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. If you find a smaller solution than one of mine, please write!

    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 an L-shaped polyomino:

    FILNPTUVWXYZ
    F * 17 5 × 3 × 2 6 × × 5 ×
    I 17 * 2 7 2 17 40 2 32 × 5 26
    L 5 2 * 2 2 9 3 2 3 7 5 9
    N × 7 2 * 2 12 6 8 × × 8 ×
    P 3 2 2 2 * 2 2 2 4 6 2 2
    T × 17 9 12 2 * 46 × 15 × 8 ×
    U 2 40 3 6 2 46 * 58 × 6 2 ×
    V 6 2 2 8 2 × 58 * 43 × 8 2
    W × 32 3 × 4 15 × 43 * × 8 ×
    X × × 7 × 6 × 6 × × * 11 ×
    Y 5 5 5 8 2 8 2 8 8 11 * 5
    Z × 26 9 × 2 × × 2 × × 5 *

    Solutions

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

    2 Tiles

    3 Tiles

    4 Tiles

    5 Tiles

    6 Tiles

    7 Tiles

    8 Tiles

    9 Tiles

    11 Tiles

    12 Tiles

    15 Tiles

    17 Tiles

    26 Tiles

    32 Tiles

    40 Tiles

    43 Tiles

    46 Tiles

    58 Tiles

    Last revised 2021-03-20.


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    Col. George Sicherman [ HOME | MAIL ]