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% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
solutions,
shortwarning
]{../../resources/ormc_handout}
\usepackage{chessfss}
\usepackage{chessboard}
\usepackage{xcolor}
\usepackage{anyfontsize}
\usepackage{afterpage}
\usepackage[hang]{footmisc}
\def\stars#1#2{%
\def\oncolor{\color{purple}}%
\def\offcolor{\color{black!40!white}}%
%
\count255=1%
\count254=#2%
\advance\count254 by -1%
\ifnum #1 > 0
% The $$ are required around \bigstar.
% the special \odot chess piece breaks
% star sizing if they are ommited.
\loop
{\oncolor $\bigstar$}%
\ifnum\count255 < #1
\advance\count255 by 1
\repeat%
\else%
{\oncolor $\bigstar$}%
\fi%
%
\ifnum \count255 < #2%
\loop
{\offcolor $\bigstar$}%
\ifnum\count255 < \count254
\advance\count255 by 1
\repeat%
\fi%
}
\def\difficulty#1#2{
\textbf{Difficulty:} \stars{#1}{#2} \par
\vspace{1mm}
}
\def\difficultynote#1#2#3{
\textbf{Difficulty:} \stars{#1}{#2} ~ #3\par
\vspace{1mm}
}
\setchessboard{
showmover=false,
borderwidth=0.5mm,
label=false,
labelleft=true,
labelbottom=true,
normalboard,
hlabelformat=\arabic{ranklabel},
vlabelformat=\Alph{filelabel}
}
\makeatletter
\cbDefineNewPiece{white}{U}
{\raisebox{1.75mm}{\cfss@whitepiececolor
$\odot$}}
{\BlackEmptySquare%
\makebox[0pt][r]{\cfss@whitepiececolor
\raisebox{1.75mm}{\makebox[1em]{$\odot$}}}}
\long\def\manyboards#1{
\if@solutions
\chessboard[setpieces = {#1}]
\hfill
\chessboard[setpieces = {#1}]
\else
\vfill
\chessboard[setpieces = {#1}]
\hfill
\chessboard[setpieces = {#1}]
\vfill
\chessboard[setpieces = {#1}]
\hfill
\chessboard[setpieces = {#1}]
\fi
}
\makeatother
\uptitlel{Advanced 2}
\uptitler{Winter 2022}
\title{Retrograde Analysis}
\subtitle{
Prepared by Mark on \today{} \\
Based on books\footnotemark{} by Raymond Smullyan
}
\begin{document}
\maketitle
\footnotetext[1]{
Most of the easy problems in this handout are from \textit{The Chess Mysteries of Sherlock Holmes}.\\
The rest are from \textit{The Chess Mysteries of the Arabian Knights}.
}
\input{parts/intro}
\section{Simple problems}
\input{parts/easy}
\section{Harder problems}
\input{parts/medium}
\section{Very difficult problems}
\input{parts/hard}
\chessboard \hfill \chessboard \par
\chessboard \hfill \chessboard \par
\chessboard \hfill \chessboard \par
\end{document}

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Advanced/Retrograde Analysis/parts/easy.tex Normal file
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% Sherlock, A little exercise
\problem{A little exercise}
\difficulty{1}{5}
Black has just moved in the game below. White started on the south side of the board.\par
What was Black's last move, and what was White's last move? \par
\manyboards{
ka8,Kc8,
Ph2,
Bg1
}
\begin{solution}
It's pretty clear that Black just moved out of check from A7.
\vspace{2mm}
How did White deliver this check? The bishop couldn't have moved to G1,
so this check must have been discovered by another piece. Since there are
no extra pieces on the board, Black must've captured this piece on his last move.
\vspace{2mm}
The only piece that could have moved from the white bishop's diagonal to
then be captured on A8 is a knight.
\vspace{2mm}
\textbf{Note:}
There are two possible answers if we don't know who started where.
If Black had started on the south side of the board, that bishop could be a promoted pawn.
\end{solution}
\vfill
\pagebreak
% Sherlock, Which color?
\problem{Which color?}
\difficulty{2}{5}
In the game below, no pieces have moved from a black square to a white square, or from a white square to a black square.
There is a pawn at G3. What color is it? \par
\hint{Again, White started on the bottom.}
\manyboards{
ke8,
Kb4,
Ug3,
Pd2,Pf2
}
\begin{solution}
The white king is the key to this solution. How did it get off of E1? \par
It must have castled kingside---castling queenside would move a rook from black to white.
\vspace{2mm}
Now, the white king is on G1. How did it get out of there? \par
It's must have moved through H2 and G3, which would be impossible if the mystery pawn on G3 was white.
Therefore, that pawn must be black.
