Tiling Right Trapezoidal Polyominoes with 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 tiling a polyomino shaped like a right trapezoid with copies of two pentominoes, using at least one of each. Such a polyomino has three straight sides, two of them parallel, and one zigzag side. For this problem, the polyomino may be triangular.

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

    See also Tiling a Right Trapezoidal Polyomino with Three Pentominoes and L Shapes from Two 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 trapezoidal polyomino:

    FILNPTUVWXYZ
    F * × × × 5 × × × × × 9 ×
    I × * × 28 3 × × × 39 × 17 ×
    L × × * 5 3 × × × 2 × 6 ×
    N × 28 5 * 3 27 × 9 × × 7 ×
    P 5 3 3 3 * 3 5 3 3 4 2 3
    T × × × 27 3 * × × 15 × × ×
    U × × × × 5 × * × × × 6 ×
    V × × × 9 3 × × * 30 × 20 ×
    W × 39 2 × 3 15 × 30 * × 7 ×
    X × × × × 4 × × × × * × ×
    Y 9 17 6 7 2 × 6 20 7 × * ×
    Z × × × × 3 × × × × × × *

    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

    9 Tiles

    15 Tiles

    17 Tiles

    20 Tiles

    27 Tiles

    28 Tiles

    30 Tiles

    39 Tiles

    Last revised 2023-12-25.

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