6

White has 20 available moves at the start of the game. After 1.a4 e5, white has 21 available moves. (The rook has gained 2 moves, while the pawn has lost 1).

What is the longest sequence of moves whereby white has at least 1 more available move than they had the previous turn, starting from the starting positon? The possibility of pawn promotion counts as four separate moves.

4
  • At most <80. If I remember correctly, the record for white mobility is exact 100, with an empty board. This can get somewhat higher if promotions count 4x - I assume you do. Apr 27, 2023 at 5:50
  • It is an interesting task but painfully complicated to investigate by hand. I expect the best answers coming form some good programmers, even if an exhasutive search is clearly out of question.
    – Evargalo
    Apr 27, 2023 at 12:56
  • If this is too daunting, maybe a first step would be to remove the black pieces and try to achieve the best possible score with only the white pieces...
    – Evargalo
    Apr 27, 2023 at 14:24
  • 3
    A more tractable problem might be add the constraint that the number of available moves must increase by exactly 1 each turn. Apr 28, 2023 at 6:24

3 Answers 3

4

EDIT: Old version below, this is a new improved version. If you have suggestions for a better final position to aim for, that allows for more White legal moves and might be reachable through this constraint, please let me know in the comments. For now I have given up on finding something better.

[FEN ""]

1. g4 b6 2. h4 e6 3. b4 Be7 4. Nh3 a5 5. f4 e5 6. c3 h6 7. Bb2 Bb7 8. Na3 e4 9. Nf2 Bc8 10. h5 Bc5 11. Rg1 g6 12. Qb1 e3 13. g5 Bd4 14. c4 d5 15. Bh3 Bf5 16. dxe3 Bc2 17. Kf1 Bd3 18. exd3 Bg7 19. Nb5 Bc3 20. a4 c5 21. Ba3 Bf6 22. Bf5 Qc7 23. Nc3 Bg7 24. Rg2 Qe5 25. Qa2 Qe7 26. Qb2 Qd6 27. Rc1 Bf6 28. Rg1 Qe5 29. Nfe4 Bg7 30. Be6 Qc7 31. Ke2 Kf8 32. d4 Ra7 33. Ra1 Nd7 34. Rh1 Ke7 35. Kf2 Qb8 36. gxh6 Rh7 37. Ke2 Qe5 38. hxg6 Qg5 39. Nd6 Qf6 40. e4 Nb8 41. Kf2 Nd7 42. Kf3 Bh8 43. Qf2 Qh4 44. Qe2 Qf6 45. Ke3 Ne5 46. g7 Rc7 47. Qc2 Ng6 48. e5 Qh4 49. Rhg1 Rc6 50. bxa5 Rc7 51. Bc8 f6 52. dxc5 Qg5 53. Bh3 Kd8 54. cxd5 N6e7 55. a6 Rb7 56. a5 Rb8 57. a7 Qg4 58. f5 Qg6 59. gxh8=Q Rg7 60. h7 Kd7 61. Rab1 Nc8 62. exf6 Nge7 63. Rb4 Rb7 64. cxb6 Rb8 65. b7 Nb6 66. a6 Rc8 67. Qf8 Rd8 68. h8=Q Rg8 69. fxe7 Qg4 70. f6 Rc8 71. b8=Q Rc6 72. dxc6+ Ke6 73. Kd2 Qf5 74. c7 Rg7 75. Qfd8 Nd7 76. f7 Kf6 77. Qh5 Rg5 78. Rc4 Kg7 79. Ke3 Nf8 80. Qxg5+ Kh8 81. Qe4 Qd5 82. Ndb5 Qa8 83. Re1 Qc6 84. Rf1 Qa8 85. Be6 Qc6 86. Ke2 Qa8 87. Bb2 Qc6 88. Ba3 Qb6 89. Qdd3 Qg1 90. Nb1 Qb6 91. Qeg2 Qc5 92. Ra4 Qg1 93. Kd2 Qh1 94. Qb6 Nd7 95. Bb4 Nf6 96. Ba3 Nh5 97. Qbc6 Nf6 98. N1c3 Ne8 99. a8=Q Qg1 100. a7 Qh1 101. Ke1 Qh2 102. Qcd6 Qh5 103. Rf2 Qh2 104. Q2g4 Qh7 105. Bc8 Qg8 106. Bc5 Ng7 107. Qgg2 Qd8 108. Rff4 Ne8 109. Qge5+ Ng7 110. Bg4 Qg8 111. Qac6 Qb8 112. Qdf6 Qg8 113. Kf1 Qb8 114. Na2 Qd8 115. Kg1 Qb8 116. Be6 Qd8 117. Ba3 Qb8 118. Qfg5 Qd8 119. Bg4 Qb8 120. Qgg6 Qd8 121. Qgd2 Qb8 122. Qf2 Qd8 123. Bd1 Qb8 124. Qg4 Qd8 125. c8=Q Qe8 126. Qge6 Qg8 127. Qeb3 Qd8 128. Qfg3 Qg8 129. Qgg2 Qd8 130. Kh1 Qg8 131. Nc7 Qe8 132. Qbb2 Qf8 133. Ne6 Qg8 134. Qbb6 Qd8 135. Rg4 Qg8 136. Qgf2 Qe8 137. Rg6 Qd8 138. Q6d7 Qg8 139. Q7d5 Kh7 140. Qdb7 Nh5 141. e8=Q Nf6 142. Qcc4 Kh8 143. Kh2 Kh7 144. Nc1 Nh5 145. Be7 Ng3 146. Qcc2 Nh1 147. Qcc3 Qh8 148. Qcd2 Qg8 149. Qec8 Qe8 150. Q7c6 Qh8 151. Qcb8 Qg8 152. Qbb1 Qe8 153. Qdf1 Qg8 154. Qfe3 Qh8 155. Qef5 Qg8 156. Q1b2 Qe8 157. Ng7 Qg8 158. a8=Q Kh8 159. Ne8+ Kh7 160. Qaa7 Ng3 161. Qac7 Nh5 162. Q7d7 Ng7 163. Qde1 Nh5 164. Qh4 Qh8 165. Qcd5 Qg8 166. Qdc7 Qh8 167. Ng7 Qe8 168. Raa6 Qg8 169. Nxh5 Qe8 170. f8=Q Qf7 171. Q8b4 Qe8 172. Ra8 Qd7 173. Qe8 Qa4 174. Qeb5 Qa1 175. Qa6 Qa4 176. Bf8+ Qd7 177. Ba4 Qf7 178. Bg7 Qf8 179. Rb8 Qe8 180. Nb3 Qg8 181. Na5 Qe8 182. Qba1 Qh8 183. Qfc2 Qg8 184. Bd7 Qf7 185. Re8 Qf8 186. Nb7 Qg8 187. Nd8 Qf8 188. Nf7 Qe7 189. Nh6 Qd8 190. Kg1 Qb8 191. Bh8 Qc8 192. Rf8 Qd8 193. Kg2 Qb8 194. Kh1 Qa7 195. Kg1 Qa8

