Solutions for The Chain of Proof is Broken
It's all about en passant and castling. If the last move in a
position must have been to castle, then the castling must have been
legal. In the prior position, however, other proof games that do not
preserve castling rights may have been possible. Similarly, if the last move
must have been en passant, the prior position might also include
possibilities where the last move was not a pawn double hop.
{A} a) 1. f3 h6 2. Kf2 h5 3. Kg3 h4
4. Kh3 e5+ 5. g4 h×g3ep+. (C+ Natch 2.0)
b) There are 21 other routes to arrive at the position immediately
prior to 5...hxg3ep+. In all of these White played 5. Kh3, and so the
e.p. would be illegal as Black's 5th move.
{B} a) 1. g4 d6 2. Bh3 Kd7 3. Kf1
Ke3 4. g5+ f5 5. g×f6ep+. (C+ Natch 2.0)
b) There are 12 other routes to arrive at the position position
immediately prior to 5. g×f6ep+. In all of these Black played
4...Ke3, and so the e.p. would be illegal as White's 6th
move. Alas I was anticipated in this one, but that's hardly
surprising.
{C} a) 1.d3 h6 2.Bxh6 Nf3 3.Bxg7 Nh7
4.Bxf8 Nxf8 5.d4 Nh7 6.d5 0-0. (C+ Natch 2.0)
b) To arrive at the position immediately prior to 6.d5, Black might
have played instead 4...Kxf8 & 5...Ke8. Alternatively 4...Rxf8
& 5...Rh8. Of course in neither case would the later castling
be legal.
{D} 1. Nf3 b5 2. Ne5 b4 3.
N×d7 b3 4. N×b8 Qd6 5. Na6 Kd7 6. a3 Kc6 7. Nb4+
Kc5 8. Na2 b×a2 9. a4 a×b1=B 10. Ra3 Ba2 11. Rd3
Bc4 12. b3 Kb4 13. Bb2 c5 14. Bf6 g×f6 15. Qa1 Bh6 16. Qd4
Be3 17. d×e3 Nh6 18. Rd1 Rg8 19. Ra1 Rg3 20. Qd1 Rh3 21.
g×h3 Bb5 22. Bg2 Be8 23. O-O. (C+ Natch 2.0) Not only a
paradox, but two capture-free roundtrips and a tempo move too!
Michel Caillaud wrote (Feb 17, 2002):
Bonsoir Nicolas et Andrew,
Non, la question de Nicolas [Dupont] n'est pas délirante et
Andrew a montré les 2 moyens de réaliser
l'idée.
La SPG en 4.5 avec pep a déjà
été composée (par Thierry Le Gleuher
et d'autres qui sont aussi "tombés" sur la même
position).
Quant au roque, il figure déjà dans des SPG assez
élaborées où le "truc" du roque sert
à présenter un thème en
évitant des démolitions par manoeuvres d'attente
indésirables.
Voici un exemple dans ma production (Probleemblad 1998) :
r1b1b3/p3pp1p/3q1p1n/2p5/Pk6/1P2P2P/2P1PPBP/R2Q1RK1 SPG 22,5
(et la position en 22,0 n'est pas une SPG exacte...)
Enfin, dans les exemples vus jusqu'ici, la position au coup n-0,5
s'obtient en n-0,5 coups mais de façon non unique. Je suis
quasi certain, mais je n'arrive pas à remettre la main sur
des exemples, que des compositeurs (Andreï Frolkin
peut-être?) se sont intéressés au cas
où la position au coup n-0,5 s'obtient en p < n-0,5
coups !!
Amicalement,
Michel
Perhaps the following page
is the one Michel is thinking about at the end. However, I haven't seen
any quantum proof games...
{E} a) 1. d4 f5 2. d3 Kf7 3. Bd2 Ke6
4. d5+ Kd6 5. Bb4+ c5 6. d×c6ep+ K×c6.
b) There are 8 other routes to arrive at the position immediately prior
to 5. d×c6ep+. In all of these Black played 5...Kd6, and so
the e.p. would be illegal as White's 6th move. (C+ Natch 2.0)
I wanted to find a paradoxical PG involving
e.p. but with no check in the final diagram. Clearly, the e.p. move
itself must give check. So the only way that this can be done is by
extending the position for at least another move, past the en passant.
Neither {A} nor {B} can be so extended. However {E} works. But can it
be done more quickly? See {F} & {G} below.
{F} a) 1. d4 c6 2. d5 c×d5
3. c4 d×c4 4. Qa4 Qa5+ 5. b4 c×b4ep+ 6. Bd2. (C+
Natch 2.0)
b) There are 6 other routes to arrive at the position immediately prior
to 5...c×b4ep+. In all of these, White played 5. Qa4, and so
the e.p. would be illegal as Black's 5th move.
{G} a) 1. b4 e5 2. Ba3 Ke7 3. Qc1 Kd3
4. b5+ c5 5. b×c5ep+ K×c6. (C+ Natch 2.0)
b) There are 9 other routes to arrive at the position immediately prior
to 5. b×c5ep+. In all of these, Black played 4. ... Kd6, and
so the e.p. would be illegal as White's 5th move..
{H}1. d3 Sh6 2.B×h6 g6
3.Qc1 B×h6 4.Kd1 Bd2 5.Q×d2 O-O. (C+ Natch 2.0) But
to arrive at the position immediately prior to 5...0-0, the game might
have gone 3. B×f8 K(R)×f8 4. Qd2 Ke8(Rh8) 5. Kd1.
Of course in neither case would the later castling be legal.
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