brtzsnr wrote:The second category is still open. As a reminder in the second category the positions must be reached by following a series of legal moves from the start position.
What do you mean by that? Practically any reasonably looking position is reachable by a series of legal moves from the initial position.
I did not look in detail at all positions listed by Peter, but they do seem to be reasonable. I'm sure most if not all of them are reachable by a series of legal moves from the initial position.
I think all positions in my list are reachable from the starting position. Take for example this position:
[D]2b1kqr1/p2p3p/3p4/p2PpP2/PpP2p2/6P1/8/RRB1KQ1N w - - 0 1
It can be reached from the starting position by the following sequence of moves:
[Event "Edited game"]
1. e4 c5 2. Ba6 bxa6 3. b4 g6 4. f4 Bh6 5. f5 Bf4 6. g3 Nf6 7. gxf4 Nh5 8.
c4 cxb4 9. Ne2 Ng3 10. Nd4 a5 11. hxg3 Nc6 12. Nxc6 Rb8 13. Nxb8 g5 14. d4
f6 15. Na6 gxf4 16. Nc5 Rg8 17. Nb7 Rh8 18. Nd6+ exd6 19. e5 fxe5 20. d5
Qf6 21. a4 Rg8 22. Nd2 Qf7 23. Rb1 Qf6 24. Ne4 Qf7 25. Rh2 Qf6 26. Nf2 Qf7
27. Nh1 Qf6 28. Qd3 Qf7 29. Qf1 Qf6 30. Ra2 Qf7 31. Raa1 Qf8
Writing a program to find such move sequences could be an interesting challenge. A* or IDA* seems like good algorithms to use, but you need a powerful heuristic function to reduce the search space enough to be able to find long move sequences.