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Position Criticality in Chess Endgames Guy Haworth and Árpád Rusz guy.haworth@bnc.oxon

Position Criticality in Chess Endgames Guy Haworth and Árpád Rusz guy.haworth@bnc.oxon.org. Árpád Rusz, Starchess World Champion. Topics. Original inspiration Example problem to solve: the Saavedra study Problems with the graph as a proof mechanism The chess variant, Chess(SP)

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Position Criticality in Chess Endgames Guy Haworth and Árpád Rusz guy.haworth@bnc.oxon

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  1. Position Criticality in Chess Endgames Guy Haworth and Árpád Rusz guy.haworth@bnc.oxon.org Pos. Crit., 2011-11-11

  2. Árpád Rusz, Starchess World Champion Pos. Crit., 2011-11-11

  3. Topics • Original inspiration • Example problem to solve: the Saavedra study • Problems with the graph as a proof mechanism • The chess variant, Chess(SP) • Algorithm and implementation • Supplementary solvable problems • Arpad’s Starchess implementation • Futures Pos. Crit., 2011-11-11

  4. The original inspiration: the Chess Win Study c7"' Kb5"' Kb4“(‘) Kc2"' Kb3"' 1w 1b 2 3 4 5 6 7 the most famous study ... Saavedra & Barbier (1895): 1.c7 Rd6+ 2.Kb5 Rd5+ 3.Kb4 Rd4+ 4.Kb3 Rd3+ 5.Kc2 Rd4! 6.c8=R!! Pos. Crit., 2011-11-11 Photo by Matt Scott, used under a CCA 2.0 Generic license

  5. What about alternative White moves? cycling move slower, equivalent progress White to move Black to move divergent progress,maybe alternative solution  Pos. Crit., 2011-11-11

  6. Chess Study solutions: effectively unique • Moves to classify and maybe ‘ignore’: • a) White moves that revisit a position, • b) White moves that allow Black to force • a position to be revisited a) 3.Kb6 b) 4.Kc3' 5...Rd4 c7"' c7"' Kb5"' Kb5"' Kb4“(‘) Kb4“(‘) Kc2"' Kc2"' Kb3"' Kb3"' 1w 1w 1b 1b 2 2 3 3 4 4 5 5 6 6 7 7 4...Rd5+" 4.Kb5 6.Kb3 Pos. Crit., 2011-11-11

  7. Time-wasting move (continued) • mainline move m is m: P1 P2 • a third class of time-wasting move: • c) Black forces the line via P2 Qc1+ Qb1+" S2' 18 19 16.Qf1+ Qb6" Qb6" Kxb6"' Kc7"' Kc6"' 14w 14b 15 16 17 18 19 • this is an example from Hornecker (2009), HHdbIV#75649 • 16. Qb6" is the only DTM-optimal move • 16. Qf1+ makes no alternative progress, is slower and … • therefore is a time-waster of type ‘c’ Pos. Crit., 2011-11-11

  8. Graph ‘proofs’ of uniqueness 19...Kg5 • Saavedra and • Barbier (1895) • Hornecker (2009) • Verdict: ‘graph proofs’ are not compact or manifestly correct 19.Qf1+ 18.Qf1+ Qf1+"' 18...Kg5 18 17.Qg2 Qc1+ Qb1+" S2' 18 19 16.Qf1+ Qb6" 17...Kf6 Qb6" Kxb6"' Kc7"' Kc6"' 18.Qa3 14w 14b 15 16 17 18 19 16...Kf5" 16.Qa5 4...Rd1" 5.Kc2"' 3.Kb6 4.Kc3' 5...Rd4 c7"' Kb5"' Kb4“(‘) Kc2"' Kb3"' 1w 1b 2 3 4 5 6 7 4...Rd5+" 4.Kb5 6.Kb3 Pos. Crit., 2011-11-11

