Author: | David Goodger <goodger@python.org> |
---|---|

Date: | 2015-02-24 |

Revision: | 600 |

Web site: | http://puzzler.sourceforge.net/ |

Copyright: | © 1998-2015 by David J. Goodger |

License: | GPL 2 |

Contents

High-quality acrylic sets of hexominoes are available from Kadon Enterprises as Sextillions.

Due to a parity imbalance, the 35 free hexominoes cannot fit into a simple rectangle without introducing holes or other irregularities.

Square: solutions incomplete

19x11 rectangles plus nubs:

45x5 rectangle with holes (design by W. Stead from Andrew Clarke's Poly Pages [Hexomino Constructions]): solutions incomplete

33x7 rectangle with a central hole (design from Andrew Clarke's Poly Pages [Hexomino Constructions]): solutions incomplete

17x15 rectangle with a central plus-shaped hole (design from Polyominoes, by Solomon W. Golomb): solutions incomplete

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

21x10 parallelogram (design from Andrew Clarke's Poly Pages [Hexomino Constructions]): solutions incomplete

35x6 parallelogram: solutions incomplete

Triangle (design from Andrew Clarke's Poly Pages): solutions incomplete

Rhombus (design by David Bird from Andrew Clarke's Poly Pages [Hexomino Constructions]): solutions incomplete

Cross (design from Andrew Clarke's Poly Pages (Hexomino Constructions)): solutions incomplete

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.

18x12: solutions incomplete

24x9: solutions incomplete

27x8: solutions incomplete

36x6: solutions incomplete

15x15 square with a central 3x3 hole (design from Kadon's Sextillions): solutions incomplete

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.

20x18: solutions incomplete

24x15: solutions incomplete

30x12: solutions incomplete

36x10: solutions incomplete

40x9: solutions incomplete

45x8: solutions incomplete

60x6: solutions incomplete

72x5: solutions incomplete

40x10 rectangle with a central hole (design from Andrew Clarke's Poly Pages [Hexomino Constructions]): solutions incomplete

Diamonds:

Square fort (design by David Bird from Andrew Clarke's Poly Pages [Hexomino Constructions]): solutions incomplete

Six crosses (design from Andrew Clarke's Poly Pages (Hexomino Constructions)): solutions incomplete

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

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