Author: Marc Bourzutschky
Date: 00:15:54 08/29/05
During the past 6 months Yakov Konoval and I have collaborated on efficient algorithms to exactly solve 7-man endgames. Yakov has come up with a program that contains every trick in the book, and then some. Many of the ideas are refinements of those used by Johan de Koning in his path breaking FEG program. Our hardware is extremely basic, about US $3,000 worth of equipment: a single 3.6 GHZ PIV, with 4 GB RAM, two 250 GB IDE hard disks, running Windows XP. To put Yakov's programming skills in perspective, his program solves the infamous 6-man krnknn endgame in just over one hour on that hardware. So far we have focused on pawnless endgames. While those may have little practical relevance, they are an important first step. Pawnless endgames will also, in the words of Tim Krabbe, show many examples of the eerie beauty of perfect but incomprehensible chess. Perhaps there will be endings that break the record of 243 moves without mate or capture in krnknn. The largest database we have generated so far is krbnkrb. This endgame is of some interest to endgame study composers, particularly the question how the outcome depends on whether the bishops have the same or opposite colors. This ending also arises occasionally in practice. For example, it occurred during the 11th Postal World Championship, in the game Buj vs. Thiele. It was adjudicated a draw. Pablo Buj was not happy and showed analysis claiming he was winning. The database indeed confirms the win, and a relatively straightforward one at that, with only 18 moves to conversion. The krbnkrb database has almost a trillion entries (462*64^5 for both White and Black.) Almost half of these entries represent legal positions, making it the largest chess endgame solved by far. Generation time was about 17.5 days, and the final size is about 168 GB. Verification took another 5 days. All the endgames are created using the DTZ metric, which for pawnless endings is the shortest distance to either checkmate or capture to a won subgame. DTZ is faster to generate than distance to mate (DTM), and also compresses better. The improvement of DTZ over DTM is roughly a factor of two in speed and size for 6-man endgames, and probably more for 7-man endgames. Generation times scale roughly with the permutation symmetries of the ending. Ball park figures are half a day for endings of the form kxxxxky, a day for kxxxkyy, two days for kxxxkyz or kxyyykz, etc. Verification usually takes between 1/2 and 3/4 of the generation time. Now some examples. I show play all the way to mate, even if there is a capture before the end. Exclamation marks are attached to moves if any other move would throw away the win. KQQQKQQ Not a terribly interesting ending in itself, but a useful reference point, perhaps for a future KQPPKQP where only queen promotions