Die Schwalbe

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1 - P0002611
Raymond Smullyan
The Chess Mysteries of Sherlock Holmes , p. 26, 1979
Ergänze einen wL auf e3 oder e4!
Monochromes Schach
play all play one stop play next play all
Forderungs-Alternative: Einfarben-Schach. Auf dem Brett steht ein unsichtbarer Läufer. Steht er auf einem schwarzen oder einem weißen Feld?
Yoav Ben-Zvi: In the book (page 26) the WB is placed between e3 and e4. It seems simpler to represent this with WB on e4, the stipulation "Monochrome chess. Is the position legal?" and a Twin b) WBe4 to e3. Proof of illegality with WB on e4: There is no piece that could perform the last capture on a dark square since no piece on the board is capable of a dark-squared capture and it could not be a piece that is missing since such a piece would itself have been captured later on a dark square. Smullyan's version uses poetic language including the metaphor of a Bear and a Lion fighting until each has entirely devoured the other. Essentially the argument uses the well-known mathemtical technique of "Infinite descent" leading to "Reductio ad Absurdum": It can be shown that assuming the existence of a "minimal" instance that has some property (such as most recent capture on a dark square) leads to the conclusion that it is not really minimal (there is a more recent capture on a dark square) so the assumption leads to self contradiction and is therefore not true. A formal definition of this method was developed by Fermat (17th Century) but it was used as far back as Euclid's "Elements" (300 B.C.). (2017-12-14)
Henrik Juel: It might be simpler to consider the retroplay:
With Le4 there is no way of producing a dark-squared man by uncapture (2017-12-14)
Brassaud: With Le4 it seems possible to producing a dark-squared man with the retroplay : 1) Lc2-e4, Kd7-e8 2) Ld3xPc2, b3xPc2 3) Ld3-e4, a4xb3 ep 4) b2-b4... But the position remains illegal as shown with the metaphor of the lion and the bear. (2017-12-15)
Henrik Juel: Good point, Francois
But the en passant trick still cannot create a full retroplay back to the initial position with 32 men
I should have written 'officer' instead of 'man' (2017-12-15)
Yoav Ben-Zvi: Henrik's modified claim that "with WBe4 there is no way of producing a dark-squared officer by uncapture" is correct but I do not see a simple way to prove it directly. A "lion and bear" type argument with retroplay could ask "Which of the pieces that move on the dark squares was the first to uncapture?". It should be noted that this piece could not be a pawn that was itself uncaptured e.p. (since the history of such a pawn does not include an uncapture) and that proving that such a piece does not exist proves illegality (since it implies that dark-squared officers such as BQd8 could not be uncaptured). A similar clarification is needed when applying the argument to forward moves.
The Homebase position of the diagram is an IC. It could be presented (optionally adding a WB on f1 and/or a BB on c8) with the stipulation "Is the position legal?" and the Twin b)-WPf2. (2017-12-17)
Keywords: Monochrome, Add pieces (L), Homebase (2)
Genre: Retro, Fairies
FEN: 4k3/8/8/8/8/8/3P1P2/4K3
Reprints: Schach mit Sherlock Holmes 1982
Input: Gerd Wilts, 1995-06-03
Last update: A.Buchanan, 2018-11-25 more...
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Gerd Wilts (1)