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1 - P1370701
James Malcom
Andrew Buchanan

chess.stackexchange.com 25/11/2019
P1370701
(36+16)
How long is the longest possible game of Horde Chess, subject to the 75-move rule?
1. b6 a6 2. b5 Ta7 3. bxa7 Sc6 4. a8=D Se5 5. Da7 Sc6 6. Db6 Se5 7. Dh6 Sc6 8. Dh5 Se5 [...] 79. b6 Sf6 80. Dh6 Sc6 81. Dh5 Sa7 [...] 154. bxa7 Sg8 155. Dh6 Sf6 156. Dh5 Sg8 [...] 229. a8=D Sf6 230. Da7 Sg8 231. Db6 Sf6 232. Db5 Sg8 [...] 303. Dh6 axb5 304. Dh5 Sf6 305. Dh6 Sg8 [...] 378. a5 Sf6 379. Dh5 Sg8 380. Dh6 Sf6 [...] 453. a6 Sg8 454. Dh5 Sf6 455. Dh6 Sg8 [...] 528. a7 Sf6 529. Dh5 Sg8 530. Dh6 Sf6 [...] 603. a8=D Sg8 604. Dh5 Sf6 605. Da6 Sg8 606. De6 Sf6 607. De5 Sg8 [...] 678. a4 Sf6 [....] 753. a5 Sg8 [....] 828. a6 Sf6 [....] 903. a7 Sg8 [....] 978. a8=T Sf6 979. Ta6 Sg8 980. Te6 Sf6 [...] 1053. a3 Sg8 [....] 1128. a4 Sf6 [....] 1203. a5 Sg8 [....] 1278. a6 Sf6 [....] 1353. a7 Sg8 [....] 1428. a8=T Sf6 1429. Taa6 Sg8 1430. Tad6 Sf6 [....] 1503. a2 Sg8 [....] 1578. a3 Sf6 [....] 1653. a4 Sg8 [....] 1728. a5 Sf6 [....] 1803. a6 Sg8 [....] 1878. a7 Sf6 [....] 1953. a8=T Sg8 [....] 2027. Ta4 bxa4 [....] 2102. Dh6 a3 [....] 2177. Dh5 a2 [....] 2252. Dh6 a1=T 2253. Dh5 Ta8 [....] 2327. Ta6 bxa6 [....] 2402. Dh6 a5 [....] 2477. Dh5 a4 [....] 2552. Dh6 a3 [....] 2627. Dh5 a2 [....] 2702. Dh6 a1=T [....] 2777. Def6 Sxf6 [....] 2852. Dh5 Sxh5 [....] 2927. b4 Sf6 [....] 3002. b5 Sg8 [....] 3077. b6 Sf6 [....] 3152. b7 Sg8 [....] 3227. b8=D Sf6 [....] 3302. Dxa8 Sg8 [....] 3377. b3 Sf6 [....] 3452. b4 Sg8 [....] 3527. b5 Sf6 [....] 3602. b6 Sg8 [....] 3677. b7 Ta7 [....] 3752. b8=D Sf6 [....] 3827. Dbxa7 Sg8 [....] 3902. b2 Sf6 [....] 3977. b3 Sg8 [....] 4052. b4 Sf6 [....] 4127. b5 Sg8 [....] 4202. b6 Sf6 [....] 4277. b7 Sg8 [....] 4352. b8=T La6 [....] 4427. Dxa6 Dc8 [....] 4501. Tbb6 cxb6 [....] 4576. Th6 b5 [....] 4651. Th5 b4 [....] 4726. Th6 b3 [....] 4801. Th5 b2 [....] 4876. Th6 b1=T [....] 4951. Db8 Dxb8 [....] 5026. Db6 Dxb6 [....] 5101. c6 Dd8 [....] 5176. c7 Sf6 [....] 5251. c8=D Tb8 [....] 5326. Dxb8 Sg8 [....] 5401. c5 Sf6 [....] 5476. c6 Sg8 [....] 5551. c7 Sf6 [....] 5626. c8=D Sg8 [....] 5701. c4 Sf6 [....] 5776. c5 Sg8 [....] 5851. c6 Sf6 [....] 5926. c7 Sg8 5927. Da7 Sf6 5928. Dca6 Sg8 [...] 6001. c8=D Sf6 [....] 6076. c3 Sg8 [....] 6151. c4 Sf6 [....] 6226. c5 Sg8 [....] 6301. c6 Sf6 [....] 6376. c7 Sg8 6377. Dca8 Sf6 [...] 6451. c8=D Sg8 [....] 6526. c2 Sf6 [....] 6601. c3 Sg8 [....] 6676. c4 Sf6 [....] 6751. c5 Sg8 [....] 6826. c6 Sf6 [....] 6901. c7 Sg8 6902. Dcb8 Sf6 [...] 6976. c8=D Sg8 [....] 7050. Dcc6 dxc6 [....] 7125. Dbc8 Dxc8 [....] 7200. D8b8 Dxb8 [....] 7275. Da8 Dxa8 [....] 7350. Dc8+ Dxc8 [....] 7425. Th5 c5 [....] 7500. Th6 c4 [....] 7575. Th5 c3 [....] 7650. Th6 c2 [....] 7725. Th5 c1=T 7726. Th6 Kd8 7727. Th5 Kc7 7728. Th6 Kb8 [...] 7800. d5 Sf6 [....] 7875. d6 Sg8 [....] 7950. d7 Sf6 [....] 8025. d8=D Sg8 [....] 8100. Dxf8 Sf6 [....] 8175. d4 Sg8 [....] 8250. d5 Sf6 [....] 8325. d6 Sg8 [....] 8400. d7 Sf6 [....] 8475. d3 Sg8 [....] 8550. d4 Sf6 [....] 8625. d5 Sg8 [....] 8700. d6 Sf6 [....] 8775. d2 Sg8 [....] 8850. d3 Sf6 [....] 8925. d4 Sg8 [....] 9000. d5 Sf6 [....] 9075. d8=D Sg8 9076. Da5 Sf6 [...] 9150. d7 Sg8 [....] 9225. d8=D Sf6 9226. De8 Sg8 [...] 9300. d6 Sf6 [....] 9375. d7 Sg8 [....] 9450. d8=D Sf6 [....] 