\end{solution}
\vfill
\pagebreak
% Arabian Knights 2
\problem{Invisible, but not invincible}
\difficulty{2}{5}
Seeing that this battle was lost, the black king has turned himself invisible. \par
Unfortunately, his position is hopeless. Mate the king in one move. \par
\hint{You don't need to find the king, you only need a checkmate.} \par
\manyboards{
Ra8,rb8,Kf8,
Nb7,Pc7,
Pa6,Rc6
}
\begin{solution}
Since it is White's move, Black cannot be in check. \par
So, either White is in check or the black king is on C8. \par
If White is in check, Black must have administered this check by moving from C8 to D7. \par
Therefore, the black king must be on C8 or D7.
\vspace{2mm}
If we capture the black rook on B8 with the pawn on C7 and promote it to a knight, the black king will be in checkmate
regardless of his position.
\end{solution}
\vfill
\pagebreak
% Sherlock, a question of survival
\problem{An empty board}
\difficulty{2}{5}
In the game below, no pieces have moved from a black square to a white square, or from a white square to a black square.
There is one more piece on the board, which isn't shown. What color square does it stand on? \par
\manyboards{
ke8,
Pd2,Pf2,
Ke1
}
\begin{solution}
Which piece performed the last capture on a black square? It couldn't have been a white pawn, which haven't moved.
It couldn't have been the white king, which is trapped; or the black king, which is restriced to white squares.
\vspace{2mm}
It must have been the piece we can't see, which therefore stands on a black square.
\end{solution}
\vfill
\pagebreak
% Sherlock, another monochromatic
\problem{The knight's grave}
\difficulty{3}{5}
In the game below, no pieces have moved from a black square to a white square, or from a white square to a black square.
The white king has made less than fourteen moves. \par
Use this information to show that a pawn was promoted. \par
\manyboards{
ke8,
Pb2,Pd2,
Ke1
}
\begin{solution}
Knights always move to a different colored square, so all four missing knights must have been captured on their home square.
What pieces captured them?
\vspace{2mm}
We can easily account for the white knights and the black knight on G8, but who could've captured the knight from B8?
The only white pieces that can move to black squares are pawns, the Bishop (which is trapped on C1), the rook (which is stuck on column A and row 1), or the king (which would need at least 14 moves to do so).
\vspace{2mm}
If this knight was captured by a pawn, that pawn would be immediately promoted. If it was captured by a piece that wasn't a pawn, that piece must be a promoted pawn.
\end{solution}
\vfill
\pagebreak
% Arabian Knights, intro (given with solution)
\problem{Promotion?}
\difficulty{2}{5}
It is White's move. Have there been any promotions this game? \par
\manyboards{
Pb2,Pe2,kf2,Pg2,Ph2,
Bc1,Kd1,Rh1
}
\begin{solution}
Since it is White's move, Black has just moved his king. Where did he move it from?
Not E1, E3, F3, or G3, since that implies Black had moved into check before. \par
\vspace{2mm}
The only remaining possibilities are F1 and G1. \par
G1 is again impossible: how would the king get there without moving into check? \par
F1, therefore, is the only choice. If we place the king on F1, we see that another piece must prevent check from the white rook.
This must have been a white black-square bishop, which moved to F2 to reveal that check, and was then captured by the black king.
\vspace{2mm}
However, there is already a white black-square bishop on the board! We can get a second only by promoting a pawn, so the answer is \say{yes.}
\end{solution}
\vfill
\pagebreak
% Sherlock Holmes, two bagatelles (1)
\problem{Whodunit}
\difficulty{2}{5}
It is Black's move. Can Black castle? \par
\manyboards{
ra8,bc8,ke8,rh8,
pa7,pc7,pe7,pg7,
pb6,pf6,ph6,
Pa3,
Pb2,Pc2,Pd2,Pe2,Pf2,Pg2,Ph2,
Bc1,Qd1,Ke1,Bf1
}
\begin{solution}
White's last move was with the pawn. \par
Black's last move must have been to capture the white piece which moved before that.
\vspace{2mm}
This piece would have to have been a knight, since the white rooks could not have got out onto the board.
It is clear that none of the black pawns captured this knight.
The black rook on A8 couldn't have captured it either, because there is no square that
the knight could have moved from to get to that position.
\vspace{2mm}
The black bishop couldn't have captured the knight either, since the only square the
knight could have come from is D6, where it would have been checking the king.
\vspace{2mm}
So, the black king or the rook on H8 made this capture. Therefore, Black can't castle.
\end{solution}
\vfill
\pagebreak
% Sherlock Holmes, two bagatelles (2)
\problem{Castle contradiction}
\difficulty{2}{5}
Neither Black nor White captured a piece on their last move. \par
It is Black's move. Can he castle? \par
\manyboards{
ke8,rh8,
pc4,
Pf3,
Pc2,Pf2,Pg2,
bd1,Rf1,Kg1
}
\begin{solution}
If White's last move was with the king, then the black rook moved to check him and Black can't castle.