are considered. The longest winning distance is 76 moves. As expected, there are often cannonades of successive checks, 73 in the maximal position below: [D] 4Qq2/5Q2/8/8/8/2q5/Q7/2K4k w - - 0 1 [Event "?"] [White "QQQ"] [Black "QQ"] [Result "1-0"] [SetUp "1"] [FEN "4Qq2/5Q2/8/8/8/2q5/Q7/2K4k w - - 0 1"] [PlyCount "167"] 1. Kd1! Qd4+ 2. Ke2! Qg4+ 3. Kd3! Qd1+ 4. Qd2 Qa3+ 5. Ke4! Qg4+ 6. Ke5! Qag3+ 7. Qff4! Qg5+ 8. Ke4! Qg6+ 9. Kd5! Qb3+ 10. Kd4! Qg7+ 11. Kc5 Qa7+ 12. Kd6! Qaa3+ 13. Ke5 Qc5+ 14. Kf6! Qbb6+ 15. Qe6! Qf8+ 16. Ke5! Qg7+ 17. Qef6 Qg5+ 18. Ke4! Qb1+ 19. Kd4! Qa1+ 20. Kc4 Qa4+ 21. Kd3! Qb3+ 22. Qfc3 Qb1+ 23. Kd4! Qb6+ 24. Ke4 Qbg6+ 25. Ke3 Qg1+ 26. Ke2! Q6g2+ 27. Kd3! Qb1+ 28. Kd4! Qb6+ 29. Qc5 Qf2+ 30. Qde3! Qfb2+ 31. Kd5! Qa2+ 32. Qfc4! Qd2+ 33. Qcd3 Qa2+ 34. Kd4 Qf6+ 35. Qee5! Qff2+ 36. Qde3 Qfb2+ 37. Qec3! Qf2+ 38. Qee3 Qf6+ 39. Kd3 Qb1+ 40. Kc4! Qa6+ 41. Kd5 Qf5+ 42. Qee5! Qf7+ 43. Ke4 Qe2+ 44. Kd4! Qef2+ 45. Qee3 Q2f6+ 46. Kd3 Qf1+ 47. Ke4 Qe6+ 48. Q5e5! Qg4+ 49. Q3f4! Qfg2+ 50. Kd4 Qd7+ 51. Qd6! Qf2+ 52. Kc4! Qa2+ 53. Kd3 Qb5+ 54. Kd4! Qa7+ 55. Ke4! Qe2+ 56. Qfe3 Qa4+ 57. Kd5 Qeb5+ 58. Qec5 Qa2+ 59. Kd4! Qf2+ 60. Qe3! Qbb2+ 61. Ke4 Qg2+ 62. Kf5! Qb1+ 63. Kf4 Qbf1+ 64. Ke5! Qg7+ 65. Kd5! Qfg2+ 66. Kc4 Qa2+ 67. Qb3 Qg4+ 68. Qcd4! Qc8+ 69. Kb4 Qb7+ 70. Q6b6 Qe7+ 71. Kc4 Qae2+ 72. Kc3 Qf3+ 73. Kb2 Qee2+ 74. Qc2 Qef2 75. Qh8+ Kg1 76. Qcxf2+ Qxf2+ 77. Qxf2+! Kxf2 78. Qg8 Ke3 79. Qg4 Kf2 80. Kc1 Kf1 81. Kd1 Kf2 82. Kd2 Kf1 83. Ke3 Ke1 84. Qg1# 1-0 KBBBBKQ This is the companion to knnnnkq I had written about earlier this year, which appeared in EG: http://www.gadycosteff.com/eg/eg156.pdf Yakov's program is a good 6 times faster and more memory efficient than my early effort. kbbbbkq with a pair of opposite colored bishops is a general win for the bishops. The longest win takes 101 moves. One is reminded of fox and geese where the plodding bishops finally prevail over the wily queen: [D] 8/3B2k1/8/6B1/7B/q6B/8/K7 w - - 0 1 [Event "?"] [White "BBBB"] [Black "Q"] [Result "1-0"] [SetUp "1"] [FEN "8/3B2k1/8/6B1/7B/q6B/8/K7 w - - 0 1"] [PlyCount "201"] 1. Kb1 Qb3+ 2. Kc1 Qc3+ 3. Kd1 Qd3+ 4. Ke1 Qe4+ 5. Kf2 Qd4+ 6. Kf3 Qd5+ 7. Ke3 Qe5+ 8. Kd2 Qd4+ 9. Kc2 Qe4+ 10. Kc1 Qc4+ 11. Kb2 Qb4+ 12. Ka2 Qc4+ 13. Ka3 Qd3+ 14. Kb4 Qe4+ 15. Kc5! Qe5+ 16. Kb6! Qb8+ 17. Kc6 Qa8+ 18. Kc7 Qa5+ 19. Kb8 Qb6+ 20. Kc8 Qa6+ 21. Kd8 Qa8+ 22. Bc8 Qd5+ 23. Bhd7 Kf7 24. Kc7! Qc5+ 25. Kb7 Qb4+ 26. Kc6 Qc4+ 27. Kb6 Qd4+ 28. Kb5 Qd5+ 29. Kb4 Qd4+ 30. Kb3 Qd3+ 31. Kb2 Qd4+ 32. Kc2 Qc4+ 33. Kd2 Qd4+ 34. Ke2 Qe4+ 35. Kf2 Kg6 36. Be7! Qd4+ 37. Ke2 Qe4+ 38. Kd2 Qd4+ 39. Kc2 Qc4+ 40. Kb2 Qe2+ 41. Kc3 Qe5+ 42. Kc4 Qe4+ 43. Kc5 Kf7 44. Bd8 Qe5+ 45. Kc4 Qe4+ 46. Kc3 Qe3+ 47. Kc2 Qe2+ 48. Kc1 Qe3+ 49. Kd1 Qf3+ 50. Ke1 Qe4+ 51. Kf1 Qh1+ 52. Kf2 Qh2+ 53. Kf3 Qh1+ 54. Ke3 Qc1+ 55. Kf2 Qc2+ 56. Kg1 Qd1+ 57. Kg2 Qd5+ 58. Kh2 Qa2+ 59. Kh3 Qb1 60. Bf5 Qh1+ 