9524. Dd6 exd6 [....] 9599. Dg8 Txg8 [....] 9674. Df8 Txf8 [....] 9749. Dd8 Txd8 [....] 9824. Th5 d5 [....] 9899. Th6 d4 [....] 9974. Th5 d3 [....] 10049. Th6 d2 [....] 10124. Th5 d1=D [....] 10199. e5 Tc6 [....] 10274. e6 Sg8 [....] 10349. e7 Sf6 [....] 10424. e8=D Sg8 [....] 10499. Dxc6 Sf6 [....] 10574. e4 Sg8 [....] 10649. e5 Sf6 [....] 10724. e6 Sg8 [....] 10799. e7 Sf6 [....] 10874. e8=D Td7 [....] 10949. Dexd7 Sg8 [....] 11024. e3 Sf6 [....] 11099. e4 Sg8 [....] 11174. e5 Sf6 [....] 11249. e6 Sg8 [....] 11324. e7 Sf6 [....] 11399. e8=D Sg8 [....] 11474. e2 Sf6 [....] 11549. e3 Sg8 [....] 11624. e4 Sf6 [....] 11699. e5 Sg8 [....] 11774. e6 Sf6 [....] 11849. e7 Sg8 11850. Df8 Sf6 [...] 11924. e8=T Sg8 [....] 11998. Dce6 fxe6 [....] 12073. Td8 Dxd8 [....] 12148. De8 Dxe8 [....] 12223. Th6 e5 [....] 12298. Th5 e4 [....] 12373. Th6 e3 [....] 12448. Th5 e2 [....] 12523. Th6 e1=D [....] 12598. Dd6 Dxd6 [....] 12673. f6 Se7 [....] 12748. f7 Sg8 [....] 12823. f8=D Da3 [....] 12898. Dxa3 Se7 [....] 12973. f5 Sc6 [....] 13048. f6 Sa7 [....] 13123. f7 Sc6 [....] 13198. f8=D Sa7 [....] 13273. f4 Sc6 [....] 13348. f5 Sa7 [....] 13423. f6 Sc6 [....] 13498. f7 Sa7 [....] 13573. f3 Sc6 [....] 13648. f4 Sa7 [....] 13723. f5 Sc6 [....] 13798. f6 Sa7 [....] 13873. f2 Sc6 [....] 13948. f3 Sa7 [....] 14023. f4 Sc6 [....] 14098. f5 D1e7 [....] 14173. Dfxe7 Sa7 [....] 14248. f8=D Sc6 14249. De1 Sa7 14250. Dfc5 Sc6 [...] 14323. f7 Sa7 [....] 14398. f8=D Sc6 [...] 14473. f6 Sa7 [....] 14548. f7 Sc6 14549. Dfe7 Sa7 [...] 14623. f8=D Sc6 [....] 14697. D8f6 gxf6 [....] 14772. D7e6 Dxe6 [....] 14847. Dc4 Dxc4 [....] 14922. Da2 Dxa2 [....] 14997. De7 f5 [....] 15072. Th5 f4 [....] 15147. Th6 f3 [....] 15222. Th5 f2 [....] 15297. Th6 f1=D [....] 15372. Df8+ Dxf8 [....] 15447. g6 Sa7 [....] 15522. g7 Sc6 [....] 15597. g8=D Sa7 [....] 15672. Dxa2 Sc6 [....] 15747. g5 Sa7 [....] 15822. g6 Sc6 [....] 15897. g7 Sa7 [....] 15972. g8=D Sc6 [....] 16047. g4 Sa7 [....] 16122. g5 Sc6 [....] 16197. g6 Sa7 [....] 16272. g7 Sc6 [....] 16347. g3 Sa7 [....] 16422. g4 Sc6 [....] 16497. g5 Sa7 [....] 16572. g6 Sc6 [....] 16647. g2 Sa7 [....] 16722. g3 Sc6 [....] 16797. g4 Sa7 [....] 16872. g5 Sc6 16873. D8c4 Sa7 [...] 16947. g8=D Sc6 16948. D8d5 Sa7 [...] 17022. g7 Sc6 [...] 17097. g8=D Sa7 17098. D8e6 Sc6 [...] 17172. g6 Sa7 [....] 17247. g7 Sc6 [....] 17322. g8=D Sa7 [....] 17396. D8g6 hxg6 [....] 17471. Da3 Dxa3 [....] 17546. Dcc5 Dxc5 [....] 17621. Ddd6+ Dxd6 [....] 17696. De7 Dxe7 [....] 17771. Th5 g5 [....] 17846. Th6 g4 [....] 17921. Th5 g3 [....] 17996. Th6 g2 [....] 18071. Tg6 g1=D [....] 18146. Tg8+ Dxg8 [....] 18221. h5 Sc6 [....] 18296. h6 Sa7 [....] 18371. h7 Sc6 [....] 18446. h8=D Sa7 [....] 18521. h4 Sc6 [....] 18596. h5 Sa7 [....] 18671. h6 Sc6 [....] 18746. h7 Sa7 [....] 18821. h3 Sc6 [....] 18896. h4 Sa7 [....] 18971. h5 Sc6 [....] 19046. h6 Sa7 [....] 19121. h2 Sc6 [....] 19196. h3 Sa7 [....] 19271. h4 Sb5 [....] 19346. h5 Sc3 [....] 19421. Dxc3 Ka8 [....] 19496. h8=D Kb7 19497. D8d4 Ka8 [...] 19571. h7 Kb7 [....] 19646. h8=D Ka8 19647. D8e5 Kb7 [...] 19721. h6 Ka8 [....] 19796. h7 Kb7 [....] 19871. h8=L D8g7 [....] 19946. Lxg7 Df6 [....] 20021. Lxf6 Ka8 [....] 20095. Db8 Kxb8 [....] 20170. Da7 Kxa7 201711. Ld8 Kb7 [...] 20245. Lb6 Kxb6 [...] 20320. Dc7 Kxc7
play all play one stop play next play all
20320.0 moves

