Parallelograms

• 4x9: 74 solutions

  

• 6x6: 156 solutions

  

• The 3x12 parallelogram has no (0) solutions.

Chevrons

• 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

  

Hexagons & Variants

• 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):

  Separated colors (1 solution)

  Joined colors (1 solution)

• 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):

  87 solutions

  2 solutions

• 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:

  4 solutions (design from Andrew Clarke's Poly Pages)	3 solutions
  4 solutions (design from Thimo Rosenkranz's pentoma.de)

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

  

• 5x2 semiregular hexagons:

  419 solutions (design from Kadon's "Iamond Hex")

  993 solutions

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

  

"Choose 9" Hexagons

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!

  9 solutions

  

• 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.

  15 solutions

  

Hexagon-Grid Shapes

• 4x3 parallelogram: 142 solutions

  

• 5x3 trapezoid: 144 solutions

  

• 3x2 semiregular hexagon: 103 solutions

  

Miscellaneous

• 4x11 trapezoid: 76 solutions

  

• 5x10 trapezoids with holes:

  68 solutions (design from Kadon's "Iamond Hex" booklet)	256 solutions
  35 solutions

• 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:

  3 joined hexagons: 640 solutions	2 solutions. This is the closest I have been able to get to an order-9 triangle, due to parity imbalances.
  Bumpy trefoil (design by Oktavian Scharek): 15 solutions	2 solutions (design from Kadon's "Iamond Hex" booklet)

• Tenyo's puzzle: 4968 solutions

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

  

• X (designed for G4G10): 11 solutions

  

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

  

• Spinners:

  751 solutions	79 solutions (design by Oktavian Scharek)
  4 solutions (design from Thimo Rosenkranz's pentoma.de)	4 solutions (design from Kadon's "Iamond Hex" booklet)
  1 solution (design from Kadon's "Iamond Hex" booklet)	1 solution (design from Kadon's "Iamond Hex" booklet)

• Deltas:

  11 solutions	232 solutions (inspired by a design by Oktavian Scharek)
  72 solutions

• Infinities:

  15 solutions (design by Oktavian Scharek)

  33 solutions (design from Kadon's "Iamond Hex" booklet)

• 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

  

• Shape with the most solutions? 14,600 solutions

  An irregular shape that is conjectured to have the most solutions:

  Christopher Monckton (creator of Eternity) asked what shape had the most solutions with the 12 hexiamonds. Michael Reid and Patrick Hamlyn worked together to find the following shape, which has 14600 solutions. With that many solutions, it's very easy to find another. Is there another shape with more solutions?

  —http://www.mathpuzzle.com/iamond.htm

  

Hexagons & Snowflakes, etc.

• 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:

  54 solutions (design from Thimo Rosenkranz's pentoma.de)	1 solution (design from Thimo Rosenkranz's pentoma.de)
  solutions incomplete

• 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):

  

Trefoils

solutions incomplete	solutions incomplete
solutions incomplete	solutions incomplete
solutions incomplete	solutions incomplete
solutions incomplete	solutions incomplete
solutions incomplete	solutions incomplete
solutions incomplete

Hexagon-Grid Shapes

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

  

Miscellaneous

• Butterfly 11x3 [11x6]: solutions incomplete

  

• Bumpy triangle: 968,744 solutions