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

Date: | 2016-12-05 |

Revision: | 643 |

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

Copyright: | © 1998-2016 by David J. Goodger |

License: | GPL 2 |

Contents

The free hexiamonds consist of 12 6-triangle polyforms, for a total of 72 unit triangles.

4x9: 74 solutions

6x6: 156 solutions

The 3x12 parallelogram has no (0) solutions.

12x3: 29 solutions

6x6: 1004 solutions

4x9: 1004 solutions

Stacked chevrons 6x6 (suggested by Dan Klarskov, under the name "polyam2"): 933 solutions

Stacked chevrons 12x3, 1: 269 solutions

Stacked chevrons 12x3, 2: 114 solutions

Stacked chevrons 12x3, 3: 46 solutions

4x10 elongated hexagon (clipped parallelogram): 856 solutions

5x8 stacked long hexagons: 378 solutions

4x12 stacked hexagons: 51 solutions

Snowflake (hexagon with tabs, radially symmetrical; a.k.a Kadon's "Iamond Hex"): 55 solutions

Kadon's "Iamond Hex" challenges (pieces come in three different colors):

Ring 2 (4-unit side-length hexagon with 2-unit hexagonal hole offset one unit from the center): 11 solutions (no solutions for the ring with a centered hole)

Crescents (4-unit side length hexagons with 2-unit hexagonal bites removed):

6x8 "coin" (5x3 elongated hexagon with central hexagonal hole): 304 solutions

"Gyroscope" (design by Oktavian Scharek): 19 solutions

Irregular hexagon 7x8 (suggested by Dan Klarskov, under the name "hexui"): 5885 solutions

Spiked hexagons:

4x3 semiregular hexagon with a 1-unit hole (design from Wolfram Mathworld): 710 solutions

5x2 semiregular hexagons:

Notched hexagon (design from Kadon's "Iamond Hex" booklet): 1 solution

Puzzles that use 9 of the 12 hexiamonds to form an regular hexagon of side length 3.

Naively, there are (12 choose 9) = 220 different sets of 9 hexiamonds. However, to form a regular hexagon, an equal number of 'up' and 'down' triangles are necessary. The F6 & P6 hexiamonds are unbalanced (two 'up' triangles and four 'down', or vice-versa), while all other hexiamonds are balanced (three of each). So puzzles must either include or exclude both F6 & P6. Therefore there are actually (10 choose 3) + (10 choose 1) = 120 + 10 = 130 possible hexiamond subsets for these puzzles.

Which 9 hexiamonds to choose? Here are some ideas.

Hexicator: The "Hexicator" is Col. George L. Sicherman's puzzle consisting of 9 hexiamonds (the club/crook/J6, pistol/signpost/H6, and shoe/hook/G6 pieces are omitted) to be formed into a regular hexagon (side length 3). Try it before you look at the 1 and only solution (picture here).

Perhaps this was the first set of 9 pieces that Col. Sicherman found that can make a hexagon in only one way. Is it the only set?

Choose-9 Hexagon 2: The pieces omitted from this puzzle are the three hexiamonds that cannot be formed with two triamonds: E6/crown, H6/pistol/signpost, and V6/lobster.

Suggested by Michael Spencer on 2013-11-24:

My local cafe has a wooden hexiamond puzzle that I thought might interest you. It is very similar to one puzzle you mention [Hexicator] but instead of omitting the G, H and J pieces it omits E, H and V (in other words, precisely those pieces that cannot be formed from two triamonds). I am now very curious to know how many solutions this version has!

Choose-9 Hexagon 3: The pieces omitted from this puzzle are the three hexiamonds that cannot be formed with three diamonds: F6/yacht, P6/sphinx, and X6/butterfly.

4x3 parallelogram: 142 solutions

5x3 trapezoid: 144 solutions

3x2 semiregular hexagon: 103 solutions

4x11 trapezoid: 76 solutions

5x10 trapezoids with holes:

5x8 stacked long butterflies: 290 solutions

4x12 stacked butterflies: 26 solutions

All the solutions are also solutions to the 4x12 stacked hexagons above, merely by rearranging the sections. In fact, the sample solutions shown here are equivalent.

Trefoils:

Tenyo's puzzle: 4968 solutions

This is the tray configuration used by the hexiamond PlaPuzzle produced by Tenyo Inc. of Japan.

Heart (design by Dan Klarskov): 4,154 solutions

Spinners:

Deltas:

Infinities:

Knobby bone (design from Thimo Rosenkranz's pentoma.de): 1 solution

Near-hexagram (design from Kadon's "Iamond Hex" booklet): 2 solutions

Trilevel (design via John Greening): 1 solution

The one-sided hexiamonds consist of 19 6-triangle polyforms, for a total of 114 unit triangles.

Long hexagon 8x3 [11x6]: solutions incomplete

Tabbed hexagon: solutions incomplete

Notched hexagon ring: solutions incomplete

Knobbed hexagon: 137 solutions

Truncated hexagram ring (design from Thimo Rosenkranz's pentoma.de): 16 solutions

Hexagrams × hexagons:

Hexagram with one omitted piece: 119 solutions (78 of which omit a symmetrical piece, 41 asymmetrical)

Solutions are possible omitting all but the "C6", "F6", & "P6" pieces. This solution omits the "O6" piece (shown to the side):

These shapes are all based on the hexagonal grid, made up of 19 6-triangle hexagons.

O'Beirne's hexiamond hexagon (19 small 6-triangle hexagons, arranged in a hexagon on a honeycomb grid: 1 hexagon surrounded by 6 hexagons surrounded by 12 in concentric rings): 124,519 solutions

The number of solutions and their distribution amongst sub-puzzles agrees with what is reported in Dancing Links by Donald E. Knuth. The puzzle and its history is briefly described in Knuth's paper, and is described in depth in "O’Beirne’s Hexiamond" by Richard K. Guy (a chapter in the book "The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner").

6x2 elongated hexagon: solutions incomplete

10x2 trapezoid: solutions incomplete

4x4 butterfly: solutions incomplete

Triangle (there are 2 too many hexagons in a length-6 triangle; I think this is the most elegant variation): solutions incomplete

Two triangles: solutions incomplete

Butterfly 11x3 [11x6]: solutions incomplete

Bumpy triangle: 968,744 solutions