Rook vs Bishop on odd-order board

Question

Suppose White has King and Rook and Black has King and Bishop on a square board of odd size (such as 9x9). Is it generally possible for White to force checkmate?

Answer 1

R vs B is already a general draw on a 7x7 board, even with a "bad" Bishop (moving on squares of the same color as all corners). Presumably this remains the case on all larger odd-order boards.

Curiously on a smaller board even the standard draw with bK in the corner next to its own Bishop doesn't work. For example, I noticed years ago that this position, which would be easily drawn on the 8x8 board, is a win for White on a board with only 6 rows:

[Title "Black to move, White wins"]
[FEN "PPPPPPPP/PPPPPPPP/8/4R3/2b5/8/2K5/k7 w - - 0 0"]

The Black Bishop is dominated; if Ba2 then Re1+ wins, and if Ka2 (or Ba6/f1) then Ra6 is mate.

In 2004 I asked Mark Bourzutschky (who was already doing exhaustive 6-man analysis), and he reported that R vs B is a general win on boards of size 6x6, 6x7, 6x8, but generally drawn on 7x7 and 6x9 and presumably all larger boards. The win on 6x8 can take as long as 110 moves!

[FEN "PPPPPPPP/PPPPPPPP/8/8/8/1k1b4/8/1KR5 w - - 0 1"]

1. Ka1! Bc4 2. Rh1 Bd5 3. Rh5 Be4 4. Rh3+ Kc2 5. Ka2 Bf5 6. Re3 Bd3 7. Rg3 Kc3 8. Ka3 Kd4 9. Kb2 Be2 10. Kc2 Bb5 11. Kd2 Ke4 12. Rb3 Bf1 13. Rb1 Ba6 14. Kc3 Be2 15. Re1 Ke3 16. Kc2 Kf2 17. Kd2 Bg4 18. Re5 Kf3 19. Kd3 Kf4 20. Re1 Bh5 21. Rg1 Bg4 22. Rh1 Be6 23. Rf1+ Ke5 24. Ke3 Kd5 25. Rd1+ Kc4 26. Ke4 Kc3 27. Rd3+ Kb4 28. Kd4 Ba2 29. Rg3 Bb1 30. Rh3 Ba2 31. Rh1 Kb3 32. Kd3 Kb4 33. Kc2 Be6 34. Rh4+ Kc5 35. Kd2 Kd5 36. Ke3 Kc5 37. Rh6 Ba2 38. Kd3 Bb1+ 39. Kc3 Kd5 40. Rh5+ Ke4 41. Rh1 Ba2 42. Re1+ Kf4 43. Kd4 Kf3 44. Rf1+ Ke2 45. Rh1 Be6 46. Ke4 Kd2 47. Rh6 Bb3 48. Kd4 Bc2 49. Rh2+ Kd1 50. Ke3 Bb1 51. Rd2+ Kc1 52. Ke2 Bg6 53. Rd6 Be4 54. Ra6 Kc2 55. Ke3 Bd5 56. Ra5 Be6 57. Rc5+ Kd1 58. Kd3 Bb3 59. Rc6 Ba4 60. Rd6 Bb5+ 61. Ke3+ Kc2 62. Rd5 Bf1 63. Rc5+ Kd1 64. Rc6 Bh3 65. Rf6 Kc2 66. Kd4 Bg4 67. Rf2+ Kb3 68. Kd3 Be6 69. Rf4 Bd5 70. Kd4 Bc6 71. Rf5 Ba4 72. Rc5 Kb2 73. Rc4 Kb3 74. Kc5 Bc6 75. Rg4 Bh1 76. Kd4 Kc2 77. Rg3 Kd2 78. Rh3 Bc6 79. Rh2+ Kc1 80. Kc3 Kd1 81. Rd2+ Ke1 82. Kd3 Kf1 83. Ke3 Kg1 84. Rf2 Bd5 85. Ke2 Bc6 86. Ke1 Be4 87. Rf4 Bc6 88. Rf6 Be4 89. Ke2 Kg2 90. Ke3 Bd5 91. Rg6+ Kh3 92. Rd6 Bg2 93. Rh6+ Kg3 94. Rg6+ Kh3 95. Kf4 Bd5 96. Rd6 Bc4 97. Kf3 Kh2 98. Rd2+ Kh3 99. Rd1 Kh2 100. Rd6 Kg1 101. Rd2 Bb3 102. Rf2 Bd1+ 103. Kg3 Bh5 104. Rf5 Bg6 105. Rg5 Bd3 106. Rd5 Be2 107. Rc5 Ba6 108. Rc1+ Bf1 109. Re1 Kh1 110. Rxf1#

Answer 2

I think it’s mostly a draw, whether or not the bishop can access the 4 corner squares.

This is because for KR to mate K relies on zugzwang. If the Black bishop is still on the board, then it can always make waiting moves.

I don’t think that White can corner, fork, pin, skewer or otherwise purloin the Black bishop, except on a very tiny board. The parity of the board size is not relevant

By comparison on an 8x8 board, white wins in 19% of positions, with 81% being draws (see Syzygy.com). Note that it’s physically impossible with this material for the bishop to deliver checkmate, so if White times out, the game is a draw.

Playing out the following win for white shows a lot of the dynamics: https://syzygy-tables.info/?fen=8/8/8/8/1R6/3K4/8/1k4b1_b_-_-_0_1 Shifting bishop to other squares e.g. h2 shows how fragile is White’s advantage. An extra row and column on the board gives White nothing.