Holey L Shapes from Two Pentominoes

  • 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.

    Here I study the problem of forming an L-shaped (six-sided) polyomino with copies of two pentominoes, using at least one of each, and allowing one-cell holes that do not touch the perimeter or one another, not even at a point. If you find a solution smaller than one of mine, please write!

    See also L Shapes from Two Pentominoes. and L Shapes from Three Pentominoes.

    Nomenclature

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

    Table

    This table shows the smallest total number of two pentominoes known to be able to tile a holey L-shaped polyomino:

    FILNPTUVWXYZ
    F * 33 16 × 15 × 10 21 × × 21 ×
    I 33 * 10 32 10 26 11 10 37 44 22 38
    L 16 10 * 10 10 16 11 10 15 26 16 22
    N × 32 10 * 10 36 26 21 × × 26 ×
    P 15 10 10 10 * 10 10 10 20 26 10 10
    T × 26 16 36 10 * 23 27 62 × 21 ×
    U 10 11 11 26 10 23 * 11 34 21 10 38
    V 21 10 10 21 10 27 11 * 11 × 11 10
    W × 37 15 × 20 62 34 11 * × 21 ×
    X × 44 26 × 26 × 21 × × * 44 ×
    Y 21 22 16 26 10 21 10 11 21 44 * 25
    Z × 38 22 × 10 × 38 10 × × 25 *

    Solutions

    So far as I know, these solutions have the fewest possible cells. They are not necessarily uniquely minimal.

    10 Cells

    11 Cells

    15 Cells

    16 Cells

    20 Cells

    21 Cells

    22 Cells

    23 Cells

    25 Cells

    26 Cells

    27 Cells

    32 Cells

    33 Cells

    34 Cells

    36 Cells

    37 Cells

    38 Cells

    44 Cells

    62 Cells

    Last revised 2023-03-10.

    Back to Polyomino and Polyking Tiling < Polyform Tiling < Polyform Curiosities
    Col. George Sicherman [ HOME | MAIL ]