# Prove the pawn's color and the knight's position

I was recently given a retrograde analysis problem to solve. Even though I've managed to at least intuitively get close to an answer, I don't know how to put it in the form of an acceptable proof. For example, I can see that the a5 bishop must be a promoted piece, but I don't see how it could have escaped White's pawn structure, unless f2 square was originally free and is now occupied by a Black pawn. But for the knight question, I don't see a clear way of distinguishing it. I am hoping that someone could show how to tackle such problems on a firm footing.

`````` [FEN "r2qk2r/1ppbp1p1/1pn2n2/b2pP3/3P4/2N3P1/PPP5/R2QK2R w - - 0 1  "]
``````

We are told that neither kings have moved yet, that there's a pawn on f2 square, and a White knight on either the f3 or f4 square.

Prove the f2 pawn's color and whether the White knight belongs on f3 or on f4.

In both parts we know one to be true, which means it suffices to disprove one possibility for each and that settles the proof.

Part one: Let's start by assuming the pawn on `f2` is white and work our way up to see if we hit any inconsistencies. So here's our diagram:

`````` [title "Assuming f2 pawn is white"]
[fen "r2qk2r/1ppbp1p1/1pn2n2/b2pP3/3P4/2N3P1/PPP2P2/R2QK2R w - - 0 1 "]
``````

First trivial observation is the fact that the black bishop on a5 is a promoted piece as the original dark square bishop could never have escaped from `f8`. This means either the `h` or `f` pawn has promoted on `g1` into a bishop that is now on `a5`. Having a white pawn on `f2` also implies that the `g3` pawn isn't the `h`-pawn having made a capture on `g3` or else the promoted bishop could never have escaped white's kingside and reach `a5`, thus, `g2-g3` is played at some point, and it was on `g2` when black made the promotion (else no escape route). This in turn means the promoted black pawn must have been the `h`-pawn. But how did it get passed the `h` pawn? We know that white is down three pieces, the two bishops and the `h`-pawn. Additionally, we know that one piece was captured on `b6` and one on `g1` (for black's h-pawn to promote). Now both `b6` and `g1` are dark squares, but 'h2' pawn couldn't have been captured on either, nor could have the `f1` bishop as it is a light square piece. Therefore, white must also have a promoted piece, as nothing else accounts for the missing captured piece, for instance a knight (or any other non-pawn) being captured on `g1` by the `h7` pawn and the `h2` having promoted into a knight. So clearly the `h` pawn hasn't been captured but has instead been promoted. But therein lies the conflict: one of the `h`-pawns must have made at least two captures to get passed the other `h`-pawn. It couldn't have been the white pawn as black has only lost one piece so far, but it cannot be the black `h`-pawn either, because it must have then made 3 captures, 2 to get passed the white `h`-pawn and one last capture on `g1` to promote into the `a5` bishop. Knowing a capture on `b6` has also taken place, this accounts for a total of 4 captures, but white has only lost 3 pieces at most. Thus, neither `h`-pawns could have legally promoted given that the `f2` pawn is white, so our starting assumption must be wrong. We conclude the `f2` pawn must be a black pawn. Notice that throughout the analysis, the position of the knight either `f3/f4,` would have made no discernible difference on the resolution of the `f`-pawn's conflict, so we were safe to do the analysis independently. The reverse is not true, namely, the knight's position is very much dependent on the fact that the pawn on `f2` is black.

Part two: Knowing now that the `f2` pawn is a black pawn, let's take the most restrictive case and assume the knight is on `f3` and perform a similar analysis to see if we hit any walls. So here's the corresponding diagram:

``````[title "Assuming the knight is on f3"]
[fen "r2qk2r/1ppbp1p1/1pn2n2/b2pP3/3P4/2N2NP1/PPP2p2/R2QK2R w - - 0 1 "]
``````