61. Kg3 Qg1+ 62. Kf3 Qf1+ 63. Ke3 Qc1+ 64. Kf2 Qc5+ 65. Kg2 Qc6+ 66. Kh2 Qd6+ 67. Kh3 Qc6 68. Kg4 Qa4+ 69. Kg5 Qc6 70. Bg3 Qc1+ 71. Bf4 Qc6 72. Bcd7 Qg2+ 73. Kh4 Qf2+ 74. Bg3 Qd4+ 75. Kh3 Qd5 76. Kg4 Qd1+ 77. Kg5 Qd2+ 78. Kh5 Qe2+ 79. Kh4 Qe3 80. Bg5 Qe2 81. Bc6 Qc4+ 82. Bfe4 Ke6 83. Be3 Qb3 84. Bef4 Qc4 85. Kg5 Qd4 86. Bf5+ Kf7 87. Be5 Qd2+ 88. Bgf4 Qd8+ 89. Kg4 Qd1+ 90. Bf3 Qf1 91. Bd5+ Ke7 92. Bfe4 Qd1+ 93. Kg5 Qg1+ 94. Kh6 Qb6+ 95. Kg7 Qa7 96. Bg5+ Kd7 97. Kf8 Qa3+ 98. Kf7 Qh3 99. Bc6+ Kc8 100. Bb7+ Kd7 101. Bec6# 1-0 For those who love the bizarre, here is the unique mutual zugzwang (out of 303) which is perfectly symmetric about the a1-h8 diagonal. All the bishops have the same color. White to move loses, but black to move only achieves a draw: [D] 6B1/1B4qB/5k2/8/3K4/8/6B1/8 b - - 0 1 KBNNNKQ This endings has about 4 times as many positions as knnnnkq or kbbbbkq. The uneasy collaboration between bishop and knight, making even kbnk a non-trivial endeavor, seems to be multiplied here to make this a very difficult win. The longest win takes 224 moves. There are over 140 checks by the queen, who buzzes about like an angry insect before Black finally succumbs: [D] 5N2/5k1N/7q/8/8/3B4/8/2K2N2 w - - 0 1 [Event "?"] [Date "????.??.??"] [White "BNNN"] [Black "Q"] [Result "1-0"] [SetUp "1"] [FEN "5N2/5k1N/7q/8/8/3B4/8/2K2N2 w - - 0 1"] [PlyCount "447"] 1. Kd1! Qh5+ 2. Ke1! Qe5+ 3. Kf2! Qd4+ 4. Ke2! Qg4+ 5. Kd2 Qb4+ 6. Ke3 Qe1+ 7. Kf4 Qf2+ 8. Ke4! Qg2+ 9. Kd4 Qb2+ 10. Ke3 Qg2 11. Nd2 Qg1+ 12. Ke2! Qh2+ 13. Kd1 Qd6 14. Kc2 Qc6+ 15. Nc4 Kg7 16. Kd2! Qg2+ 17. Kc3! Qd5 18. Bc2 Qc6 19. Kd4! Qa6 20. Ne3! Qb6+ 21. Ke4 Qc6+ 22. Kf4 Qh6+ 23. Kf3 Qh3+ 24. Ke4 Kh8 25. Bd1 Qh1+ 26. Bf3 Qb1+ 27. Kf4 Qb8+ 28. Kg4 Qb4+ 29. Kg3 Qb8+ 30. Kf2 Qb2+ 31. Be2 Qb8 32. Bc4 Qh2+ 33. Kf3 Qh1+ 34. Kf4 Qh4+ 35. Ke5 Qe7+ 36. Kd4 Qh4+ 37. Kd3 Qg3 38. Bd5 Qe1 39. Ng2 Qg3+ 40. Ke2 Kg7 41. Ne3 Qh2+ 42. Kf3 Qh5+ 43. Kg3 Qe5+ 44. Kf2 Qh2+ 45. Bg2 Qh4+ 46. Kf3 Qh5+ 47. Kf4 Qh6+ 48. Ke4 Qh4+ 49. Kd3 Qh5 50. Bc6 Qa5 51. Bd5 Qb5+ 52. Ke4 Qe2 53. Be6 Qe1 54. Kd3 Qb4 55. Bf5 Qb5+ 56. Ke4 Qc6+ 57. Kf4 Qh6+ 58. Kf3 Qh5+ 59. Kg3 Qe2 60. Ng4 Qe1+ 61. Kg2! Qd2+ 62. Nf2! Qd5+ 63. Be4! Qc4 64. Kg3 Qc7+ 65. Kg4 Qc4 66. Kh4 Qe2 67. Kg3! Qc4 68. Kg2 Kg8 69. Kf3 Qf7+ 70. Kg3 Qc7+ 71. Kh3 Qc3+ 72. Nd3 Qd4 73. Bf5! Qe3+ 74. Kg4 Qg1+ 75. Kf4 Qh2+ 76. Ke4 Qg2+ 77. Ke5 Qg7+ 78. Kd6 Qd4+ 79. Kc6 Qa4+ 80. Kd5 Qb3+ 81. Ke5 Qb8+ 82. Kd4 Qa7+ 83. Nc5 Kh8 84. Be4 Qa1+ 85. Kd5 Qa5 86. Kc6 Qa3 87. Bd3 Qa2 88. N5e6 Qa3 89. Bb5 Qa8+ 90. Kc5 Qa3+ 91. Kb6 Qe3+ 92. Kb7 Qb3 93. Nc7 Qb4 94. Kc8 Qc5 95. Bd7 Qd6 96. Bf5 Qc5 97. Be4 Qb4 98. Bb7 Qa3 99. Ba8 Qb4 100. Nce6 Qc3+ 101. Kd7 Qd3+ 102. Ke8 Qb5+ 103. Ke7! Qa5 104. Bc6 Qa3+ 105. Ke8 Qa6 106. Bf3 Qb5+ 107. Ke7! Qb4+ 108. Kd7 Qb5+ 109. Kd6 Qd3+ 110. Bd5! Qa6+ 111. Ke5 Qa1+ 112. Kf5 Qf1+ 113. Nf4 Kg7 114. Be4 Qf2 115. Kg4 Qg1+ 116. Kf3 Qf1+ 