Another checkmate in 1

White to move and #1. It's a bit tricky 😁

En passant mate

black made 6 pawn captures
white has 9 units on board
Bf1 was not captured by pawns
no spare captures available

white made 7 pawn captures
black has 8 units on board
there is one spare capture

what was black's last move?
not Kd7-c7 because Ph7 was from d2, and white would have to take back d5xc6, and then a 2nd capture to fix the pawn structure, but there is not one available
not Kb7-c7 for the same reason
not Kd8-c7 or Kc8-c7 because the white Rook can't retract the check
not e7xf6 because then the Bishop could not have left f8 [Bb8 cannot have been promoted because there are no spare captures to get a black pawn through]

As noted earlier, black has no spare captures. This rules out potential last moves such as a6xb5, Kd8xNc7 and Kc8xBc7.

the only move left is b7-b5

1.axb6#

The solution is correct but there is a flaw in the puzzle design. It's a detail but a significant one considering the precision of all retro counters. The white rook on h8 ought to be on g8 or f8. The reason is that the last black move Kc8-c7 is made impossible by the checking rook on h8 which has no retraction move. If that rook is placed on f8/g8 the checking would be the same but there would be a new retraction option Rh8xf8+ (or Rxg8+). But considering that all missing material is accounted for makes a capturing retraction impossible! A very nice addition to the failed retraction collection.

Btw, the black retraction Kc8xBc7 preceded by Bd8-c7+ fails on insufficient material on the white side. That was already the case in the original and does not change.

d2 - 4 captures
f2 - 1 capture
h2 - 1 capture

What did I miss?

The missing white pawn - from b2 - must have promoted so it could be sacrificed to a black pawn. It had capture a black pawn - either a7 or c7 - to get through.

.And there are some more small improvements which lead to this diagram:

.

Source is Sam Loyd, American Chess Journal 1876.

Original task is "White to mate in how many?" The task was set in the 19th century, but it would catch someone using an engine today to claim a M3 solution!

For anyone not familiar with the rules on whether an e.p. capture is permitted in such a position, see this blog: Chess problem conventions re castling and capturing en passant. This Sam Loyd problem shows how this convention has been around for a very long time.

Actually, Sams construction does nor show that, since it does not matter whether you justify his e.p. by analysis or by convention. The outcome would be the same if the e.p. convention were 'castling-like'. But I suppose the publication came with a narrative to explain that a law-abiding citizen really ought to make a decent retro-analysis before e.peeing!

The e.p. convention is more relevant for the countless number of puzzle and game diagrams with neighbouring pawns of opposite color on the 4th or 5th rank with e.p. potential. Jokers on chess.com take advantage of these common situations to deceive us.

Another technical note on Loyd's original puzzle. Clearly in his time much less attention was paid to the details of a retrograde construction. A century later, the same Sam would not have missed that a knight on b8 has double the value of a bishop with regard to 'retro check domination'. Plus the added value to the e7xf6 retraction try.

Ah, that's nice - though parts are lost in the mist! And how is there even a pawn on a5? It explains why the retrograde problem is a bit poor. Sam Loyd was more interested in entertaining the public than in subscribing to standards in one category or another.

Indeed, there is also a mate-in-3! I verified by engine

Are you sure you're looking at Loyd's diagram correctly? It DOES have neighbouring pawns of opposite colours on a5 and b5. This is a standard retro-analytical problem that follows the e.p. convention, which states that an e.p. capture as the key-move is permitted only if such a move can be proved legal by retro-analysis.

Yes! What I tried to convey is that it doesn't matter what the convention says. The conventions only come in when there is uncertainty. Considering that retro-analysis removes all doubt, the instructions in the e.p. convention do not apply here.

So this problem is not a good handle for explaining the e.p. convention but it can be used to clarify that every puzzle position (plus its solution) requires a proof game. Which is not particular for e.p. dependent puzzle solutions.

Thx. It's amazing how they wrote their retro-analysis in those days! It's how I did it as a teen after making my first retro-puzzle. Today they won't even waste as many words on it as BigDoggProblem did!

That's such a bizarre way of looking at the e.p. convention. Of course the convention matters in Loyd's problem – it permits the e.p. key-move based on retro-analysis. Suppose the convention were different, such as that an e.p. key is never permitted, regardless of retro-analysis. In that case, the problem has no M1 solution and is pointless.

Your point seems to be that if the castling convention is analogously applied to e.p. – namely that the special move is allowed unless proven illegal by retro-analysis – then the Loyd would still be solvable by the M1 e.p. capture. That's trivially true, but this objection equally applies to countless retro problems that correctly demonstrate the basic e.p. convention. Here is a random bunch of problems that show an e.p. key correctly and require retro-analysis as per the convention.

https://www.yacpdb.org/#577074

https://www.yacpdb.org/#573384

https://www.yacpdb.org/#572520

https://www.yacpdb.org/#346946

https://www.yacpdb.org/#548264

Are you saying that all of these problems are like Loyd's in being "not a good handle for explaining the e.p. convention", because they also have "certainty"? Yes or No?

If yes, your issue seems to be with the e.p. convention itself that's commonly accepted by all composers like these ones, as opposed to an issue with the Loyd problem in particular. If you think these are all bad examples of the basic e.p. convention, can you give a single good example and explain how it differs? (Don't give any example of the advanced conventions like AP and RS that are also covered in my blog, since the topic here is the basic e.p. convention as illustrated by the Loyd.)

If no – i.e. you think the linked problems are good examples – then what are the specific features of Loyd's that set it apart from these other e.p. problems? You've only made vague claims about the Loyd that make little sense to me. For all I know, you think the same claims apply to a whole branch of e.p. problems like the ones I linked.

It's not bizarre, it is exactly right. The standard conventions cannot deny e.p. right as this leads to contradictions. For instance: A position is stalemate unless you play a provable e.p. move. Denying that by convention leads to stalemate which is illegal as there is no proof game for it.

The castling convention is just one model. There is an infinite amount of (insane) ways to create a mathematically consistent convention, like "you are permitted to play e.p. only if it is not impossible and there is a knight on h5". I said "insane". The point is that all thinkable consistent conventions are no match for proof game logic which is required as the foundation for all retrograde analysis. If you have no clue what the active e.p. convention is, you can still solve Loyd's problem.

Note that this is the core of page 1 of mathematical Model Theory. It exclusively concerns itself with the free space left by a formal system with missing information. In chess, information goes missing when you take a diagram out of a game context - which had full information - and have a need to fill in the gaps. That is what the majority of conventions are about. In math they are addressed as "choice rules".

I will look at your examples later but it is obvious that you can discuss the e.p. convention when you are discussing e.p. proof as it is a related subject. But as a solver you always need to make a retro-analysis first to find out if any choices are left to decide by convention.