Pos. Crit., 2011-11-111

Position Criticalityin Chess Endgames

Guy Haworth and Árpád Ruszguy.haworth@bnc.oxon.org

Árpád Rusz, Starchess World Champion

Pos. Crit., 2011-11-112

Topics

Original inspirationExample problem to solve: the Saavedra studyProblems with the graph as a proof mechanismThe chess variant, Chess(SP)Algorithm and implementationSupplementary solvable problemsArpad’s Starchess implementationFutures

Pos. Crit., 2011-11-113

The original inspiration: the Chess Win Study

Pos. Crit., 2011-11-114

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!!

1w 1b 2 3 4 5 76Kb3"'Kb5"' Kb4“(‘) Kc2"'c7"'

Photo by Matt Scott, used under a CCA 2.0 Generic license

What about alternative White moves?

Pos. Crit., 2011-11-115

White to move

Black to move

cycling move slower, equivalent progress

divergent progress, maybe alternative solution

Chess Study solutions: effectively unique

Pos. Crit., 2011-11-116

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

1w 1b 2 3 4 5 76

5...Rd44.Kc3'

4.Kb54...Rd5+"

Kb3"'Kb5"' Kb4“(‘) Kc2"'c7"'

1w 1b 2 3 4 5 76

3.Kb6

6.Kb3

Kb3"'Kb5"' Kb4“(‘) Kc2"'c7"'

a)

b)

Time-wasting move (continued)

Pos. Crit., 2011-11-117

14w 14b 15 16 17 18 19

S2' 18 1916.Qf1+

Kxb6"' Kc7"' Kc6"'

Qb1+"

Qb6"

Qb6"

Qc1+

• mainline move m is m: P1 P2

• a third class of time-wasting move:c) Black forces the line via P2

this is an example from Hornecker (2009), HHdbIV#7564916. Qb6" is the only DTM-optimal move16. Qf1+ makes no alternative progress, is slower and …

therefore is a time-waster of type ‘c’

• Saavedra and Barbier (1895)

• Hornecker (2009)

• Verdict: ‘graph proofs’ are not compact or manifestly correct

Graph ‘proofs’ of uniqueness

Pos. Crit., 2011-11-118

1w 1b 2 3 4 5 76

3.Kb6

6.Kb3

4...Rd1" 5.Kc2"'5...Rd44.Kc3'

4.Kb54...Rd5+"

Kb3"'Kb5"' Kb4“(‘) Kc2"'c7"'

14w 14b 15 16 17 18 19

16.Qa516...Kf5"

S2' 18 19

17...Kf616.Qf1+

17.Qg2

18.Qf1+

18...Kg5

19...Kg519.Qf1+

18.Qa3 Kxb6"' Kc7"' Kc6"'

Qf1+"'

Qb1+"

Qb6"

Qb6"

Qc1+18

Can White win, without repeating position?

Pos. Crit., 2011-11-119

• 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 Pto be ‘drawn’ rather than

won for White

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 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)

Pos. Crit., 2011-11-1110

SP

IP1

IP2positions with changed

values or depths

positions with changed values

depth = d plies

TP

?

??

No effect on positions which are no deeper

other irrelevant positions,e.g., with Pawns indifferent positions

Ignorable positions: Black wins and draws, positions no deeper than p SP Unreachable positions, e.g., Pawns elsewhere

Pos. Crit., 2011-11-1111

}EPositions SP and no deeper than shallowest in SP

draws

wins for Black

SPIP1

TPthe

remainingpositions

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., 2011-11-1112

Starchess: the start position and moves

Pos. Crit., 2011-11-1113

Video

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., 2011-11-1114

Pos. Crit., 2011-11-11

Starchess: a Vital type B zugzwang

15

Video

Pos. Crit., 2011-11-11

a) 1.R34

b) 1.R26

c) 1.Q21 ...

d) 9 ply

e) , LP

f) 11 ply

Starchess: ‘Reflective’ Vital type B zugzwangs

16 Video

Starchess: ‘longest transit’ Vital type B zug

Pos. Crit., 2011-11-1117

Video

A B1-M zug which fails to be a Vital B1 zug

Pos. Crit., 2011-11-1118

Video

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., 2011-11-1119