117. Ke3 Kh6 118. Nd5! Qg1+ 119. Ke2 Qh2+ 120. Kd3 Qb8 121. Ndf6 Qb3+ 122. Ke2 Qc4+ 123. Kf2 Qd4+ 124. Kf3 Qd1+ 125. Kf4 Qd2+ 126. Kg4 Qe3 127. Bf3 Qg1+ 128. Kf5 Qg3 129. Ke4 Qh4+ 130. Ke3 Qe1+ 131. Kf4 Qc1+ 132. Kf5 Qc5+ 133. Ke4 Qb4+ 134. Kd3 Qd6+ 135. Ke3 Qc5+ 136. Ke2 Qb5+ 137. Kf2 Qb2+ 138. Kg3 Qe5+ 139. Kg2 Qb2+ 140. Kh3 Qc3 141. Kg4 Qc4+ 142. Ne4 Kg7 143. Kg3 Qc7+ 144. Kh3 Qc8+ 145. Bg4 Qa6 146. Ng3 Kg8 147. Bf5 Qh6+ 148. Kg4 Qg7+ 149. Kh4 Qe7+ 150. Kh3 Qe5 151. Be4 Qd4 152. Bf3 Qd3 153. Bg2 Qd2 154. Ne4 Qe3+ 155. Kg4 Qg1 156. Kf3 Qd1+ 157. Kf2 Qc2+ 158. Kg3 Qc7+ 159. Kf3 Qb7 160. Kg4 Qc8+ 161. Kh5 Qc2 162. Bf3 Kg7 163. Neg5 Qh2+ 164. Kg4! Qg1+ 165. Kh3 Qf1+ 166. Bg2 Qd3+ 167. Kh4 Qd2 168. Nfe6+ Kh6 169. Kg3 Qe3+ 170. Kg4 Qe2+ 171. Bf3 Qc4+ 172. Kg3 Qc3 173. Nf4 Qd4 174. Nd5 Qc4 175. Ndf6 Qc1 176. Be4 Qg1+ 177. Kh4 Qf2+ 178. Kg4 Qg1+ 179. Kf5 Qf2+ 180. Nf3 Qc5+ 181. Ne5 Qf2+ 182. Ke6 Qb6+ 183. Bc6 Qc5 184. Bd5 Qc8+ 185. Ned7 Kg6 186. Be4+ Kh6 187. Ng4+ Kh5 188. Ne5 Kh4 189. Nf6 Kg5 190. Bc6 Kf4 191. Nd5+ Kg5 192. Ne7 Qa6 193. Nf6 Qa2+ 194. Ned5 Qg2 195. Bb5 Qh3+ 196. Ke7 Qf5 197. Kd6 Qb1 198. Nf3+ Kh6 199. Bd7 Qa2 200. Ne5 Qe2 201. Nf4 Qd2+ 202. Nfd3 Qe2 203. Bc6 Qa2 204. Ne4 Qg8 205. Ke7 Kg7 206. Ndc5 Qf8+ 207. Kd7 Qb8 208. Ne6+ Kh8 209. Nd6 Kg8 210. Ndf7 Qa7+ 211. Ke8 Qb8+ 212. Ke7 Qb4+ 213. Kf6 Qh4+ 214. Neg5 Qf2+ 215. Bf3 Qb6+ 216. Nc6 Qb2+ 217. Nfe5 Qf2 218. Ne7+ Kf8 219. Nf5 Qb6+ 220. Bc6 Qd8+ 221. Kg6! Qa5 222. Nef7 Qb6 223. Nh7+ Kg8 224. Ne7# 1-0 kbnnnkq contains 3 full-point mutual zugzwangs, where whoever is to move loses. Such zugzwangs are extremely rare in endings without pawns. Here is one of the 3 cases: [D] 8/8/5N2/8/6N1/2kq1N2/8/2KB4 w - - 0 1 White to move loses: [Event "?"] [Date "????.??.??"] [White "BNNN"] [Black "Q"] [Result "0-1"] [SetUp "1"] [FEN "8/8/5N2/8/6N1/2kq1N2/8/2KB4 w - - 0 1"] [PlyCount "8"] 1. Nh6 Qe3+! 2. Kb1 Qb6+ 3. Bb3 Qxb3+ 4. Kc1 Qc2# 0-1 Black to move from the same position also loses: [Event "?"] [Date "????.??.??"] [White "BNNN"] [Black "Q"] [Result "1-0"] [SetUp "1"] [FEN "8/8/5N2/8/6N1/2kq1N2/8/2KB4 b - - 0 1"] [PlyCount "134"] 1... Qc4 2. Ne3 Qc6 3. Ne1 Kd4+ 4. N1c2+ Ke5 5. Nfd5 Qg6 6. Be2 Qh7 7. Bf3 Qd3 8. Kb2 Kd6 9. Ndb4 Qg6 10. Kc3 Qg1 11. Nc4+ Ke6 12. Nd4+ Kf7 13. Nd6+ Kg6 14. Ne4 Qa1+ 15. Kc4 Qa4 16. Bg4 Kg7 17. Bf5 Kh6 18. Nc5 Qa8 19. Be4 Qa7 20. Nde6 Kh5 21. Nc2 Qa2+ 22. Kc3 Qb1 23. Ncd4 Qa2 24. Bf5 Qf2 25. Ne4 Qe3+ 26. Kc4 Qa3 27. Kd5 Qa5+ 28. Kd6 Qb4+ 29. Ke5 Qb8+ 30. Kf6 Qh8+ 31. Kf7 Qa8 32. N4g5 Qh8 33. Ndf3 Qc3 34. Nfd4 Qa5 35. Bh3 Kh4 36. Nf5+ Kh5 37. Ne4 Qc7+ 38. Ne7 Qb7 39. Bf5 Kh4 40. Nf8 Kh5 41. Nd7 Kh4 42. Kg8 Qa8+ 43. Kh7 Qa7 44. Be6 Qb7 45. Ng6+ Kh5 46. Nd6 Qc7 47. Nf7 Qc3 48. Nde5 Qc4 49. Bd5 Qe4 50. Bc6 Qf5 51. Bf3+ Qg4 52. Nh4 Qxf3 53. Nexf3! Kg4 54. N7g5 Kf4 55. Ng6+ Kg4 56. Kg7 Kg3 57. Kf6 Kg2 58. Nf4+ Kf2 59. Kg6 Kg3 60. Nd3 Kg4 61. Nd2 