Since it seems at least some people are enjoying these I'm continuing the series.
This one is not quite as elegant, but still interesting.
Can White Castle?
I present to you, pure inefficiency:
1. Nf3 f6 2. Nh4 Kf7 3. Nf5 Ke6 4. Nd6 Ke5
5. Nc4+ Kf4 6. Nb6 Nh6 7. Nxc8 Ng4 8. Nb6 Nc6
9. Nxa8 Nd4 10. Nb6 Nf3+ 11. exf3 Ne5 12. Bd3 Nc4
13. Bg6 hxg6 14. Na4 Rh5 15. Nac3 Rh6 16. Qe2 Rh3
17. Qe4+ Kg5 18. Qf5+ gxf5 19. Na3 Ne3 20. Ncb5 Kf4
21. Nd4 Nc4 22. Nab5 Ne5 23. Na3 Nd3+ 24. cxd3 Ke5
25. Nc6+ Kd5 26. Nb4+ Kd4 27. Nd5 Kc5 28. Nc3 Kb4
29. Ne4 fxe4 30. Nc2+ Kc5 31. Na3 exd3 32. Nb5 Rh4
33. Nc3 Rc4 34. Nd1 Rxc1 35. Rb1 Rc4 36. Rc1 Kb4
37. Rc2 Rh4 38. Ne3 Rh3 39. Nd1 dxc2 40. Nc3 Kc4
41. Nd1 Kd3 42. Nc3 Rg3 43. Nd5 Rg4 44. Ne3 Rg5
45. Nd1 Rg3 46. Ne3 c1=R+ 47. Nd1 Kc2 48. h3 Ra1
49. h4 Kc1 50. Nc3 Rh3 51. Ne4 Rh2 52. Nc3 Kc2+
53. Nb1 Kd3 54. h5 Kd4 55. h6 Ke5 56. h7 Ke6
57. h8=B Kf7 58. d3 Ke8 59. O-O
It can definitely be done faster, but this works.
Yes! White can castle in that game.
White had castle variant: Apronus PGN Viewer
1. Nc3 Nc6 2. Nd5 Ne5 3. Nb6 Nf3+ 4. exf3 f6 5. Bd3 Kf7 6. Bg6+ hxg6 7. Qe2 Nh6 8. Qe4 Ng4 9. Nxc8 Ne5 10. Nb6 Nd3+ 11. cxd3 Rh5 12. Qf5 gxf5 13. Ne2 Rh4 14. Ng3 Rc4 15. Ne4 fxe4 16. Nxa8 exd3 17. Nb6 Ke6 18. Nd5 Ke5 19. Ne3 Kd4 20. Nd1 Rxc1 21. h3 Rc4 22. Rc1 Rc5 23. Rc2 dxc2 24. Ne3 Kd3 25. h4 c1=R+ 26. Nd1 Ra1 27. h5 Kc2 28. h6 Kc1 29. h7 Rh5 30. h8=B Rh2 31. O-O Kc2 32. Nc3 Kd3 33. Nb1 Kd4 34. d3 Ke5 35. Re1+ Kf5 36. Re4 Kg6 37. Kf1 Kf7 38. Ke1 Rh5 39. Rh4 Ke8 40. Rh1 Rh2
White haven't castle variant: Apronus PGN Viewer
1. Nc3 Nc6 2. Nd5 Ne5 3. Nb6 Nf3+ 4. exf3 f6 5. Nxc8 Kf7 6. Nb6 Ke6 7. Nxa8 Nh6 8. Nb6 Ng4 9. Bd3 Ne5 10. Bg6 hxg6 11. Nd5 Nd3+ 12. cxd3 Ke5 13. Qe2+ Kd4 14. Qe4+ Kc5 15. Qf5 gxf5 16. Ne2 Rh4 17. Ng3 Rc4 18. Ne3 Rxc1+ 19. Nd1 Rc4 20. Ne4+ fxe4 21. Ne3 exd3 22. Nd1 Kd4 23. Rc1 Rc5 24. Rc2 dxc2 25. Ne3 Kd3 26. Nd1 Rf5 27. Nc3 c1=R+ 28. Nd1 Ra1 29. h3 Kc2 30. h4 Kc1 31. Nc3 Rg5 32. Nb1 Rg3 33. Nc3 Rh3 34. Nd1 Rh2 35. Nc3 Kc2+ 36. Nb1 Kd3 37. h5 Kd4 38. h6 Ke5 39. d3 Ke6 40. h7 Kf7 41. h8=B Ke8 42. O-O
Wrong answer (I made a boneheaded mistake, kindly pointed out in comments by @RShields, to which I have drawn attention below.)
The WBh8 obviously got there by promotion.
The only WP able to reach h8, given the existing WP structure, is the h-pawn.
There is no way the BQR can possibly have escaped from behind the pawns.
Therefore at least one of the BRs currently on the board is promoted.
If it promoted on f1, g1, or h1 then the WK or WR must have moved and W can't castle.
If it ever reached d2 then the WK must have moved and W can't castle.
If it promoted on d1 then the WK must have moved and W can't castle.
So if W can castle, some BP went from c2 to either b1 or c1 and promoted to a R, still on the board.
The only available BP is the h-pawn. To get to c2 it must have made 5 captures.
W is only missing 6 pieces, so all of them must have been taken by that pawn. <-- This is the boneheaded mistake; between writing the previous line and writing this one the number 5 metamorphosed in my brain into the number 6. D'oh!
Those captures all have to have been on white squares.
But one of W's missing pieces is the WQB (which, note, is not the thing now on h8) which can't possibly have been captured on a white square.
Hence W cannot castle.