Considering how fast the number of n-minoes grows with increasing n, I should have expect the problem of dividing an n*n square into n n-minoes to have even faster growth in number of solutions.
If a computer can find a solution in under a second, the 9*9 square packing of tetrominoes and pentaminoes sounds like a good first build for such puzzles.Looking at Wikipedia's article on the nonominoes
, I see that the vast majority are asymmetric, and only six have higher symmetry. The D4 nonominoes are obviously the 3*3 square and the cross formed by two I pentominoes intersecting at their middle square, but I can't easily visualize the D2 nonominoes without a description, nor does the article mention if any of the nonominoes with holes are symmetric. If we require the solution to include both D4 nonominoes, any D2 nonominoes that lack a hole, and restrict the choice for the remainding pieces to the symmetric nonominoes, does the problem become tractable while still producing a solution? If all 4 of the D2 nonominoes lack holes, are their any solutions containing both D4, all 4 D2, and one each with a line of reflection aligned to the grid, a line of reflection at a 45 degree angle to the grid and point symmetry?
I would ask for a link to Jaap's solver, but I have a feeling I wouldn't be able to use it.