Kh4 62. Nde4 Kg4 63. Kf6 Kh5 64. Ndf2 Kh6 65. Kf7 Kh5 66. Kg7 Kh4 67. Nf3+ Kh5 68. Nf6# 1-0 (KBBNNKQ and KBBBNKQ have not been generated yet.) KRRRKQB This turns out to be another marathon, with the win taking up to 200 moves. In the line below it is entertaining to watch what happens after move 200. Play can only be understood by realizing that the algorithm prefers a capture in one move to a mate in 2 moves, so the bizarre moves are fully correct from a game theoretic point of view. Of course, play before move 200 may well be beyond human understanding, with the rooks finally prevailing after a stately progression like in a Japanese tea ceremony: [D] 8/4R3/8/1R5b/5R2/8/7k/K6q w - - 0 1 [Event "?"] [Date "????.??.??"] [White "RRR"] [Black "QB"] [Result "1-0"] [SetUp "1"] [FEN "8/4R3/8/1R5b/5R2/8/7k/K6q w - - 0 1"] [PlyCount "437"] 1. Ka2! Qa8+ 2. Kb3! Qd8 3. Rfe4! Qd3+ 4. Kb4! Qd6+ 5. Rc5 Qb6+ 6. Kc4 Qa6+ 7. Kc3 Qa3+ 8. Kd4 Qb4+ 9. Rc4! Qd6+ 10. Kc3 Bg6 11. Re2+! Kg1 12. R7e6! Qa3+ 13. Kd4! Qd3+ 14. Kc5! Qa3+ 15. Kc6 Qa6+ 16. Kd5 Qa8+ 17. Kd4 Qa1+ 18. Ke3 Bh5 19. Rd2! Qe1+ 20. Kd3! Qf1+ 21. Kc3! Qc1+ 22. Rc2 Qa3+ 23. Kd4! Qa7+ 24. Rc5 Qd7+ 25. Ke5! Bg4 26. Rc1+! Kg2 27. R5c2+ Kg3 28. Rc3+! Kh4 29. Rec6! Qg7+ 30. Kd5 Qf7+ 31. Kd4 Qd7+ 32. Kc5 Qe7+ 33. Kb6 Qb4+ 34. Ka7 Qa4+ 35. Ra6 Qd4+ 36. Rc5 Be2 37. Rc6! Qa4+ 38. Kb7! Qb3+ 39. Rb6! Qf3+ 40. Rcc6 Qf7+ 41. Rc7 Qd5+ 42. Kb8! Qg8+ 43. Rc8 Qg3+ 44. Kb7 Qa3 45. Rh1+ Kg5 46. Kb8! Qg3+ 47. Rc7! Qe5 48. Rg1+! Kh5 49. Rc1 Qe8+ 50. Rc8 Qe5+ 51. Kb7 Qd5+ 52. R8c6 Qf7+ 53. Rc7 Qd5+ 54. Kb8 Qg8+ 55. Rc8 Qg3+ 56. Kb7 Qf3+ 57. R1c6 Qe4 58. Rc7! Bf3 59. Rb5+! Kh4 60. Kb6! Qd4+ 61. Rcc5! Qf6+ 62. Ka7 Qd4 63. Ra5 Be4 64. Ra3! Bd3 65. Rc3 Qa4+ 66. Kb8! Qb4+ 67. Rb7 Qf4+ 68. Rcc7 Qf8+ 69. Ka7! Qf2+ 70. R7c5 Qa2+ 71. Kb8 Qg8+ 72. Rc8 Qg3+ 73. Ka7 Qf2+ 74. Ka8 Qa2+ 75. Kb8! Qh2+ 76. R8c7 Qd6 77. Rbb3! Be2 78. Rc6 Qf4 79. Rc2 Qd4 80. R2c5 Kg4 81. Rcc3 Qd6 82. Rc6 Qd8+ 83. Rc8 Qd5 84. Rb4+ Kg3 85. Rg6+ Kf2 86. Rf6+! Bf3 87. Rbb6 Qe5+ 88. Rbd6 Kg3 89. Re6 Qb2+ 90. Rb6! Qd4 91. Rce8 Bh5 92. Rb3+ Kf2 93. Rf8+ Kg2 94. Rg8+ Kf2 95. Rbb6 Qf4+ 96. Red6 Bf3 97. Re8 Kg3 98. Kc7 Qc1+ 99. Kd7 Qc4 100. Kd8 Qh4+ 101. Rf6 Qg5 102. Kc7 Qc5+ 103. Kb8 Qd4 104. Rg6+ Bg4 105. Rgd6 Qf4 106. Kc7 Qf7+ 107. Kd8! Qf5 108. Rbc6 Qg5+ 109. Re7 Kh3 110. Rh6+ Bh5 111. Rhf6 Kh2 112. Rd7 Bf3 113. Ra6! Qg8+ 114. Kc7 Qc4+ 115. Kb6 Qb4+ 116. Ka7 Qc5+ 117. Rab6 Be4 118. Rg7 Qa3+ 119. Ra6! Qc5+ 120. Kb8 Qe5+ 121. Rad6! Bf5 122. Rh6+ Bh3 123. Rgh7 Qb5+ 124. Rb6 Qe5+ 125. Kb7 Qd5+ 126. Kc7 Qc5+ 127. Rhc6 Qe5+ 128. Kb7 Qd5 129. Rb4 Kg3 130. Rg7+ Kf2 131. Kb6! Bf5 132. Rgc7 Bd3 133. Rc8 Qa2 134. Re8 Kg2 135. Rd6 Qa6+ 136. Kc7! Qa7+ 137. Rb7 Qc5+ 138. Rc6! Qd5 139. Rbb6 Kg3 140. Rd6 Qc5+ 141. Kd8 Qg5+ 142. Re7 Qg8+ 143. Kc7 Bf5 144. Rbc6 Qh8 145. Rd5 Qf8 146. Rc3+! Kf4 147. Rd4+ Kg5 148. Rce3! Qc8+ 149. Kb6! Qb8+ 150. Rb7 Qh8 