Holeless Pentahex Compatibility

Introduction

A pentahex is a figure made of five regular hexagons joined edge to edge. There are 22 such figures, not distinguishing reflections and rotations.

Dr. Erich Friedman's Math Magic for September 2004 shows compatibility figures for pairs of pentahexes (and many other polyforms). Here are minimal known compatibility figures for pairs of pentahexes. Joe DeVincentis and Dr. Friedman found many solutions. Dr. Andrejs Cibulis was the first to solve some of the hardest compatibilities.

For compatible pairs of pentahexes with holes allowed, see Pentahex Compatibility.

For compatible pairs of pentahexes with vertical mirror symmetry, see Pentahex Compatibility with Vertical Symmetry.

For compatible pairs of pentahexes with an odd number of tiles, see Pentahex Odd Pairs.

  • Nomenclature
  • Solutions

    • Nomenclature

    I adopt Dr. Friedman's nomenclature:

    Solutions

    In the table, green figures indicate solutions that are minimal even without the condition of holelessness. Below I show only holeless solutions that differ from those shown on Pentahex Compatibility.

     ACDEFHIJKLNPQRSTUVWXYZ
    A*?22229222223212??68324
    C?*14102910232239222363?103
    D214*333222222222?232322
    E2103*32282322228??182222
    F2233*31023222323?2122682
    H29323*102222223823162362
    I910221010*610232885??55?22
    J2228226*222223232212322
    K232232102*22222215332222
    L222322222*222222422823
    N2222223222*233322242322
    P23222222222*223?222222
    Q392232822232*222642224
    R2222238322322*52326422
    S122283852223325*?23147522
    T?2???2?315222?22?*32??2?
    U?32?23?234226323*10??62
    V663181216523242423210*10?22
    W83222251222222614??10*662
    X3?3263?328322475???6*24
    Y210228622222222226262*2
    Z432222222322422?22242*

    3 Tiles

    4 Tiles

    5 Tiles

    6 Tiles

    8 Tiles

    9 Tiles

    10 Tiles

    12 Tiles

    14 Tiles

    15 Tiles

    16 Tiles

    22 Tiles

    75 Tiles

     

    Last revised 2026-06-27.

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    Col. George Sicherman [ HOME | MAIL ]