Two-Pentomino Holey Balanced Rectangles

  • Introduction
  • Nomenclature
  • Table
  • Solutions

    • Introduction

    A pentomino is a figure made of five squares joined edge to edge. There are 12 such figures, not distinguishing reflections and rotations. They were first enumerated and studied by Solomon Golomb.

    It has long been known that only four pentominoes can tile rectangles:

    For other rectangles that these pentominoes tile, see Mike Reid's Rectifiable Polyomino Page.

    Rodolfo Kurchan's online magazine Puzzle Fun studied the problem of tiling some rectangle with two different pentominoes, in Issue 19, and revisited the problem in Issue 21. The August 2010 issue of Erich Friedman's Math Magic broadened this problem to use two polyominoes of any size, not necessarily the same.

    In Two-Pentomino Balanced Rectangles I study the related problem of tiling some rectangle with two pentominoes, using the same number of copies of each. Here I study the same problem, allowing one-cell holes in the rectangle. The holes must lie in the interior of the rectangle and must not touch one another.

    Thanks to Jenard Cabilao for suggesting this variant.

    Bryce Herdt first solved the case of I+X, and suggested that U+X ought to have a solution.

    Nomenclature

    I use Solomon W. Golomb's original names for the pentominoes:

    Table

    This table shows the area of the smallest rectangle known to be tiled by two pentominoes, using at least one of each, with optional one-cell holes.

    FILNPTUVWXYZ
    F * 42 20 × 40 × 21 20 × × 40 ×
    I 42 * 20 42 20 21 33 42 42 576 40 77
    L 20 20 * 20 20 42 20 20 20 98 32 42
    N × 42 20 * 20 66 20 20 × × 40 ×
    P 40 20 20 20 * 21 20 21 40 150 20 21
    T × 21 42 66 21 * 63 64 117 × 20 ×
    U 21 33 20 20 20 63 * 44 192 176 21 66
    V 20 42 20 20 21 64 44 * 66 532 42 20
    W × 42 20 × 40 117 192 66 * × 42 ×
    X × 576 98 × 150 × 176 532 × * 99 ×
    Y 40 40 32 40 20 20 21 42 42 99 * 40
    Z × 77 42 × 21 × 66 20 × × 40 *

    Solutions

    So far as I know, these solutions have minimal area. They are not necessarily uniquely minimal. For a given area, the number of holes is minimal.

    Area 20

    Area 21

    Area 32

    Area 33

    Area 40

    Area 42

    Area 44

    Area 63

    Area 64

    Area 66

    Area 77

    Area 98

    Area 99

    Area 117

    Area 150

    Area 176

    Area 192

    Area 532

    Area 576

    Last revised 2021-05-12.

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    Col. George Sicherman [ HOME | MAIL ]