151. Rg3+ Kh5 152. Rd5 Qb2+ 153. Kc6 Qf6+ 154. Kc5! Qf8+ 155. Kb5 Qe8+ 156. Kb4! Qe4+ 157. Kc5! Qc2+ 158. Kd6 Kh4 159. Re3 Qc8 160. Rbb3 Qf8+ 161. Kc7 Qc8+ 162. Kb6 Qb8+ 163. Kc5 Qf8+ 164. Kc4 Qc8+ 165. Kb4 Qf8+ 166. Kc3 Qc8+ 167. Kb2 Qc2+ 168. Ka3! Qc1+ 169. Kb4! Qc8 170. Ree5 Bg4 171. Rbd3 Qb7+ 172. Rb5 Qe4+ 173. Kc3! Qc6+ 174. Rec5 Qh1 175. Rcd5 Qc1+ 176. Kb4 Qe1+ 177. Rd2 Bf3 178. Rd6 Kg3 179. Rc5 Qb1+ 180. Ka4 Qa1+ 181. Kb3 Qb1+ 182. Rb2 Qh7 183. Rd4 Qb7+ 184. Kc3 Qe7 185. Rcc4 Qe1+ 186. Kb3 Bd1+ 187. Ka2! Qa5+ 188. Kb1! Qf5+ 189. Re4 Qf3 190. Ka2 Qf7 191. Re3+ Bf3 192. Rc2 Qa7+ 193. Ra3! Qf7 194. Rc1 Qe6 195. Rb3 Qd5 196. R4c3 Qa8+ 197. Ra3 Qd5+ 198. Ka1 Qe5 199. Rf1 Qf6 200. Rxf3+ Kh4 201. Rxf6 Kh5 202. Ra4 Kg5 203. Rh3 Kxf6 204. Rh6+ Kg7 205. Rf4 Kg8 206. Rf7 Kxf7 207. Ra6 Ke8 208. Ra7 Kd8 209. Kb2 Kc8 210. Rh7 Kd8 211. Kc3 Ke8 212. Kd4 Kf8 213. Ke5 Ke8 214. Kf6 Kd8 215. Ke6 Kc8 216. Kd6 Kb8 217. Kc6 Ka8 218. Kb6 Kb8 219. Rh8# 1-0 KRBNKRB Now finally the largest ending of the lot. To make it more manageable, we broke it into two independent components, one with the bishops having the same color (sc), the other with the bishops having opposite colors (oc). sc has the longer winning line of 226 vs. 191 for oc. However, the sc ending is "simpler" in the sense that it compresses better (75 GB for sc, 92 GB for oc). What this means exactly is not clear to us yet. Below a 226 move winning line. Particularly striking is the retreat by the white king to h1 on move 28, where he remains for over a hundred moves until finally reemerging on move 134. It is almost as if the white monarch lets his minions play cat and mouse with his black counterpart, who gets chased all around the board in the meantime: [D] 8/8/1b6/8/4N2r/1k6/7B/R1K5 w - - 0 1 [Event "?"] [Date "????.??.??"] [White "RBN"] [Black "RB"] [Result "1-0"] [SetUp "1"] [FEN "8/8/1b6/8/4N2r/1k6/7B/R1K5 w - - 0 1"] [PlyCount "499"] 1. Rb1+! Kc4 2. Nd2+! Kd3 3. Nf3! Rh6 4. Kd1! Ba5 5. Rb3+! Bc3 6. Ne5+! Ke4 7. Ng4! Rg6 8. Nf2+! Kd4 9. Rb8! Re6 10. Rd8+! Kc4 11. Rc8+! Kd4 12. Ng4 Re1+ 13. Kc2 Re2+ 14. Kb3! Rb2+ 15. Ka3! Rc2 16. Bg1+ Kd3 17. Rd8+! Ke4 18. Nf2+ Kf5 19. Rd3! Rc1 20. Rf3+! Ke6 21. Nh3! Be5 22. Kb3! Rb1+ 23. Kc2 Rb2+ 24. Kd1! Rb1+ 25. Ke2 Rb2+ 26. Kf1 Rb1+ 27. Kg2 Rb2+ 28. Kh1 Rb4 29. Ng5+! Kd5 30. Rd3+! Kc4 31. Rd1! Rb3 32. Re1! Kd5 33. Rf1 Ra3 34. Rf2! Rb3 35. Rf5 Rb4 36. Nf3 Ke4 37. Rf7! Rb2 38. Ng5+! Kd5 39. Ra7 Bd6 40. Ra5+! Kc4 41. Ne4! Bb8 42. Ra8 Bc7 43. Nf6 Kb5 44. Rg8! Kc6 45. Rg6! Kd6 46. Ng4+! Kd5 47. Ne3+! Ke4 48. Re6+! Kf3 49. Nd5 Ba5 50. Re3+! Kg4 51. Re4+! Kg3 52. Re8 Kf3 53. Rf8+ Kg3 54. Rf1 Bb4 55. Rf7! Bd6 56. Rd7 Bb8 57. Rd8 Be5 58. Re8! Bd6 59. Re6 Bb8 60. Re3+ Kg4 61. Ra3 Kf5 62. Ra4! Rd2 63. Ne3+! Ke6 64. Nc4 Rc2 65. Ra6+ Kd7 66. Ne3! Rc6 67. Ra4! Bc7 68. Ng4! Rc2 69. Re4 Bd6 70. Nf6+! Kc7 71. Re6! Bc5 72. Bh2+! Kb7 73. Ne4 Bb6 