symmetry, four copies can be arranged to create full symmetry.

    Table

    This table shows the smallest number of tiles known to suffice to construct a polyomino with diagonal symmetry, tilable by the tetromino and the pentomino.

     FILNPTUVWXYZ
    I16 2016 2016 2016 208 10???128 160×32 40?
    L8 108 108 108 108 1024 308 104 58 1016 208 1016 20
    N8 10?16 2024 308 10???8 108 108 10?
    Q32 4016 2016 2016 204 5??4 5?×16 204 5
    T16 2016 2016 2032 4016 20???16 20?16 20?

    Solutions

    Last revised 2023-12-12.

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    Col. George Sicherman [ HOME | MAIL ]