Previous version:

And here is a more serious attempt, managing to promote all 8 pawns. In particular every pair of half moves adds exactly one more legal move:

[fen ""]
1. b3 d5 2. f4 g5 3. c4 Bh6 4. b4 Bh3 5. a4 b5 6. Ra2 f5 7. g3 e5 8. Ra3 Na6 9. Rd3 Bg2 10. Na3 Bf3 11. Rc3 Nc5 12. e4 d4 13. d3 Bg2 14. h3 Qd5 15. Qc2 Bg7 16. Qe2 Nf6 17. Be3 Kf8 18. Kd1 h6 19. Qb2 Qc6 20. g4 Nh5 21. Ke1 Qd5 22. Rh2 a5 23. h4 Ng3 24. Nf3 Qb7 25. Ng1 Bf3 26. Rc1 Nb3 27. h5 Qd5 28. Ra1 Bh1 29. Rd1 Nc5 30. Nf3 Ne2 31. Qa2 Ng1 32. Rc1 c6 33. Qb2 Kf7 34. bxa5 Bf8 35. Ra1 Nb7 36. Kd1 Rb8 37. a6 Nd8 38. Qb4 Ke6 39. Raa2 Qd7 40. a7 Bg2 41. Nb1 Qf7 42. c5 Bh3 43. Qc3 Qf6 44. Na3 Qg6 45. Nb1 Bd6 46. axb8=Q Rf8 47. Qa5 Nf7 48. Qc3 Ke7 49. Qa8 Rh8 50. Qc8 Qh7 51. Qd8+ Ke6 52. Na3 Qg6 53. Nc4 Re8 54. Rab2 Rh8 55. Rb1 Bb8 56. Ra2 Rh7 57. hxg6 Bc7 58. Bf2 Bb6 59. Be3 Ba7 60. g7 Bb6 61. Rb3 Rh8 62. Rb1 Bg2 63. Bf2 h5 64. Bg3 Nh3 65. Bf2 Bh1 66. Be3 Ba7 67. axb5 Rf8 68. Nb6 Nh8 69. Qd6+ Kf7 70. Nc4 Bb6 71. Qd7+ Kf6 72. g8=Q Rd8 73. Qc7 Rd6 74. Qe8 Ba7 75. bxc6 Rd5 76. Qb7 Rd6 77. c7 Re6 78. Bg1 Rd6 79. Be3 Rd5 80. Ke1 Bb8 81. Qb6+ Kg7 82. Qbe6 h4 83. Kd1 Bg2 84. Rb7 Bh1 85. Rb1 Kh7 86. Qf8 Rd8 87. cxb8=Q Rd7 88. Qff6 Rd5 89. Qee8 Rd8 90. Bg1 Rd5 91. exd5 e4 92. dxe4 Ng6 93. Qee7+ Kh6 94. c6 d3 95. Qff7 Nh8 96. Qfg8 Ng6 97. Bf2 Nh8 98. exf5 d2 99. Kxd2 Ng1 100. Bg2 Ng6 101. Nxg5 Nh8 102. Kd3 Nh3 103. Nf3 Nf7 104. c7 Nd8 105. cxd8=Q Ng1 106. Qgf7 Nh3 107. Qdd7 Ng5 108. Qfg8 Ne6 109. Qda7 Nf8 110. Ke2 Ne6 111. Qa4 Ng5 112. Nh2 Nh7 113. Qc6+ Nf6 114. d6 Nh5 115. d7+ Nf6 116. Qca8 Nh5 117. d8=Q Nf6 118. Rc2 Nh5 119. Qaa7 h3 120. Qdd5 Ng7 121. Qbc8 Ne8 122. Ra2 Nd6 123. Qa4 Ne4 124. Nf3 Ng5 125. Nd6 h2 126. Ne4 Nf7 127. Q8c7 Ng5 128. Ng3 Nh7 129. Qcb8 Nf8 130. f6 Ng6 131. Qa6 Nh4 132. Qh8+ Kg6 133. Ng5 Nf5 134. Qa4 Ng7 135. f7 Nh5 136. f8=Q Ng7 137. Rab2 Nf5 138. Qfc8 Ng7 139. Rxh1 Ne8 140. Rbb1 Nd6 141. Nf5 Nc4 142. Nf3 Ne3 143. Bh3 Ng2 144. Nh6 Nh4 145. g5 Kh5 146. Qb6 Nf5 147. Bg3 Ne3 148. f5 Nc2 149. f6 Na3 150. Qhh7 Nc2 151. Qhe4 Na3 152. f7 Nc2 153. Rb2 Na1 154. f8=Q Nc2 155. Rg1 Ne3 156. Qb1 Ng2 157. Qff6 Ne3 158. Qff4 Nf1 159. Rd2 Ne3 160. Ra2 Nf1 161. Bh4 Ne3 162. g6+ Nf5 163. Ng5 Nd6 164. Rg2 Nb7 165. Rxh2 Nd8 166. Kf1 Nb7 167. Rae2 Na5 168. g7 Nb7 169. Kg2 Na5 170. Rh1 Nb3 171. Kf2 Na5 172. g8=Q Nb3 173. Qgh8 Nd2 174. Qhh7 Nf1 175. Kg2 Nd2 176. Qhh8 Nc4 177. Qcb8 Na5 178. Bf5 Nb7 179. Qbc8 Na5 180. Bh7 Nb3 181. Qhf8 Na5 182. Qfg7 Nb7 183. Ng4 Nd8 184. Qh8 Nf7 185. Qhg8 Nh6 186. Qgg7 Ng8 187. Qh8 Nh6
