To answer the question, which convex forms have tilings with the double DiDom tile set, seem easy at first sight. But if I looked deeper into the material, I found lots of difficulties and there are many more solutions than I thought. Finally the problem remains unsolved although there are partial results.

The first approach is to find all forms of area 52, which are enclosed by Doms in the primary grid. I call those regular where the vertices of the resulting forms fit to the base grid too. Finding all regular forms is relatively easy to achieve by using a straightforward recursive algorithm. I did not find a reasonable method to find all irregular forms, but I can show some examples ( [Logelium] More DiDom Kite Rings).

The next step is to find all kite rings with all positions, which can live inside the forms found. (see some of them: [Logelium] DiDom Kite Rings). Each ring is tilted by the angle λ in positive or negative direction. Note that forms can contain more than one ring may be different too. This leads to an enormous number of variants illustrated by the ring selection.

Each form must have at least a solution with Kites, Dominoes and Doms before we can think of solutions with DiDoms. This is for forms with an odd number of cells obviously not the case. Cells are all units of the base grid which are at least partially covered by the form.

The pictures show a kite ring example and the corresponding positions of kites, doms and dominoes.

To calculate solution with the Logelium solver, we have to create a form definition file for each ring constellation. These files are of course generated and sent to the solver by some scripts. The following table shows the accumulated results of the findings.

 Type ConvexeForms OddCells NoRing WithSolution SimpleRing SimpleSolution DoubleRing DoubleSolution Tetragon 29 0 17 9 210 77 173 0 Pentagon 94 13 14 38 2167 430 714 7 Hexagon 302 69 27 77 7366 1032 1926 25 Heptagon 429 171 2 55 9533 597 1050 26 Octagon 357 155 1 53 8375 667 1056 13 Nonagon 145 80 0 13 2919 124 191 3 Decagon 57 35 0 4 1065 41 78 0 Undecagon 3 1 0 0 92 0 0 0 Dodecagon 2 0 0 2 103 18 0 0 Sum 1418 524 61 251 32830 2986 5188 74

»Convex Forms« is the total number of convex forms, »Odd Cells« counts the forms with an odd number of cells. If this number is odd, there is no solution even with Doms only. The forms »No Rings« have no kite rings at all, because they are no narrow. The number of forms, which have at least one kite ring with solutions, is counted in »With solution«. We can observe that the solutions concentrate on very few forms..

The convex form of a category have »Simple Ring« simple ring positions, where »Simple Solution« is the number of ring constellations with solutions. The double ring configurations are »Double Ring« with »Double Solution«.

The numbers are not cleared of symmetrical configurations. Hopefully the tricky scripts have no bugs, so that a ring configuration escaped the net.

### I cannot present all forms and solution here, so I choose “one dog from each village” .

The given numbers of solutions belong to the shown ring configuration only.

The 1-2 kite ring has 36 possible positions in a 4*13 rectangle. If we discard all, which a symmetry redundant, there are five different positions. The pictures show those rings tilted +λ on the left side. The total number of solutions of all ring positions is1227 * 4.

 1175 solutions with all the same kite position 20 solutions 2 kite positions 12 solutions 2 kite positions 20 solutions 2 kite positions no solution

This is one of the 38 convex pentagon forms with solutions. I choose two of 25 solvable ring configurations of this form.

 161 solutions 20 solutions
 There are 81 * 2 solutions of this symmetrical arrangement. The outer form has 22 more solvable ring configurations, but this is the only fully symmetric. This diamond has19 solutions.
 This completely symmetrical heptagon with two kite rings has 12 * 2 solutions. A not very spectacular form with 40089 solutions.
 A diamond with 39 solutions. With 100 solutions a nice s-symmetrical form. A good starting point for constructing irregular forms. (13 solutions)
 There are only a few symmetrical nonagons. This has 27 solutions.
 A superb form with three solutions.

There are no solvable Undecagons. The only solvable Dodecagon is extensively explained in the section [Logelium] About  Kites. Forms with even more vertices are impossible as you can easily see.

Impressum