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What is the longest forced checkmate where the winning side has only a rook? I know this position where White has a mate in 67:

[FEN "2K4k/7p/8/8/8/8/3R4/n7 w - - 0 1"]

1. Kd7 Nb3 2. Rb2 Nd4 3. Rg2 h5 4. Kd6 Kh7 5. Ke5 Nb3 6. Rb2 Nc5 7. Rb5 Nd3+ 8. Kf6 Kh6 9. Rb8 Kh7 10. Rc8 Nf4 11. Kg5 Ne6+ 12. Kf5 Nd4+ 13. Ke4 Ne6 14. Ke5 Ng5 15. Kf5 Nf3 16. Rd8 Kh6 17. Rd6+ Kg7 18. Rd3 Ne1 19. Rg3+ Kh6 20. Kf6 Nc2 21. Rg2 Nd4 22. Rg8 Kh7 23. Re8 Nb3 24. Kg5 Kg7 25. Re7+ Kf8 26. Kf6 Nc5 27. Rc7 Ne4+ 28. Kf5 Nd6+ 29. Ke6 Ne4 30. Rc4 Ng3 31. Kf6 Ke8 32. Rd4 Nf1 33. Ke6 Ne3 34. Rh4 Ng2 35. Re4 Kf8 36. Kf6 h4 37. Re5 Nf4 38. Rf5 Ng2 39. Rf3 Ke8 40. Kf5 Ke7 41. Rh3 Kd7 42. Ke4 Ne1 43. Rh2 h3 44. Rxh3 Nc2 45. Rb3 Kd6 46. Kd3 Ne1+ 47. Ke3 Nc2+ 48. Ke4 Kd7 49. Rb6 Ke7 50. Rb2 Na3 51. Kd3 Kd6 52. Rb3 Nc2 53. Kxc2 Kd5 54. Re3 Kd4 55. Re1 Kc4 56. Rd1 Kb4 57. Rd4+ Kb5 58. Kb3 Kc5 59. Rd1 Kb5 60. Rc1 Ka5 61. Kc4 Kb6 62. Kb4 Ka6 63. Kc5 Kb7 64. Kb5 Ka7 65. Kc6 Ka8 66. Kc7 Ka7 67. Ra1#
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If you care not for uniqueness, Blathy has you covered.

[Title "Ottó Titusz Bláthy, The Chess Amateur 08/1922, Mate In 74"]
[FEN "rb6/k1pn2p1/2K1p1p1/4p1p1/4p3/4p3/b5n1/r5R1 w - - 0 1"]

1. Rxa1 Nc5 2. Rxa2+ Na6 3. Rb2 Nc5 4. Rxg2 Na6 5. Ra2 g4 6. Rb2 Nc5 7. Re2 Na6 8. Ra2 g3 9. Rb2 Nc5 10. Re2 Na6 11. Ra2 g5 12. Rb2 Nc5 13. Re2 Na6 14. Ra2 g4 15. Rb2 Nc5 16. Re2 Na6 17. Ra2 g6 18. Rb2 Nc5 19. Re2 Na6 20. Ra2 g5 21. Rb2 Nc5 22. Re2 Na6 23. Ra2 g2 24. Rb2 Nc5 25. Rxg2 Na6 26. Ra2 g3 27. Rb2 Nc5 28. Re2 Na6 29. Ra2 g4 30. Rb2 Nc5 31. Re2 Na6 32. Ra2 g2 33. Rb2 Nc5 34. Rxg2 Na6 35. Ra2 g3 36. Rb2 Nc5 37. Re2 Na6 38. Ra2 g2 39. Rb2 Nc5 40. Rxg2 Na6 41. Ra2 e2 42. Rb2 Nc5 43. Rxe2 Na6 44. Ra2 e3 45. Rb2 Nc5 46. Re2 Na6 47. Ra2 e4 48. Rb2 Nc5 49. Re2 Na6 50. Ra2 e5 51. Rb2 Nc5 52. Re2 Na6 53. Ra2 e2 54. Rb2 Nc5 55. Rxe2 Na6 56. Ra2 e3 57. Rb2 Nc5 58. Re2 Na6 59. Ra2 e4 60. Rb2 Nc5 61. Re2 Na6 62. Ra2 e2 63. Rb2 Nc5 64. Rxe2 Na6 65. Ra2 e3 66. Rb2 Nc5 67. Re2 Na6 68. Ra2 e2 69. Rb2 Nc5 70. Rxe2 Na6 71. Rb2 Nc5 72. Rb1 Nb3 73. Rxb3 Ka6 74. Ra3#

If you do, this is your answer.

[Title "Milan Velimirovic, C6743 The Problemist 03/1983, Version C. J. Morse, Special Prize, Mate In 69"]
[FEN "4R3/pn1p1p1p/p2n4/8/8/p4p2/7p/5K1k w - - 0 1"]

1. Re1 Ne4 2. Rxe4 Nc5 3. Re1 Nd3 4. Rb1 a2 5. Ra1 f2 6. Rd1 Nb2 7. Rc1 Nd3 8. Ra1 a5 9. Rd1 Nb2 10. Rc1 Nd3 11. Ra1 a4 12. Rd1 Nb2 13. Rc1 Nd3 14. Ra1 a3 15. Rd1 Nb2 16. Rc1 Nd3 17. Ra1 a6 18. Rd1 Nb2 19. Rc1 Nd3 20. Ra1 a5 21. Rd1 Nb2 22. Rc1 Nd3 23. Ra1 a4 24. Rd1 Nb2 25. Rc1 Nd3 26. Ra1 d6 27. Rd1 Nb2 28. Rc1 Nd3 29. Ra1 d5 30. Rd1 Nb2 31. Rc1 Nd3 32. Ra1 d4 33. Rd1 Nb2 34. Rc1 Nd3 35. Ra1 f6 36. Rd1 Nb2 37. Rc1 Nd3 38. Ra1 f5 39. Rd1 Nb2 40. Rc1 Nd3 41. Ra1 f4 42. Rd1 Nb2 43. Rc1 Nd3 44. Ra1 f3 45. Rd1 Nb2 46. Rc1 Nd3 47. Ra1 h6 48. Rd1 Nb2 49. Rc1 Nd3 50. Ra1 h5 51. Rd1 Nb2 52. Rc1 Nd3 53. Ra1 h4 54. Rd1 Nb2 55. Rc1 Nd3 56. Ra1 h3 57. Rd1 Nb2 58. Rc1 a1=Q 59. Rxa1 Nd3 60. Rb1 a2 61. Ra1 a3 62. Rd1 a1=Q 63. Rxa1 a2 64. Rd1 Nb2 65. Rc1 a1=Q 66. Rxa1 Nd3 67. Rb1 Ne1 68. Kxf2 d3 69. Rxe1#
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    That's remarkably close to the 5-man tablebase maximum of 67 quoted by the OP. This #69 has a unique solution, which the OP did not require -- I followed the "tablebase" analysis, and learned that the solution is unique for White's first 32 moves(!) but then there are two choices to force mate in 35 starting from Kf6,Rd4/Ke8,Rd4,Nf1. I was not able to find online the longest wins with KR/KNPP or KR/KPPP, but those must be known, and possibly one is over 67. Probably by now the maximal DTM for each 7-man endgame is known too.
    – Noam D. Elkies
    Commented Jun 10 at 21:14

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