Special conventions in retro-analytical problems

In an earlier Walkabout, we examined Two problem conventions re castling and capturing en passant. These important conventions determine if the two special moves in chess are permitted when they are an apparent option in a problem diagram. Refer to that blog for two demonstrative problems, but as a recap, the rules are: (1) Castling is assumed to be legal, unless it can be proved that the king or rook must have moved previously, and (2) An en passant capture of an adjacent pawn on the fifth rank is assumed to be illegal, unless it can be proved that the previous move must have been a double-step by that pawn. Such proofs form the basis of many retro-analytical problems, where we try to deduce the prior play that led to the diagram. Now, there are a few additional conventions that elaborate on the two principal ones, covering some situations where the legality of more than one special move is in question. These rules have technical-sounding names but they give rise to an intriguing sub-genre of retro compositions, examples of which can be quite entertaining and, at times, even controversial.

The special conventions are called Retro-Strategy (RS), A Posteriori (AP), and Partial Retrograde Analysis (PRA). Though all are briefly described in Article 16 of the Codex for Chess Composition (official rules formulated by the WFCC), these ideas are much easier to grasp when illustrated with specific problems. Here I shall cover the first two types, Retro-Strategy and A Posteriori. The examples used are all found in Chess Problems Out of the Box by Werner Keym, an excellent book I reviewed some time ago.

To force mate in two moves, White must put the h1-rook on f1 to threaten 2.Rf8, with either 1.Rhf1 or 1.0-0. But apparently Black can defeat either move with 1…0-0-0 – castling is allowed since Black’s previous move could have been …c7-c6 or …h7-h6, meaning Black’s king and rook need not have moved before. Let’s retro-analyse the position further. Assume for now that white castling is legal as well, which indicates the king and h1-rook have never moved. That would also imply the original rook from a1 couldn’t have left its corner area (due to the blocking king and pawns), and the rook on f3 is actually promoted. Regardless of where the promotion took place, for the f3-rook to have escaped from the 8th rank, it had to use a file not blocked by a black pawn. So the piece must have visited either a8, e8, or f8 (where it attacked e8), in all cases dislodging the black king or rook from its initial square. Thus black castling is proved to be illegal if the f3-rook is promoted and white castling is still possible. However, what if the f3-rook is the original rook from a1? In that case, there’s no reason to assume Black has moved either the king or rook, but then White must have moved the king previously to let the a1-rook out, and so cannot castle now.

In other words, we have shown that castling rights are mutually exclusive: if White can castle, then Black can’t, and if Black can castle, then White can’t. The Retro-Strategy convention dictates that in such a situation, the player who has the turn is permitted to castle. Consequently, this mate-in-2 problem is solved by 1.0-0! (2.Rf8) – White “proves” that castling is still legal by doing it, and thereby renders black castling illegal, due to the mutually exclusive rights. If White plays 1.Rhf1? instead, that fails to take advantage of this “first come, first served” rule, and Black can legally refute with 1…0-0-0!

The A Posteriori convention brings about some very curious compositions, such as this helpmate. (Unlike Retro-Strategy, which automatically applies in problems, the A Posteriori condition is usually specified in a problem’s stipulation – as “AP” – to indicate it’s in effect.) Here White wants to start with an en passant capture of the d5-pawn, to enable the e5-pawn to promote on the third move and mate. But the standard convention would forbid such a capture, because Black’s last move wasn’t necessarily d7-d5 – it could have been a king or rook move. Now suppose Black castles at some point in the solution (as is permitted under the basic rule); the move would establish that Black actually had not moved the king or rook previously and, by implication, Black’s last move to reach the diagram was d7-d5 after all. Thus, employing A Posteriori logic – the term means “from the later” – we will play 0-0-0 later and that retroactively “proves” Black had made the pawn double-step. The unique last move hence legalises 1…exd6 e.p.! The solution continues 2.0-0-0 dxe7 3.Rf8 exf8=Q. Note that after 1…exd6 e.p., alternative lines like 2.Ra7 dxc7 3.Ra6 c8=Q and 2.Kd8 dxe7+ 3.Kc8 e8=Q are invalid, because Black didn’t castle and therefore the initial en passant capture was never legalised!

When the A Posteriori principle is applied to directmate problems and the two sides are acting in opposition, the results can be even more bizarre. In this mate-in-3 position, Black’s threat of 1…a1=Q+ is so strong that it requires White to start with a checking key, but 1.Bxd5+?, say, fails to 1…Kxg3 2.0-0 (threat: 3.g8=Q) a1=Q!, or 2.g8=Q+ Kf4! White’s only chance is to capture the d5-pawn en passant (to both check and open the rank for the queen), but then we need to determine if …d7-d5 was the only possible last move. We can quickly discount …bxa6 as the last move since that would have prevented the white bishop from ever reaching a8. That leaves only (1) the d5-pawn and (2) the black king as candidates for making the previous move.

(1) White is missing only three units (two knights and a bishop), and Black’s tripled-pawns on the a-file account for all of their captures. That means there’s no spare white unit for the d5-pawn to have captured and the last move wasn’t …cxd5 or …exd5. The black pawn couldn’t have just come from d6 either, because then White would’ve had no way of delivering the bishop check from a8 (it wasn’t a bishop promotion since all eight white pawns are present). If the d5-pawn has just moved from d7, though, then White could have made the discovered check, Rc6-h6+. Therefore …d7-d5 is a potential last move. (2) If Black has just moved the king to g2, it couldn’t have come from f3 because of an impossible check by the e2-pawn. Likewise, the piece didn’t come from h3 or h2 because it would be in an impossible double-check by the rooks. But the king could have just arrived from g1, where it would be in a legal check by the h1-rook; the latter gave that check by moving along the h-file. Therefore …Kg1-g2 is another potential last move.

Since there are two potential last moves, not just …d7-d5, it seems that the en passant capture should be disallowed. But let’s look at the alternative move …Kg1-g2 more closely. It implies that White had just moved the h1-rook to check, as mentioned, and such a rook move would have ruled out the possibility of white 0-0 later on. However, since Black’s last move wasn’t necessarily …Kg1-g2 (it could have been …d7-d5), we can still assume White has castling right (as per the basic convention). Furthermore, if White does castle in the solution, that will demonstrate that the h1-rook had never moved and couldn’t have delivered a check to the black king on g1. And that means the king actually didn’t arrive from g1, for that would imply another kind of impossible check by the h1-rook. Thus, White can invoke the A Posteriori principle to disprove …Kg1-g2 as a legitimate last move, by playing 0-0 in the future. Now that only …d7-d5 remains as a viable last move, 1.cxd6 e.p.+! is legalised as the key. Strangely enough, after the forced 1…Kxg3, White cannot mate with 2.Qg5 – it ends the play prematurely! Rather, 2.0-0! is essential for the en passant key to be legal in the first place, and then 2…a1=Q 3.Qg5 or 2…Kg4 3.g8=Q.