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

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

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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 Arpads Starchess implementation Futures Pos. Crit., 2011-11-11 3

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The original inspiration: the Chess Win Study Pos. Crit., 2011-11-11 4 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!! 1w1b234576 Kb3 "' Kb5 "' Kb4 () Kc2 "' c7 "' Photo by Matt Scott, used under a CCA 2.0 Generic license

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

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Chess Study solutions: effectively unique Pos. Crit., 2011-11-11 6 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 1w1b234576 5...Rd4 4.Kc3' 4.Kb5 4...Rd5+ " Kb3 "' Kb5 "' Kb4 () Kc2 "' c7 "' 1w1b234576 3.Kb6 6.Kb3 Kb3 "' Kb5 "' Kb4 () Kc2 "' c7 "' a) b)

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Time-wasting move (continued) Pos. Crit., 2011-11-11 7 14w14b1516171819 S2'1819 16.Qf1+ Kxb6"'Kc7"'Kc6"' Qb1+" Qb6" Qc1+ mainline move m is m: P 1 P 2 a third class of time-wasting move: c) Black forces the line via P 2 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

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Saavedra and Barbier (1895) Hornecker (2009) Verdict: graph proofs are not compact or manifestly correct Graph proofs of uniqueness Pos. Crit., 2011-11-11 8 1w1b234576 3.Kb6 6.Kb3 4...Rd1 " 5.Kc2 "' 5...Rd4 4.Kc3' 4.Kb5 4...Rd5+ " Kb3 "' Kb5 "' Kb4 () Kc2 "' c7 "' 14w14b1516171819 16.Qa5 16...Kf5" S2'1819 17...Kf6 16.Qf1+ 17.Qg2 18.Qf1+ 18...Kg5 19...Kg5 19.Qf1+ 18.Qa3 Kxb6"'Kc7"'Kc6"' Qf1+"' Qb1+" Qb6" Qc1+ 18

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Can White win, without repeating position? Pos. Crit., 2011-11-11 9 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... EGT SP what has changed... (EGT, EGT SP ) ? Define this position P to be drawn rather than won for White

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The impact of defining some extra draws 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 Whites win from a position P in TP through a position in SP... if and only if P is a draw in Chess(SP) Pos. Crit., 2011-11-11 10 SP IP1 IP2 positions with changed values or depths positions with changed values depth = d plies TP ? ? ? No effect on positions which are no deeper

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

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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 12

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

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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 14

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

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Pos. Crit., 2011-11-11 a)1.R 34 b)1.R 26 c)1.Q 21... d) 9 ply e), LP f) 11 ply Starchess: Reflective Vital type B zugzwangs 16 Video

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

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

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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 19

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