Some Fully Solved Heptiamond Shapes

The twenty-four Heptiamonds can form many interesting shapes but it takes a lot of time to discover all solutions for most of them. Here I will present only three shapes for which I have been able to calculate all solutions.

List of shapes on this page:

• Hexagon with Seven Holes - 686 solutions;
• Hexagon with Seven Holes Second Form - 6002 solutions;
• Gasket with Three Holes - 9 solutions;
• Tetrahex Twin Shape - 3 solutions.

Hexagon with Seven Holes

This shape has been discovered by J.H.Hindriks and be seen on his website or on David Goodger's Puzzler. It has exactly 686 solutions that took my computer more nearly 24 hours to calculate. The shape has axial and six-rotary symmetry which means that each solution has 12 symmetric twins. With these in mind, it is easy to see that the X piece can be placed in just seven different positions. Each of them leads to solutions, as is shown below.

First position for X piece -- 87 solutions.

Second position for X piece -- 10 solutions (least of all positions).

Third position for X piece -- 68 solutions.

Fourth position for X piece -- 19 solutions.

Fifth position for X piece -- 228 solutions.

Sixth position for X piece -- 12 solutions.

Seventh solution for X piece -- 262 solutions (most of all positions). The solution on Puzzler belongs to this category.

Hexagon with Seven Holes - Second Form

This shape is also an invention of J.H.Hindriks but unlike the previous one, there are only four possible positions for the X piece. On the other hand, the total number of solutions is nearly ten times bigger: 6002.

First position for X piece gives 1494 solutions.

Second position for X piece gives 2799 solutions, which is the biggest number.

Third position for X piece gives the least number of solutions, only 415.

Fourth position for X piece gives 1295 solutions.

Gasket with Three Holes

This shape comes from Puzzler and it has three very big holes which leads to a very smaller number of solutions. In fact, there are only 9 of them. Further more, these divide into three groups of, one of just one solution, and two of four solutions each. I have shown below a representative of each group. Symmetric regions, which can be rotated to give the other solutions in each group, are coloured.

Mirror reflections of the two red regions give three more solutions in each case. In his list of terms, J.H.Hindriks calls the upper region a Clover Leaf and the lower one a Gemstone. Notice that the only difference between these two groups is the three-piece regions in the lower left corner, composed of the G, N and T pieces in two distinctively different ways.

This solution forms a group on its own.

A Tetrahex Twin Shape 

David Goodger has remarked that the total surface of the twenty-four Heptiamonds is exactly equivalent to 28 hexagons each comprising six triangles. This means that shapes formed with the Tetrahexes can be copied with Heptiamonds. It would be interesting to the have the complete list of such shapes. I think it would be of human size. For the time being, I took a Tetrahex shape with a unique solution and high symmetry (discovered by Abaroth and present on Puzzler too). It turned out that the Heptiamond twin has exactly three solutions, which is the least I know for a Heptiamond shape at all. Are there symmetric Heptiamond shapes with just one solution?

Abaroth's Tetrahex shape and its unique solution is on the right. On the left is one solution with Heptiamonds and another one, which can be transformed by mirroring the red regions. All in all, only three Heptiamond solutions.