9
  • Might 200 be possible with more delays in piece advancing? Apr 28, 2023 at 22:14
  • 1
    Actually, I take that back. To my knowledge the presumed (maybe proved?) upper bound for number of legal moves for a player is 218, which would imply that 200 moves here can not be possible. And also, that 195 is actually quite close already to the limit, although I think 196 is definitely possible from what I've got for 195.
    – koedem
    Apr 29, 2023 at 7:12
  • 1
    @RewanDemontay what I meant to say is, it is thought that there exists no position with more than 218 legal moves for one side. Since each pair of plies must add at least one legal move, after 199 moves White would need to have at least 199 + 20 = 219 legal moves which would be impossible. So based on that, the upper limit might be 198 moves, although it is of course not clear whether that is reachable in practice.
    – koedem
    Apr 29, 2023 at 9:19
  • 1
    @RewanDemontay but why does it matter? Each pair of half moves gains one legal move. So the length of this game is exactly the number of legal moves in the final position - 20. It doesn't matter much how we get there, the first 150 moves or so are not very important. However, reaching a position with a very high number of legal moves is the challenging bit.
    – koedem
    May 1, 2023 at 11:10
  • 1
    @RewanDemontay Think about it logically. If I gain one legal move for every pair of half moves that happen, then that is clearly the best strategy. I can't gain less than that, otherwise I violate the rules of the challenge. And I shouldn't do more than that, otherwise I waste potential moves I could use. So since I gain exactly one legal move, the final number of moves must be the number of legal moves in that final position, minus the 20 legal moves I started out with. Thus, when I only gain one legal move each round, the only thing that matters is how many legal moves I end up with.
    – koedem
    May 1, 2023 at 15:25
3