It is the most restrictive consideration as it forces the `f7` pawn to have made at least 2 captures to reach `f2,` because the white knight is assumed to be on `f3`, which means that `f7` pawn must have just made a capture on `f2` from `e3` (that it reached from yet another capture). So far we have accounted for 4 white pieces having been captured and the `h7` and `h2` having promoted for black and white respectively. The only1 way the latter could have happened is if the `h7` pawn (i) made two captures to go around the white `h`-pawn, or (ii) made only one capture from `h3` (so taking something on `g2`) while white's `h2` pawn still untouched. (i) is impossible by just considering the piece count, because so far there has been one capture on `b6`, two captures by `f7` pawn and two by the `h7` pawn, and that is already more captures than the number pieces white has lost. So only option is (ii): we know that the `h7` pawn must have captured a non-pawn piece on `g2` (as the `g3` white pawn couldn't have been from either the `h` pawn which promoted, or the `f2` pawn as black only lost one piece and that was the original `f8` bishop on its original square). This in turn implies that `g2-g3` must have been played by white before `h7` captured on `g2`. But that means the promoted black bishop could not have escaped white's kingside via the `g3-e5` diagonal anymore, therefore, it must have escaped from the `f2-d4` diagonal instead, which settles the fact that the `f7` pawn couldn't have captured a pawn on `f2` (from `e3`), or else the bishop could never have reached `a5`, had there still been a pawn on `f2` just presumably captured by black's `f7` pawn. Now here's the dilemma: what on earth happened to white's `f2` pawn in all this? Can we logically account for it knowing black just took a non-pawn piece on `f2` and there's a white knight lying on `f3` (starting assumption)? It definitely couldn't have been the piece that the `f7` pawn captured to reach the `e-file` either as white only made one capture and that was on `f8`, so `f2`-white-pawn couldn't have magically been on the `e`-file. Trivially, it couldn't have been captured on `b6` or `g2` either. It couldn't have been promoted either as it would have forced black's king to move (disallowed in the starting description). So `f2` pawn couldn't have been promoted, or been accounted for by any of the four captures performed by black, but somehow it isn't on the board, thus, we can conclude that we've reached an inconsistency again, meaning there's no logical set of events that would legally reach the position given above with the knight standing on `f3` which was our original assumption, so the knight must be on `f4.`

Come to think of it, there's a much shorter way of realising that the knight cannot be on `f3` without focusing on the fate of the white `f2`-pawn: quick recap of captures: we know a dark square piece has been captured on b6, two non-pawn pieces captured one on e-file and one on `f2,` one non-pawn piece also captured on `g2`. But how many non-pawn pieces has white lost? The two bishops and another non-pawn (we cannot know which because we don't know what piece the `h2` pawn promoted into) on the `e`-file. That's 3, so there's no other non-pawn left for black to have just captured on `f2` (we already know it couldn't have been a pawn on `f2` captured). And that's another formulation of the same inconsistency. Notice that if we are to assume that the non-pawn captured on `f2` was a piece that resulted from the promotion of the `f2`-pawn, then it implies that the `f2` pawn reached `f7` with a check and was not captured by anything thus allowed to promote, but that's impossible as we know neither kings have moved yet, in other words, if the `f2` pawn were to ever promote it must have gone passed the `f7` square without being captured, which is impossible. Anyhow, let's end with the final correct diagram:

`````` [title "Solution"]
[fen "r2qk2r/1ppbp1p1/1pn2n2/b2pP3/3P1N2/2N3P1/PPP2p2/R2QK2R w - - 0 1 "]
``````

If you're interested in these kinds of discussions, have a look into the books of Raymond Smullyan, I even recall similar puzzles in one of his books but I have no copies at my disposal at the moment to check for you, maybe someone else can verify.

1: as the white pawn couldn't have made two captures to go around the black `h` pawn, because white has only made one capture in this game and that has been the `f8` bishop on its original square.

• This is problem M9 (p.149) of Raymond Smullyan's The Chess Mysteries of Sherlock Holmes. Your analysis of both questions agrees with Smullyan's. Commented Jan 6, 2018 at 11:25
• @RosieF haha great, thanks a bunch for checking it and good find! In fairness, it wasn't much of a guess for me because the Smullyan's books are the only retrograde books I know :P Commented Jan 6, 2018 at 12:39
• @Phonon : ​ I don't immediately see that the Ba5 can't have been the `d` pawn. ​ ​ ​ ​
– user2668
Commented Mar 9, 2018 at 21:48
• @RickyDemer the `d` pawn? that pawn is still on the board... I reckon you meant something else. Commented Mar 9, 2018 at 22:34
• @Phonon : ​ ​ ​ How do you know? ​ (Black's `f`-pawn is not accounted for, and could have gone f7-e6-d5.) ​ ​ ​ ​ ​ ​ ​ ​
– user2668
Commented Mar 9, 2018 at 22:37