.. -*- coding: utf-8 -*- ================================== Hexominoes: Puzzles & Solutions ================================== :Author: David Goodger :Date: $Date: 2015-02-24 14:21:02 -0600 (Tue, 24 Feb 2015) $ :Revision: $Revision: 600 $ :Web site: http://puzzler.sourceforge.net/ :Copyright: |c| 1998-2015 by David J. Goodger :License: `GPL 2 <../COPYING.html>`__ .. image:: images/puzzler.png :align: center .. sidebar:: Also see: * `Polyominoes: Puzzles & Solutions `_ * `Pentominoes: Puzzles & Solutions `_ * `Solid Pentominoes: Puzzles & Solutions `_ * `Pentacubes: Puzzles & Solutions `_ * `Polycubes: Puzzles & Solutions `_ * `An Introduction to Polyominoes `_ * `Notes on Polyominoes `_ * `Polyform Puzzler: Puzzles & Solutions `_ * `Polyform Puzzler FAQ `_ (`polyform details `__, `numbers of polyforms `__, `interpreting solution files `__) .. contents:: High-quality acrylic sets of hexominoes are available from Kadon Enterprises as `Sextillions`_. .. _Sextillions: http://gamepuzzles.com/polycub2.htm#SX Hexominoes =========== Squares & Rectangles -------------------- Due to a `parity imbalance`_, the 35 free hexominoes cannot fit into a simple rectangle without introducing holes or other irregularities. .. _parity imbalance: http://en.wikipedia.org/wiki/Hexomino#Packing_and_tiling - Square: `solutions incomplete <../solutions/ominoes/hexominoes-square.txt>`__ .. image:: images/ominoes/hexominoes-square.png - 19x11 rectangles plus nubs: .. list-table:: :class: borderless * - .. figure:: images/ominoes/hexominoes-rectangle-plus-nub-1.png design from `Tenyo "Pla-Puzzle" / "Beat the Computer" / "Mystery in a Case" `__ no. 600 (now discontinued); `solutions incomplete <../solutions/ominoes/hexominoes-rectangle-plus-nub-1.txt>`__ * - .. figure:: images/ominoes/hexominoes-rectangle-plus-nub-2.png `solutions incomplete <../solutions/ominoes/hexominoes-rectangle-plus-nub-2.txt>`__ - 45x5 rectangle with holes (design by W. Stead from `Andrew Clarke's Poly Pages [Hexomino Constructions]`_): `solutions incomplete <../solutions/ominoes/hexominoes-holey-rectangle-1.txt>`__ .. image:: images/ominoes/hexominoes-holey-rectangle-1.png - 33x7 rectangle with a central hole (design from `Andrew Clarke's Poly Pages [Hexomino Constructions]`_): `solutions incomplete <../solutions/ominoes/hexominoes-holey-rectangle-2.txt>`__ .. image:: images/ominoes/hexominoes-holey-rectangle-2.png - 17x15 rectangle with a central plus-shaped hole (design from `Polyominoes`, by Solomon W. Golomb): `solutions incomplete <../solutions/ominoes/hexominoes-holey-rectangle-3.txt>`__ .. image:: images/ominoes/hexominoes-holey-rectangle-3.png Parallelograms -------------- As demonstrated below, the 35 hexominoes can fit into parallelograms with odd-length bases. The `parity imbalance`_ that prevents simple rectangles also prevents solutions to parallelograms with even-length bases (e.g. 30x7 & 42x5). - 15x14 parallelogram (design from `Polyominoes`, by Solomon W. Golomb): `solutions incomplete <../solutions/ominoes/hexominoes-parallelogram-15x14.txt>`__ .. image:: images/ominoes/hexominoes-parallelogram-15x14.png - 21x10 parallelogram (design from `Andrew Clarke's Poly Pages [Hexomino Constructions] `_): `solutions incomplete <../solutions/ominoes/hexominoes-parallelogram-21x10.txt>`__ .. image:: images/ominoes/hexominoes-parallelogram-21x10.png - 35x6 parallelogram: `solutions incomplete <../solutions/ominoes/hexominoes-parallelogram-35x6.txt>`__ .. image:: images/ominoes/hexominoes-parallelogram-35x6.png Misc ---- - Triangle (design from `Andrew Clarke's Poly Pages `_): `solutions incomplete <../solutions/ominoes/hexominoes-triangle.txt>`__ .. image:: images/ominoes/hexominoes-triangle.png - Rhombus (design by David Bird from `Andrew Clarke's Poly Pages [Hexomino Constructions]`_): `solutions incomplete <../solutions/ominoes/hexominoes-rhombus.txt>`__ .. image:: images/ominoes/hexominoes-rhombus.png - Cross (design from `Andrew Clarke's Poly Pages (Hexomino Constructions)`_): `solutions incomplete <../solutions/ominoes/hexominoes-cross-1.txt>`__ .. image:: images/ominoes/hexominoes-cross-1.png Hexominoes-Plus =============== Also known as Kadon's Sextillions_, these are the hexominoes with a second N06 ("short N") piece called S16 ("short S"), for a total of 36 pieces, avoiding the `parity imbalance`_ and allowing the construction of simple rectangles. Rectangles ---------- - 18x12: `solutions incomplete <../solutions/ominoes/hexominoes-plus-18x12.txt>`__ .. image:: images/ominoes/hexominoes-plus-18x12.png - 24x9: `solutions incomplete <../solutions/ominoes/hexominoes-plus-24x9.txt>`__ .. image:: images/ominoes/hexominoes-plus-24x9.png - 27x8: `solutions incomplete <../solutions/ominoes/hexominoes-plus-27x8.txt>`__ .. image:: images/ominoes/hexominoes-plus-27x8.png - 36x6: `solutions incomplete <../solutions/ominoes/hexominoes-plus-36x6.txt>`__ .. image:: images/ominoes/hexominoes-plus-36x6.png Misc ---- - 15x15 square with a central 3x3 hole (design from Kadon's `Sextillions`_): `solutions incomplete <../solutions/ominoes/hexominoes-plus-square.txt>`__ .. image:: images/ominoes/hexominoes-plus-square.png One-Sided Hexominoes ===================== These are just like regular hexominoes, except that non-isomorphic reflections (different shape when flipped over) are treated as separate pieces, and pieces are not allowed to be flipped. Rectangles ---------- - 20x18: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-20x18.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-20x18.png - 24x15: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-24x15.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-24x15.png - 30x12: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-30x12.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-30x12.png - 36x10: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-36x10.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-36x10.png - 40x9: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-40x9.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-40x9.png - 45x8: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-45x8.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-45x8.png - 60x6: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-60x6.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-60x6.png - 72x5: `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-72x5.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-72x5.png - 40x10 rectangle with a central hole (design from `Andrew Clarke's Poly Pages [Hexomino Constructions]`_): `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-holey-rectangle-1.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-holey-rectangle-1.png Misc ---- - Diamonds: .. list-table:: :class: borderless * - .. figure:: images/ominoes/one-sided-hexominoes-diamond-1.png design from `Andrew Clarke's Poly Pages (Hexomino Constructions)`_; `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-diamond-1.txt>`__ * - .. figure:: images/ominoes/one-sided-hexominoes-diamond-2.png `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-diamond-2.txt>`__ - Square fort (design by David Bird from `Andrew Clarke's Poly Pages [Hexomino Constructions] `_): `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-square-fort.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-square-fort.png - Six crosses (design from `Andrew Clarke's Poly Pages (Hexomino Constructions) `_): `solutions incomplete <../solutions/ominoes/one-sided-hexominoes-six-crosses.txt>`__ .. image:: images/ominoes/one-sided-hexominoes-six-crosses.png Cornucopia Puzzle ================= Invented by Stewart T. Coffin, this is a semi-arbitrary subset of the hexominoes. Many Cornucopia puzzles described in Coffin's `Puzzling World of Polyhedral Dissections `_ then only use a subset of this subset. From the set of hexominoes, eliminate all pieces having reflexive or rotational symmetry and all those containing a 2 x 2 square because they are less desirable for various reasons already explained. The remaining 17 pieces are the set of Cornucopia pieces. -- `The Puzzling World of Polyhedral Dissections, by Stewart T. Coffin `__ These are all the pieces listed in `An Introduction to Polyominoes: Hexominoes `__ with 8 aspects which do not contain 2x2 blocks. The Polyform Puzzler hexomino names of the pieces are listed under the first puzzle below. * 17x6 rectangle (suggested by Dan Klarskov): `162,086 solutions <../solutions/ominoes/cornucopia-17x6.txt>`__ .. image:: images/ominoes/cornucopia-17x6.png The hexomino piece names of the Cornucopia puzzle are (in roughly equivalent relative positions to the pieces in the puzzle image above):: F06 N16 F36 F16 G06 T16 H06 L06 F26 W26 J06 U06 Z16 M06 W06 Y06 V06 .. |c| unicode:: U+00A9 .. copyright sign