Scaled Two-Pentomino 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.

    Here I show which pairs of pentominoes can tile a rectangle, using the pentominoes at various sizes. If you find a solution better than one of mine, please write!

    See also Balanced Two-Pentomino Rectangles.

    Nomenclature

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

    Table

    This table shows the fewest scaled pentominoes known to be able to tile a rectangle, using at least one of each pentomino. The rectangles are not necessarily the smallest that can be tiled, only the smallest that can be tiled with the fewest tiles.

    FILNPTUVWXYZ
    F * 11 4 × 6 × 6 4 × × 5 ×
    I 11 * 3 9 3 7 6 5 11 15 6 9
    L 4 3 * 4 3 8 4 4 4 5 6 7
    N × 9 4 * 4 10 4 4 × × 6 ×
    P 6 3 3 4 * 3 4 3 4 5 3 3
    T × 7 8 10 3 * 56 7 16 × 4 ×
    U 6 6 4 4 4 56 * 3 × 3 6 ×
    V 4 5 4 4 3 7 3 * 30 × 6 3
    W × 11 4 × 4 16 × 30 * × 10 ×
    X × 15 5 × 5 × 3 × × * 5 ×
    Y 5 6 6 6 3 4 6 6 10 5 * 5
    Z × 9 7 × 3 × × 3 × × 5 *

    Solutions

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

    Solutions labeled BH are by Bryce Herdt.

    3 Tiles

    4 Tiles

    5 Tiles

    6 Tiles

    7 Tiles

    8 Tiles

    9 Tiles

    10 Tiles

    11 Tiles

    15 Tiles

    16 Tiles

    30 Tiles

    56 Tiles

    Last revised 2021-03-21.


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    Col. George Sicherman [ HOME | MAIL ]