Puzzling Stack Exchange is a question and answer site for those who create, solve, and study puzzles. It only takes a minute to sign up.
Sign up to join this community
Catastrophe! During a last-minute mad scramble, a White pawn was struck off the board into oblivion. Neither player can remember where to place it. The piece count is looking bad for Black. Additionally, White is threatening a mate in one.
Two things may save Black yet.
In how many ways can Black legally place the missing White pawn to create a saving grace?
Wolfgang Dittmann, Die Schwalbe 11/10/1968
Suppose White has castling rights to threaten 0-0-0#, the only mate in one when the position is considered in isolation. Then the pieces on a1, e1, g2 and g7 must not have moved, while that on h3 must have made exactly one move (h3). It is easy to see that the only legal way the Black king could have reached g1 given these fixed piece constraints is through g3 and h2, while the rook on h1 is a promoted pawn. The number of Black pawn shifts across files, each of which can only be effected by a capture, is at least 6 (see loopy walt's answer), but White had 11 pieces before the incident, which is a contradiction. Thus White must have no castling rights and Black may place the white pawn in any empty legal position that does not prevent the mate in one – 39 positions in all, less d2, e2 and f2.
Proof game for the position as displayed (without the extra White pawn):
1. e4 d5 2. exd5 Nf6 3. h3 Na6 4. Be2 Kd7 5. d6 Ke6 6. Bb5 Kf5 7. Nf3 Kf4 8. Nd4 Rb8 9. f3 Kg3 10. Rf1 Kh2 11. Rf2 Kg1 12. Re2 h5 13. Na3 h4 14. d3 Nc5 15. Bf4 Na6 16. Bg3 hxg3 17. f4 Nc5 18. Nf5 Na4 19. Nh6 Nc5 20. Ng4 Na6 21. Nh2 gxh2 22. Bc4 h1=R 23. Nb5 Rh5 24. Nd4 Nd5 25. Qd2 Nb6 26. Be6 Na4 27. Bg4 f5 28. Nb5 fxg4 29. Nd4 g3 30. Nf5 N6c5 31. Nh6 Rd5 32. Ng4 Rd4 33. Nh2 gxh2 34. Qc3 Bg4 35. Qc4 Bh5 36. Qe6 Bg4 37. Re5 Bd1 38. Qf6 exf6 39. Rg5 fxg5 40. f5 Nb6 41. a3 Nbd7 42. b3 Ne5 43. c3 Ng6 44. fxg6 Ne6 45. a4 Be7 46. b4 Nf8 47. c4 Nh7 48. gxh7 Rf4 49. h8=B Bb3 50. d7 Qf8 51. d8=Q a6 52. a5 b6 53. b5 c6 54. c5 Ba4 55. axb6 Bb3 56. bxa6 Rc8 57. Qc7 Bd6 58. Qxc6 Qf5 59. Qe8 Rd8 60. c6 Qb5 61. a7 Qa4 62. a8=N Qa7 63. b7 Qa4 64. b8=N Qa7 65. c7 Qa4 66. c8=B Rd7 67. Qxd7 Qc4 68. Qa4 Qg8 69. Qxb3 Bb4+ 70. Qxb4 Rf6 71. d4 Rf4 72. d5 Rf6 73. d6 Rf4 74. d7 Rc4 75. d8=R Rc5 76. Rd2 Rb5 77. Rc2 Qc4 78. Qxc4 Rf5 79. Qd5 Rf8 80. Qe6 Rg8 81. Qxg8 g4 82. O-O-O#
Can white castle?
only if the black king and rook (as a pawn) came in via h2. For this 3 black pawns on top of the native g pawn must have made their way to the g-file (min 4 white pieces captured) and two of them on to the h-file (another 2 white pieces). But counting the knocked-over pawn white still has 11 of their pieces.
Proof game showing that apart from that there are no restrictions
1. h3 h6 2. Nf3 f5 3. Ng5 hxg5 4. b4 e5 5. Bb2 e4 6. Be5 d5 7. Bg3 f4 8. Rh2 fxg3 9. a4 d4 10. b5 d3 11. cxd3 Bf5 12. dxe4 Ne7 13. exf5 Ng6 14. fxg6 Rh7 15. gxh7 gxh2 16. Qb3 Ke7 17. Qd5 g4 18. Qxb7 Kd6 19. Kd1 Kc5 20. Kc2 Kb4 21. Qxa7 Nd7 22. Qxc7 Rc8 23. Kd3 Kb3 24. Qe5 Rc1 25. Qe6+ Kb2 26. Qg8 Rxf1 27. h8=B Rh1 28. b6 Kc1 29. b7 Kd1 30. a5 Ke1 31. Kc2 Kf1 32. Kd1 Kg1 33. Ke1 Nc5 34. Nc3 Nd7 35. a6 Nb6 36. a7 Nc8 37. bxc8=B Qe7 38. Nb5 Qd7 39. Nc7 Bd6 40. Na8 Bb8 41. axb8=N
White still threatens mate in two (assuming 1 sec = 2 moves):
Rf2 Ke2 or Re2 Kd2 or Bb7 Ke2 or or Bxg7 Bd4 or Qd5 Ke2 or Qd5 Qd4 or Qf7 Qf2 or Qf7 Qf1 etc. The only pawn preventing all of them is one on e2
