It's the same problem as with generating good sudoku boards. It's not easy, and there's not many publicly available solutions, but solutions exist.
A common opinion is that a good board is solvable without the use of backtracking. A set of known techniques should be enough to solve the board. To validate if a board is "fun" you need to have a program that can solve the board using these known techniques. Making that program is much harder than just making a general solver. And then you need to find the boards that can be validated as fun. Either you search through random boards, or you get clever...
I've noticed that puzzles that can be solved with CP-SAT's presolver so that the SAT search does not even need to be invoked basically adhere to this (no backtracking, known rules), e.g.:
#Variables: 121 (91 primary variables)
- 121 Booleans in [0,1]
#kLinear1: 200 (#enforced: 200)
#kLinear2: 1
#kLinear3: 2
#kLinearN: 30 (#terms: 355)
Presolve summary:
- 1 affine relations were detected.
- rule 'affine: new relation' was applied 1 time.
- rule 'at_most_one: empty or all false' was applied 148 times.
- rule 'at_most_one: removed literals' was applied 148 times.
- rule 'at_most_one: satisfied' was applied 36 times.
- rule 'deductions: 200 stored' was applied 1 time.
- rule 'exactly_one: removed literals' was applied 2 times.
- rule 'exactly_one: satisfied' was applied 31 times.
- rule 'linear: empty' was applied 1 time.
- rule 'linear: fixed or dup variables' was applied 12 times.
- rule 'linear: positive equal one' was applied 31 times.
- rule 'linear: reduced variable domains' was applied 1 time.
- rule 'linear: remapped using affine relations' was applied 4 times.
- rule 'presolve: 120 unused variables removed.' was applied 1 time.
- rule 'presolve: iteration' was applied 2 times.
Presolved satisfaction model '': (model_fingerprint: 0xa5b85c5e198ed849)
#Variables: 0 (0 primary variables)
The solution hint is complete and is feasible.
#1 0.00s main
a a a a a a a a a a *A*
a a a b b b b *B* a a a
a a *C* b d d d b b a a
a c c d d *E* d d b b a
a c d *D* d e d d d b a
a f d d d e e e d *G* a
a *F* d d d d d d d g a
a f f d d d d d *H* g a
*I* i f f d d d h h a a
i i i f *J* j j j a a a
i i i i i k *K* j a a a
Great theory. But as you can see in the picture:
1. This thing is very carefully crafted.
2. The holes have different sizes.
I don't think this is by chance. There must be a reason for this and the explanation, be it coin counting, knitting or whatever, has to take this into account.
With so many different coin sizes and types in the empire, I think this makes most sense.
Wikipedia also mentions this:
> Several dodecahedra were found in coin hoards, suggesting either that their owners considered them valuable objects, or that their use was connected with coins — as, for example, for easily checking coins fit a certain diameter and were not clipped.