Surround Numbers for Polyhexes

A polyhex is a plane figure formed by joining equal regular hexagons edge to edge. The surround number of a polyhex is the least number of copies of the polyhex that can enclose it without gaps.

Strong surround numbers for polyominoes were proposed by Jaime Poniachik in Issue 8 of Puzzle Fun. He asked for the smallest polyominoes with a given strong surround number. In Issue 10, Rodolfo Kurchan extended the problem to polyiamonds, polyhexes, and polyaboloes. He also investigated the smallest polyforms that cannot surround themselves, and the smallest holeless such polyforms. This page extends his results.

Here I show minimal surrounds for holeless polyhexes with up to 6 cells, the smallest polyhexes with given surround numbers up to 8, and the smallest holeless polyhexes that cannot surround themselves. The solutions for minimal surrounds are not necessarily unique.

See also

  • Strong Surround Numbers for Polyaboloes
  • Strong Surround Numbers for Polycairos

    • Minimal Surrounds

    Monohex

    Dihex

    Trihexes

    Tetrahexes

    Pentahexes

    Hexahexes

    Minimal Polyhexes with Given Surround Numbers

    Rodolfo M. Kurchan found all these solutions with up to 6 copies.

    3 Copies, 4 Cells

    This solution is a tetrad. As such, it was found by Scott Kim in 1977.

    4 Copies, 2 Cells

    5 Copies, 3 Cells

    6 Copies, 1 Cell

    7 Copies, 7 Cells

    8 Copies, 8 Cells

    9 Copies, 9 Cells

    10 Copies, 9 Cells

    Minimal Holeless Polyhexes that Cannot Surround Themselves

    The first example was found by Rodolfo Kurchan.

    Last revised 2025-01-18.

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    Col. George Sicherman [ HOME | MAIL ]