  9. Can White win, without repeating position? • declare this positions drawn! • pos. P becomes a refuge for Black • give Black a Get out of Jail card • let SP be the set of such refuges • this defines Chess variant Chess(SP) • compute the Chess(SP) EGT ... EGTSP • what has changed ... (EGT, EGTSP) ? Define this position P to be ‘drawn’ rather than won for White Pos. Crit., 2011-11-11

  10. The impact of defining some extra draws positions with changed values or depths TP • SP is the set of positions newly defined to be draws • the set SP defines a ‘designer’ Chess Variant Chess(SP) • TP is the set of positions whose Chess(SP) values we ‘target’ • When the Chess(SP) values of positions in TP are known ... stop! • Black can force White’s win from a position P in TP through a position in SP ... if and only if P is a draw in Chess(SP) IP2 ? positions with changed values IP1 ? ? SP depth = d plies No effect on positions which are no deeper Pos. Crit., 2011-11-11

  11. Ignorable positions: Black wins and draws, positions no deeper than p  SPUnreachable positions, e.g., Pawns elsewhere other irrelevant positions, e.g., with Pawns in different positions } TP the remaining positions IP1 SP E Positions SP and no deeper than shallowest in SP draws wins for Black Pos. Crit., 2011-11-11

  12. Starchess Statistics • 1,626,168,997 different sub-6-man positions • 9,967,573 type B1-M zugs i.e. White (to move) has a win, but has a quicker win with btm • 7,168,489 type B1-M zugs have a Pawn and zug-depth = 1 or = 2 • 2,799,084 type B1-M zugs needed to be examined • the goal was to find ‘Vital’ B1 zugs where Black forces through the btm pos. • 2,751,547 type Pawnless B1-M zugs had zd = 1 • 47,537 type B1-M zugs required more examination • 910 B1-M zugs are Vital B1 zugs • 6 Vital B1 zugs are ‘reflective’ ... Involving symmetry • we will see them all: 2 with zd=1, 1 with zd=2, 1 with ‘Limping Pawn’ • The computation was done 50x faster than calculating EGTs Pos. Crit., 2011-11-11

  13. Starchess: the start position and moves Video  Pos. Crit., 2011-11-11

  14. Guide to the Starchess videos following • In the top left panel are indicated: • the Starchess values and depths if White to move, or Black to move • the Starchess values and depths after the various moves available • In the bottom left panel are indicated: • the Starchess(SP) values and depths if White to move, or Black to move • the Starchess(SP) values and depths after the various moves available • Thus, it is possible to see: • if a position is a type B1-M zugzwang in Starchess • If a position is a vital type B zug (if so, the position is a draw, wtm, when the shallower btm position is set to draw) • Top right is the opening position of some line illustrated Pos. Crit., 2011-11-11

  15. Starchess: a Vital type B zugzwang  Video Pos. Crit., 2011-11-11

  16. Starchess: ‘Reflective’ Vital type B zugzwangs • 1.R34 • 1.R26 • 1.Q21 ... • d) 9 ply • e) , LP • f) 11 ply Pos. Crit., 2011-11-11  Video

  17. Starchess: ‘longest transit’ Vital type B zug  Video Pos. Crit., 2011-11-11

  18. A B1-M zug which fails to be a Vital B1 zug  Video Pos. Crit., 2011-11-11

  19. Going forward ... • Compose some 250 Starchess studies based on zug transits • Identify positions with An Essentially Unique Winning Move • Identify sequences of such positions • Audit 70,799 study positions in the Endgame Table Zone (ETZ) • modified Chess EGT generators are in prospect • sub-6-man Pawnless positions, sub-6-man Pawnful etc. • Implement Chess(SP) in a way in which it can be widely used • by composers, validators, judges of and the audience for studies • Adopting various sets of positions as SP • Exploiting prior per-endgame-slice WDL and EGT information • Devise a structure for Study Expositions which ... separates out the ‘core story’ and essential aesthetics from ... the detail lines which are needed for technical completeness Pos. Crit., 2011-11-11

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