Here's a first attempt of mine:

[FEN ""]

1. b3 a6 2. g3 a5 3. b4 b6 4. c3 c6 5. c4 b5 6. bxa5 d5 7. g4 f5 8. a6 h5 9. a7 d4 10. axb8=R Kd7 11. cxb5 Ba6 12. bxa6 g5 13. gxf5 Qc7 14. f6 Kd6 15. f3 Bg7 16. Kf2 h4 17. fxe7 Kc5 18. Ke1 Bf8 19. Kf2 Rxa6 20. e8=Q h3 21. d3 Ne7 22. a3 Ng6 23. Ke1 Nh4 24. Kf2 Kd6 25. Ke1 g4 26. Kf2 gxf3 27. Ke1 fxe2 28. Kf2 Nf3 29. Kxf3 e1=R 30. Kf4 Ra4 31. Kf3 Rxa3 32. Kf2 Rxa1 33. Kg3 Ra3 34. Kf2 c5 35. Kg3 c4 36. Kf2 cxd3 37. Kg3 Ra1 38. Kf2 Bh6 39. Kg3 Bg7 40. Kg4 Rg8 41. Na3 Rb1 42. Nf3 Kc5 43. Nxb1 Kd6 44. Na3 Qc3 45. Nxd4 Rh8 46. Kg3 Kc5 47. Kg4 Rg8 48. Nc4 Rf8 49. Kxh3 Rf2 50. Kg3 Rexf1 51. Kh4 Kd5 52. Bd2 Qb2 53. Nd6 Qa1 54. Rxf1 Kc5 55. Kh3 Rf5 56. Kg2 Qb1 57. Kg3 Rf6 58. Kg2 Qa1 59. Bg5 Rf5 60. Kg3 Rf8 61. Nc4 Kd5 62. Bd2 Rg8 63. Bg5 d2 64. Rf2 Rh8 65. Kg2 Bh6 66. Kg3 Kc5 67. Rf6 Qb1 68. Kg2 Qa1 69. Bxd2 Bg7 70. h4 Kd5 71. Rb3 Bf8 72. Bf4 Kc5 73. Nc2 Qc1 74. Kf2 Bh6 75. Kg2 Bg5 76. Kf2 Qb1 77. Kg2 Qa1 78. Ra6 Bd8 79. Kf2 Bf6 80. Qed7 Bg5 81. Be5 Rh6 82. Qf3 Bc1 83. Kg1 Rh8 84. Rb8 Qa4 85. h5 Qa2 86. h6 Qa1 87. h7 Rf8 88. Kf2 Rg8 89. Ke1 Qa2 90. Kf1 Bg5 91. hxg8=R Be7 92. Rh6 Bd8 93. Ke1 Bc7 94. Rb3 Bb8 95. Rbxb8 Qa3 96. Rb1 Qa6 97. Kd2 Qb7 98. Ra1 Qa8 99. Nb4 Qa7 100. Kc2 Qa6 101. Rag1 Qa5 102. Ra1 Qa8 

Which, incidentally, breaks the supposed 100 move barrier. :)

1
  • 1
    I should have clarified that "100" was the value for the standard set. Since you have a second queen on board, breaking the 100 is not a surprise :-) (Which in no way should belittle your record attempt!) Apr 28, 2023 at 18:13
1

Here is my starting bid of 48 moves.

[FEN ""]

1. b4 Nc6 2. b5 Nf6 3. b6 d6 4. g4 Rg8 5. g5 Rh8 6. g6 Rg8 7. gxh7 Ne4 8. bxa7 Rb8 9. a4 e6 10. a5 Bd7 11. a6 Be7 12. axb7 Ra8 13. h4 Nf6 14. h5 Ne4 15. h6 Ng5 16. hxg7 Rh8 17. c4 Rg8 18. f4 Rh8 19. d4 Rg8 20. e4 Rh8 21. f5 Rg8 22. c5 Rh8 23. d5 Rg8 24. e5 Ne4 25. cxd6 Rh8 26. dxc7 Rg8 27. dxc6 Nc5 28. Kf2 Rh8 29. Qc2 Rg8 30. fxe6 Rh8 31. Rh2 Rg8 32. Rh5 Rh8 33. Rh4 Qc8 34. Ra5 Bd8 35. Raa4 f6 36. exf6 Rg8 37. e7 Rh8 38. cxd7+ Kf7 39. Nf3 Ke6 40. Kg3 Kd6 41. f7 Kc6 42. gxh8=Q Kb6 43. Qhb2+ Kc6 44. e8=R Kd5 45. Re6 Be7 46. f8=Q Bg5 47. Bxg5 Na6 48. h8=Q Nb8
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