Begin by promoting a White pawn as quickly as possible, because until that happens, White is just haemorrhaging pawn moves. Now can promote a second pawn more slowly. This promoted piece is captured by Black pawn to open a file, and the first file's promotions can happen.

Scheme: axb to let all wPs from the a-file promote, then bxa+bxa to let all remaining wPs from the b-file promote, then cxb to let all wPs from the c-file promote, and so on.

One player is the active one, moving pawns. This player can also capture spare officers with officers. But pawns can never be captured before they promote (hugely expensive) and there are only 2 captures by wPs and 9 captures by bPs.

There are in principle 188+36+48+15=287 marks (captures & pawn moves). However, we lose 9 for Black pawn captures, 2 for White pawn captures, and 5 for pawn moves leading to the first White promotion. So there are really just 271 marks. So if the count begins after 4. a8=?, we have a total length of 20328.5 moves minus 0.5 for each parity switch.

So how many parity switches must there be? White is the first active player, then need two switches to promote the pawns from any file. So that's 16 switches, and White has promoted all his h-file pawns. A 17th switch is required for Black to capture these.

What impact does dead position have? If it applies literally, then Black cannot win if he has lost all his units, because he can't capture the last White unit. So let's suppose that Black winning "counts" as checkmate for DP purposes. That gives a total of 20320.0 moves.
James Malcom: Thanks for adding this in Andrew. (2019-12-18)
A.Buchanan: You’re welcome James and welcome to PDB :-) (2019-12-18)
Olaf Jenkner: It's his 7th problem in the PDB, welcome again. (2019-12-19)
Henrik Juel: I hope the authors will supply a more detailed solution than just the number
Curious that the answer is 75x269 (2019-12-19)
Mario Richter: Henrik, in case you just can't wait, follow the original discussion at
https://chess.stackexchange.com/questions/27037/longest-possible-game-of-horde-chess

Btw. my own estimate was approx. 20400 (my game starting with 1. b6 Sc6 2. bxa7 Tb8 3. a8=S), so most probably I was counting the "marks" wrongly ... (2019-12-19)
Mario Richter: Quote: "However, we lose 12 for Black pawn captures"

Stupid question: why 12? I'm counting only 9:axbxa, bxa, cxb, ..., hxg.
Scheme: axb to let all wPawns from the a-file promote, then bxa+bxa to let all remaining wPawns from the b-file promote, then cxb to let all wPawns from the c-file promote, a.s.o. (2019-12-20)
A.Buchanan: @Mario you’re right. There’s a lot more parity handovers, but it gives is a chance to clear up the board better. I wonder if it’s possible to use the animation + multiple diagrams to show selected parts of the solution? (2019-12-20)
A.Buchanan: Playing out the moves and doing the counting both lead to the same total now: 20320.0 moves. (2019-12-21)
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Keywords: Horde Chess, 75 move rule, Dead Position
Genre: Mathematics
FEN: rnbqkbnr/pppppppp/8/1PP2PP1/PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP
Input: A.Buchanan, 2019-12-17
Last update: A.Buchanan, 2019-12-21 more...
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