Hexiamond Pair Tri-Oddities

Introduction

A hexiamond is a plane figure made of six equilateral triangles joined edge to edge. There are 12 such figures, not distinguishing reflections and rotations. They were first enumerated by T. H. O'Beirne.

A tri-oddity is an arrangement of a number of copies of a polyform, not a multiple of 3, with ternary symmetry. It is a variant of an oddity.

Many polyforms cannot have ternary symmetry. Polyiamonds can, along with polyhexes and polycubes.

Here I show minimal known tri-oddities formed by copies of two hexiamonds, using at least one of each. If you find a smaller solution, or solve an unsolved case, please let me know.

  • Arbitrary Solutions
  • Holeless Solutions

    • Arbitrary Solutions

    Table

    This table shows the fewest tiles known to construct a tri-oddity with two given hexiamonds.

    AEFHILOPSUVX
    A * 5 4 8 — 4 4 10 13 13 — 13
    E 5 * 5 5 5 5 4 5 5 5 5 5
    F 4 5 * 7 7 4 4 5 8 7 7 11
    H 8 5 7 * 10 4 4 7 7 4 7 10
    I — 5 7 10 * 7 4 4 8 8 — 4
    L 4 5 4 4 7 * 4 5 7 2 7 7
    O 4 4 4 4 4 4 * 4 4 4 4 4
    P 10 5 5 7 4 5 4 * 4 4 4 7
    S 13 5 8 7 8 7 4 4 * 7 7 13
    U 13 5 7 4 8 2 4 4 7 * 5 8
    V — 5 7 7 — 7 4 4 7 5 * 4
    X 13 5 11 10 4 7 4 7 13 8 4 *

    Solutions

    These solutions are minimal. They are not necessarily uniquely minimal.

    2 Tiles

    4 Tiles

    5 Tiles

    7 Tiles

    8 Tiles

    10 Tiles

    11 Tiles

    13 Tiles

    Holeless Solutions

    Table

    This table shows the fewest tiles known to construct a holeless tri-oddity with two given hexiamonds.

    AEFHILOPSUVX
    A * 5 4 8 — 4 4 10 16 13 — ?
    E 5 * 5 5 5 5 4 5 5 5 5 5
    F 4 5 * 7 7 4 4 5 10 7 11 16
    H 8 5 7 * 10 4 4 7 7 4 7 14
    I — 5 7 10 * 7 4 4 13 13 — 4
    L 4 5 4 4 7 * 4 7 7 4 7 7
    O 4 4 4 4 4 4 * 4 4 4 4 4
    P 10 5 5 7 4 7 4 * 4 4 4 7
    S 16 5 10 7 13 7 4 4 * 7 10 19
    U 13 5 7 4 13 4 4 4 7 * 7 10
    V — 5 11 7 — 7 4 4 10 7 * 4
    X ? 5 16 14 4 7 4 7 19 10 4 *

    4 Tiles

    5 Tiles

    7 Tiles

    8 Tiles

    10 Tiles

    11 Tiles

    13 Tiles

    14 Tiles

    16 Tiles

    19 Tiles

    Last revised 2023-04-28.

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    Col. George Sicherman [ HOME | MAIL ]