When presenting a shape that can be tiled with pentominoes or hexiamonds, it is customary to indicate the total number of non trivial solutions, i.e. without mirror and rotation derivates. In 1958 Dana Scott discovered that a chess board with removed center has exactly 65 pentomino tilings and many more shapes have been thoroughly solved since then.
Various polyform researchers have looked for the shapes with maximum tilings with pentominoes and hexiamonds. Although there are no mathematical proofs, the numbers are 16720 (pentominoes -- discovered by Aad van de Wetering) and 14600 (hexiamonds -- discovered by Mike Reid and Patrick Hamlyn). This means that any pentomino or hexiamond shape can be exhaustively solved in the matter of minutes, depending on available hardware.
However, when we move to heptiamonds, things become incredible more complicated. Patrick Hamlyn has estimated that the following heptiamond shape has 10^14 solutions, which is far too many to calculate (or save and manipulate):
However, there exist shapes which can be fully solved, especially symmetrical holey ones. Here I intend to compile a list of such shapes. Many of them are bi-tilable. It is impossible to provide the actual solutions, but I'll send them on request (contact details at the bottom of this page).
| Shape | Solutions | Notes |
|---|---|---|
| 3 | Heptiamond version of a tetrahex shape by Abaroth. In tetrahexes there is just one solution. Out of the 11 ternary bi-tilable shapes, this one is the only to have six holes. | |
| 9 | Discovered on David Goodger's site where it was not fully solved at the time. I messaged him the total number of solutions which he checked and published. | |
| 157 | The hole in the middle can be filled with hexiamonds. | |
| 380 | The hole in the middle can be filled with hexiamonds. | |
| 4 | Bi-tilable with tetrahexes. The shape itself and its tetrahex solution were provided by George Sicherman. Out of the 11 ternary bi-tilable shapes, this one is the only to have holes of two hexes. | |
| 9 | Bi-tilable with tetrahexes. The hole in the middle is 14 hexagons (= 84 triagles) wide, which is exactly one half of the size of the set itself. | |
| 70 | Bi-tilable tetrahex shape with six holes and mirror symmetry. Found on Abaroth's site. | |
| 141,995 | This shape was proposed and fully solved by Patrick M. Hamlyn in a little less than 33 hours. It is the first heptiamond shape with a big number of solutions to be fully solved. The shape can also be seen on David Goodger's site. | |
| 1 | Bi-tilable shape with a hole of 19 hexagons, discovered by George Sicherman. This is the biggest known bi-tilable farm as of September 2020. Its perimeter is 86, which is just 9 under the absolute theoretical maximum of 95 for bi-tilable shapes. | |
| 6,003 | Similar to this but with more space on the edge. discovered this shape on David Goodger's site where J. H. Hindriks is indicated as author. The number of solutions has been checked by Patrick M. Hamlyn. | |
| 27 684 | ||
| 13,602 | One in two holes is close to the edge. This shape has six symmetrical forms (three from rotations and double that because of axial symmetry) and thus has roughly twice as many trivial solutions as this shape. | |
| 14,864 | Four out of six holes are close to the edge. This shape has four symmetrical forms (two axes of symmetry) and thus has roughly 60,000 trivial solutions. | |
| 686 | I discovered this shape on David Goodger's site where J. H. Hindriks is indicated as author. Compare to this. | |
| 3,358,174 | The solutions for this shape were calculated by Patrick M. Hamlyn in less than 4 hours and a half. Out of the 333 heptahexes, all of which can be tiled with heptiamonds, this one is very likely with the least number of solutions. | |
| 5,582,395 | The tilings of this highly symmetric shape have been counted by Patrick M. Hamlyn who invested 272 hours of CPU work in it. The shape has been designed by Johannes H. Hindriks, as reported by David Goodger. | |
| 2,719 | A side 2 decahex with a dodecahex hole that can hold the hexiamonds. | |
| 37,136,072 | Another shape that has been fully solved by Patrick M. Hamlyn. It is very particular in that it is composed of two unconnected highly symmetrical shapes. Rotating or mirroring the flowers independantly doesn't count as a new solution. Hamlyn has found that there are 24C12 = 2,704,156 ways of selecting 12 pieces. 1,894,774 of these tile a flower shape, 228,167 of them uniquely. Of the 1,894,774 we have 569,621 where the other 12 also tile the shape. Of the 228,167 we have 7,973 where the other 12 also tile uniquely (one such serves as a visual illustration here). Notice that if the flowers touch, sharing just a single triangle edge, the number of solutions for the shape thus obtained would be 144 times greater because the two shapes will no more be able to be rotated or mirrored. And this means more than 5 billions solutions! | |
| 16 | ||
| 3 | ||
| 2 | Bi-tilable. Swap {A,P} with {S,T} to obtain the other solution. | |
| 675 | This shape has been fully solved by Wen-Shan KAO (高文山). It is remarkable in that it has 12-fold symmetry and 12 holes. | |
| 18,166,625 | Tilings were counted by Patrick Hamlyn in nearly 128 hours. The shape is an example of the similar hole problem and has ternary symmetry. It has been discovered by Mike Reid, as reported on the Poly Pages. | |
| 25,081,583 | Tilings were counted by Patrick Hamlyn in nearly 141 hours. The shape is an example of the similar hole problem and has ternary symmetry. |