Beautiful retrograde analysis problem by Alexsey A. Troitsky I saw this beautiful, subtle &amp; accessible problem a couple of days ago, and wanted to share it with you folks. (It's by the ground-breaking endgame theorist who e.g. invented the "Troitsky line" in KNNvKP analysis.)

[title "A.A.Troitsky - Bohemia 1910"]
[fen "1B6/bp3ppp/BR1pPp2/RPp1kP2/bpK5/3P2P1/1PP3PN/5nrr w - - 0 1"]  


White to play and mate in 2?

The forward play is pretty obvious. But, as is common in retrograde analysis, you need to justify the forward play by figuring out what was Black's last move.

Even if/when a speedybrain supplies a full answer, I hope that you can still enjoy solving the puzzle.
Short answer: last move was -1. c7-c5, and before that -1...Rc6-b6, the white bishop returns to c1 and -d2-d3 unlocks the cage, enabling -Kd3-c4 and everything unlocks. -1. c6-c5 doesn't work because the white bishop will definitely end up trapped behind the black bishop in the northern dark squares and can't get out - -c6-c7 will leave the two bishops on d8/e7/f8 and not a7/b8 because of tempo concerns (retro-opposition), and they can't leave.

(Notice the use of minus signs, white moves noted with ..., and I said the white bishop "returns" to c1 - I'm retracting moves from the diagram position, which makes this kind of problem much easier.)



Long answer: Now how to explain this without fluff or skipping too many "obvious" details? From a pedagogical point of view, I will insist on talking about this problem using retractions rather than the usual forward chess moves; it's easier and far clearer to play moves backwards from the diagram than it is to create some arbitrary position, assert that it's legal then play forward moves to reach the diagram. It also ensures we don't miss anything - easier to check every possible retraction from the diagram than to check that every possible chess position leading to the diagram has ends with a certain move.

Those who already understand retraction can probably see the cage themselves, skip the first two bolded points and jump right to the section -1. c7-c5.

Pawn captures, basics of retraction explained

Standard approach is to look at pawn structure. Each side has 14 units and 8 pawns. White's captures were hxg3 and axb, black's were exf6 and axb. So from the diagram, we can't magically retract (take back) an uncapture immediately.

Also, the fact that all bishops are on the board and each side has their full 8 pawns means we can't retract exf6, d7-d6 or ...d2-d3 without the corresponding bishop being at home. (In the forward direction, it means that the pawn move was played while the bishop was at home and hadn't yet moved.)

A side can retract themselves into check but the other side must retract immediately to remove the check (once again this is seen in reverse - in the forward direction, one side checks and the king moves away.) For instance -1. c6-c5 Kb3-c4 would be possible, except that black can't remove the check from Ba4. -1. Kf4-e5 is impossible as white needs to retract -1...hxg3 to answer (pawn structure.) -1. Ne3 is impossible as one cannot deliver check by retracting (in the forward direction, that would be black moving away from check.)

What is a retrocage?

Usually, once all pieces have enough freedom of retraction, the position shouldn't pose any legality problems. We therefore want to retract moves with the goal of freeing as many pieces as possible. By considering all retractions (it's not hard once you're used to it, like seeing legal chess moves), we notice a few things:


The kings can't retract at the moment.
The southeast (bottom-right) corner unlocks with retracting a knight move, impossible for now because of the kings.
The west (left) side, specifically the 8 pieces in a rectangle except bBa7 and wRb6, is a cage that unlocks only with -Bb3-a4, which is again impossible because of the kings.


So to unlock the position, we need to free the kings by giving them a retraction. So a bishop has to go home to c8, f8 or c1 to enable a pawn retraction to (respectively) d7, e7 or d2. The black light bishop is trapped on a4, so that leaves just the two dark bishops.

Position is legal: -1. c7-c5

We retract for example -1. c7-c5 Rc6-b6 -2. Be3-a7 Bb8-a7 -3. Bh6-e3 Be3-a7 -4. Bg5-h6 Bc1-e3 -5. Bh6-g5 d2-d3 (the white dark bishop came home) -6. Bg5-h6 Kd3-c4 and the cage unlocks. The white king moving frees everything, as explained above.

En passant is legal: -1. c6-c5? is impossible

The crux of the matter. If we try to retract this, then the two dark bishops are locked in step on the 6 dark squares in the north. They can't pass each other, nor can they lose a tempo since there are no other legal retractions except -c7-c6, which incidentally commits the bishops to the dark squares to the left or right of c7. There are only 3 cases:


Both bishops on the left (bBa7/wBb8). This is impossible - the bishops can't lose tempo, and with the bishops in such a position (i.e. right now, the only chance) it's white to retract, not black.
Both bishops on the right (bBd8/wBf8). Once -c7-c6 is retracted, the bishops are trapped in these squares. The only way to let them out is to retract e7xf6 (impossible; the black bishop can't find itself on the right of the white one to be at home on f8), or to retract -d7-d6 (also impossible; the black light bishop on a4 can't come home before the cage is unlocked.)
Black bishop on the left, white bishop on the right. The same problem as in (2), we can't get either bishop home as they're each trapped on the wrong side of the pawn wall.


With no other way to unlock the position, we conclude that -1. c6-c5 is impossible.

Conclusion:
Therefore -1. c7-c5 was the last move (no other possibilities), so en passant is provably legal and forward mate is 1. bxc6 e.p.+ Bb5 2. Raxb5#.