74. Be5 Bc5 75. Rh6 Bb6 76. Nc3 Ba5 77. Nd5 Rc5 78. Rh7+! Ka6 79. Rd7! Kb5 80. Bg3! Kc6 81. Nf6! Rg5 82. Rd6+! Kb5 83. Rd3! Rg6 84. Ne4 Kc6 85. Rd1! Re6 86. Ng5 Rg6 87. Bf4 Rf6 88. Nh3 Rg6 89. Rc1+ Kd7 90. Nf2 Ke6 91. Rb1! Kf5 92. Bh2! Rb6 93. Rc1! Rb5 94. Rc8! Bb6 95. Rf8+! Ke6 96. Nd3! Rb3 97. Nf4+! Ke7 98. Rh8 Rb2 99. Nd3! Rd2 100. Ne5! Bc5 101. Bf4 Re2 102. Nd3 Be3 103. Bg3 Rd2 104. Bh4+ Kd6 105. Be1! Rc2 106. Rh7 Ra2 107. Bg3+ Ke6 108. Rh5! Bd4 109. Be1 Ba7 110. Rg5 Bd4 111. Nf4+ Kf7 112. Rf5+ Ke8 113. Nd3 Ra3 114. Rf3! Ra2 115. Rf4 Ba7 116. Nc1 Rb2 117. Rf5 Kd8 118. Nd3 Ra2 119. Bg3 Kd7 120. Rh5 Be3 121. Rd5+ Kc8 122. Rf5 Rd2 123. Rf8+ Kb7 124. Rd8 Ka7 125. Be1 Rc2 126. Re8 Bb6 127. Nb4 Rc4 128. Re6 Kb7 129. Nd5! Ba5 130. Bg3 Rc5 131. Re7+ Kc6 132. Nf6 Rf5 133. Re6+ Kc5 134. Kg2 Bc3 135. Ne8 Kd5 136. Nc7+! Kc4 137. Re7 Rf6 138. Ne6 Bd2 139. Ng7 Bh6 140. Ne8 Rf8 141. Nd6+ Kd5 142. Re4 Bd2 143. Rg4 Be3 144. Rh4 Kc5 145. Ne4+ Kc6 146. Nc3 Kc5 147. Re4 Bd4 148. Ne2 Bg7 149. Nf4 Rf7 150. Bf2+ Kc6 151. Rc4+ Kb5 152. Rc5+ Ka4 153. Ne6 Re7 154. Nc7 Rd7 155. Bg3 Kb4 156. Rc6 Rd3 157. Bd6+ Kb3 158. Ne6 Bd4 159. Bf4 Be3 160. Be5 Ba7 161. Nf4 Rd2+ 162. Kf3 Bd4 163. Bb8 Bg1 164. Ne2 Rd3+ 165. Kg4 Rd2 166. Re6 Bc5 167. Bf4 Rd7 168. Kf3 Kc4 169. Re4+ Kd5 170. Nc3+ Kc6 171. Re6+ Kb7 172. Ne4 Rd3+ 173. Kg4 Rd5 174. Rh6 Ba3 175. Rh3 Ra5 176. Rc3 Bf8 177. Rc7+ Ka6 178. Be3 Re5 179. Kf3 Bb4 180. Bd4 Re6 181. Rc5 Ba5 182. Rd5 Bb6 183. Ba1 Re8 184. Rd1 Rd8 185. Nc5+! Kb5 186. Nd7 Bc7 187. Rb1+ Kc6 188. Nf6! Bd6 189. Ke4 Bc5 190. Rc1 Kb6 191. Be5 Kc6 192. Nh5 Kb5 193. Ng7 Rd2 194. Ne6 Re2+ 195. Kf5 Bf2 196. Nc7+ Kb4 197. Nd5+ Kb3 198. Rb1+ Kc4 199. Ke6 Rc2 200. Rb4+ Kd3 201. Ra4 Rd2 202. Kf5 Bc5 203. Bg7 Rc2 204. Bf6 Rd2 205. Nf4+ Ke3 206. Rc4 Bd6 207. Rc3+ Kf2 208. Ne6 Re2 209. Ng5 Kg2 210. Bd4 Rd2 211. Rc4 Bg3 212. Ne4 Ra2 213. Rc1 Bb8 214. Kg4 Ba7 215. Nc3 Ra5 216. Rc2+ Kf1 217. Bf6 Bb8 218. Bh4 Ba7 219. Bg3 Ra1 220. Kf3 Bd4 221. Bh4 Be5 222. Be7 Re1 223. Bc5 Ra1 224. Rf2+ Kg1 225. Re2+ Bd4 226. Bxd4+ Kf1 227. Rf2+ Kg1 228. Ra2+ Kh1 229. Rxa1+ Kh2 230. Rh1+ Kxh1 231. Ne4 Kh2 232. Ng3 Kh3 233. Bg1 Kh4 234. Ne4 Kh5 235. Bd4 Kg6 236. Nd6 Kg5 237. Kg3 Kh5 238. Bf6 Kg6 239. Be7 Kh5 240. Bd8 Kg6 241. Kh4 Kh6 242. Be7 Kg6 243. Bg5 Kh7 244. Kh5 Kg7 245. Be7 Kh8 246. Kh6 Kg8 247. Kg6 Kh8 248. Nf7+ Kg8 249. Nh6+ Kh8 250. Bf6# 1-0 The longest line for the opposite color case seems to strongly suggest that the ending is a general win, since white starts with his pieces rather awkwardly placed and still wins. The poor steed in particular is marooned in the h1 corner all of the first 142 moves, before finally entering the fray to great effect: [D] 8/b2r4/1R4B1/8/8/4k3/8/2K4N w - - 0 1 [Event "?"] [Date "????.??.??"] [White "RBN"] [Black "RB"] [Result "1-0"] [SetUp "1"] [FEN "8/b2r4/1R4B1/8/8/4k3/8/2K4N w - - 0 1"] [PlyCount "383"] 1. Rb3+! Kf4 2. Rb4+! Ke5 3. Rb5+! Rd5 4. Rb3! Kf4 5. Rb4+! Rd4 6. Rb7! Ra4 7. Kc2! Ra2+ 8. Kb3! Rg2 9. Rf7+! Ke3 10. Re7+! Kf4 11. Bc2 Bb8 12. Bd1 Rd2 13. Rf7+! Ke3 14. Rf3+ Kd4 15. Bc2 Rg2 16. Rd3+! Kc5 17. Rd1 Rh2 18. Bf5! Re2 19. Bd3! Rh2 20. Rf1 Kd4 21. Bf5! Ke3 22. Bd7! Ke2 23. Bb5+ Ke3 24. Bc6 Ke2 25. Rc1 Kd3 26. Rc5 Ke3 27. Rc4 Kd3 28. Rg4 Rd2 29. Rg5 Kd4 30. Rd5+ Ke3 31. Rf5! Rh2 32. Kc3 Rh6 33. Ba8 Ra6 34. Rf3+! Ke2 35. Rf2+! Ke3 36. Bg2 Ra4 37. Rf3+! Ke2 38. Rf7 Bd6 39. Bd5! Rh4 40. Rf2+! Ke1 41. Ra2 Be5+ 42. Kb3! Rd4 43. Be6! Rd2 44. Ra5! Bf6 45. Bf5 Bh4 46. Ra4 Rd5 47. Bc2! Rh5 48. Ra2! Kf1 49. Bd3+! Kg1 50. Be4! Re5 51. Ba8! Rc5 52. Ra4 Be1 53. Re4! Kf1 54. Re3 Bh4 55. Kb4 Rc1 56. Re6 Rc7 57. Re4! Be1+ 58. Kb5! Rd7 59. Bc6 Rd3 60. Re6 Rc3 61. Bd5 Bh4 62. Re4 Rh3 63. Kc6 Rh2 64. Re5 Rb2 65. Kc5 Rh2 66. Kb5 Bf2 67. Kb4 Bh4 68. Re3 Rd2 69. Kc4 Be1 70. Re6 Rc2+ 71. Kd4 Rd2+ 72. Kc5 Bh4 73. Kc4 Re2 74. Rg6 Rd2 75. Bf3 Rd8 76. Rh6 Be1 77. Re6 Rd2 78. Bb7 Bh4 79. Re4 Be1 80. Ba6 Rd6 81. Bb5 Rd1 82. Re6 Rb1 83. Bc6 Rb4+ 84. Kc5 Rb1 85. Kd4 Rb4+ 86. Ke3 Rb3+ 87. Kf4 Rb4+ 88. Kf5 Rd4 89. Re8 Rh4 90. Bf3 Ra4 91. Re3 Ra7 92. Rd3 Ra4 93. Rb3 Ra7 94. Rb2 Ra5+ 95. Kf4 Ra4+ 96. Be4 Bh4 97. Rc2 Be1 98. Ke5 Rb4 99. Kf5 Rb5+ 100. Kf4 Rb4 101. Ra2 Bh4 102. Ke3 Rb3+ 103. Kd4 Ke1 104. Kd5 Re3 105. Bf5 Kf1 106. Kd4 Re1 107. Be4 Kg1 108. Bc6 Rc1 109. Rg2+ Kf1 110. Be4 Rd1+ 111. Ke5 Be1 112. Ra2 Rd2 113. Ra1 Ke2 114. Bf5! Bf2 115. Ra3 Kd1 116. Ra4 Kc1 117. Rb4! Re2+ 118. Kf4! Bh4 119. Kf3 Re5 120. Be4! Kd2 121. Rc4! Ra5 122. Rc2+! Kd1 123. Rh2 Ra3+ 124. Kg4! Ra4 125. Kf4 Be1 126. Ke3! Ra5 127. Rg2 Rg5 128. Bf3+ Kc1 129. Re2 Bh4 130. Kf4 Rc5 131. Re4 Rc2 132. Kg4! Bf2 133. Kh3 Kb2 134. Bd1 Rd2 135. Be2! Be1 136. Bh5 Bf2 137. Kg2 Bc5+ 138. Be2 Rd5 139. Rg4 Rf5 140. Bf3 Be7 141. Re4 Rg5+ 142. Bg4 Bd6 143. Nf2 Bc5 144. Nh3 Rd5 145. Re6 Rd6 146. Re5 Rd2+ 147. Be2 Rc2 148. Re8 Kb3 149. Re4 Kc3 150. Nf4 Kd2 151. Nd5 Rb2 152. Bc4 Kd1+ 153. Kg3 Rd2 154. Nc3+ Kc2 155. Nb5 Kc1 156. Re1+ Kb2 157. Kf3 Rc2 158. Bd5 Rh2 159. Be6 Rc2 160. Rh1 Be7 161. Bf7 Rc5 162. Rh2+ Rc2 163. Rh6 Rc1 164. Ke4 Bc5 165. Rh2+ Rc2 166. Rh8 Rf2 167. Be6 Re2+ 168. Kd5 Be7 169. Rb8 Kc1 170. Nc3 Rb2 171. Rg8 Rb6 172. Ne4 Kc2 173. Bf7 Kb3 174. Nc5+ Kb4 175. Rg4+ Ka3 176. Nd3 Kb3 177. Rc4 Rf6 178. Be6 Rh6 179. Rc7 Bf8 180. Ke5+ Ka4 181. Rb7 Rh5+ 182. Ke4 Rh6 183. Bc4 Rh4+ 184. Kd5 Rh5+ 185. Kd4 Rg5 186. Ra7+ Ra5 187. Nb2+ Kb4 188. Rb7+ Ka3 189. Rb3+ Ka2 190. Rb5+ Ka3 191. Rxa5+ Kb4 192. Ra4# 1-0 Here a list of the other endings generated, with the longest winning lengths shown in parentheses: krrrkqn (131), knnnknn (93), krrkbbb (50), krrknnn (49), kqqkrrr (40), kbbbknn (36), krrrkrr (34), kbbbkbb (32), knnnkbb (30). Some of the remaining endings we are considering are kbbbnkq, kbbnnkq, krrbkrr, kbnnknn, kbnnkbn, krnnkbb. -Marc
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