\vspace{2mm}
If White's last move wasn't with the king, White must have castled. \par
What was Black's last move? \par
If it was with the king or rook, Black can't castle.
\vspace{2mm}
It could not have been with the bishop, since then White would have had no move immediately before that.
Now, suppose Black moved his pawn. Then White's preceding move must have been with the pawn from E2,
capturing a piece on F3. This means that the bishop on D1 is a promoted bishop. The promoting pawn must
have come from D7, passed D2, checked the white king, making it move!
This contradicts our assumption that White has just castled.
\end{solution}
\vfill
\pagebreak
% Arabian Knights, intro (given with solution)
\problem{A matter of order}
\difficulty{3}{5}
A black bishop captured a White piece earlier in this game. \par
Which bishop was it, and what did it capture? \par
\manyboards{
ra8,qd8,ke8,
pa7,pc7,pd7,pf7,ph7,
pb6,nc6,pe6,nf6,ph6,
Bb5,be5,
Pe4,bg4,
Pc3,Nf3,
Pa2,Pb2,Pc2,Qe2,Pf2,Pg2,Ph2,
Kc1,Rd1,Rh1
}
\begin{instructornote}
\textbf{Hints to give:} (these should be answered in order)
\begin{itemize}
\item How many pieces does Black have? Where were the missing ones captured?
\item Which pieces is White missing? Where could they have been captured?
\item How did those white pieces get to the place they were captured?
\end{itemize}
\end{instructornote}
\begin{solution}
First, notice that the pawn on C3 came from D2 by capturing a piece. \par
This must have been a black rook, which is the only missing black piece.
\vspace{2mm}
This black rook couldn't have moved there before the black pawn on G7 captured a white piece on H6.
This piece couldn't have been the missing white bishop, because that bishop would still be trapped by the pawn on D2.
Therefore, the missing white knight was captured on H6.
\vspace{2mm}
The only other missing white piece is the black-square bishop, which must have been captured by the black bishop on E5.
\end{solution}
\vfill
\pagebreak

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% Arabian Knights 5
\problem{The hidden castle}
\difficulty{7}{7}
There is a white castle hidden on this board. Where is it? \par
None of the royalty has moved or been under attack. \par
\manyboards{
nb8,qd8,ke8,ng8,rh8,
pa7,pb7,pc7,pf7,pg7,
pe6,pf6,ph6,
Pa4,Bc4,Pe4,
Pc3,
Pb2,Pd2,Pf2,Pg2,
Qd1,Ke1
}
\begin{solution}
See \say{The Hidden Castle} in \textit{The Chess Mysteries of the Arabian Knights}.
\end{solution}
\vfill
\pagebreak
% Arabian Knights 6
\problem{Who moved last?}
\difficulty{7}{7}
After many moves of chess, the board looks as follows. \par
Who moved last? \par
\manyboards{
ka8,Kc8,bf8,rh8,
pb7,pc7,pf7,pg7,
Ba6,
Pe4,
Pa2,Pb2,Pd2,Pg2,Ph2,
Ra1,Nb1,Bc1,Qd1,Rh1
}
\begin{solution}
See \say{A Vital Decision} in \textit{The Chess Mysteries of the Arabian Knights}.
\end{solution}
\vfill
\pagebreak
% Arabian Knights 3
\problem{The king in disguise}<kingdisguise>
\difficulty{7}{7}
The white king is exploring his kingdom under a disguise. He could look like any piece of any color.\par
Show that he must be on C7.
\manyboards{
qa8,nb8,be8,Qg8,kh8,
pa7,Pb7,pc7,Nd7,pe7,Pf7,ph7,
pa6,Pc6,Pg6,
ra5,pb5,Rd5,Ph5,
Pa4,Nc4,Pe4,Bg4
}
\begin{solution}
Black is in check, so we know that it is Black's move and White is not in check.\par
Assume the white king is not on C7. Where else could he hide?
First, we exclude the black pawns on A6, A7, and B5, since the white king would be in check in any of those positions. \par
\vspace{2mm}
The pawn on A6 came from B7 by capturing one piece, and the pawn on B5 came from D7 by capturing two.
(Note that this may not be true if we don't assume the pawn on C7 is real.)
We've counted three captures, all on white squares, so the white black-square bishop must have been captured seperately.
\vspace{2mm}
Thus, at least four white pieces have been captured. White has 12 pieces on the board,
so the white king must be disguised as a white piece if he isn't on C7.
If we Exclude a few more pieces in check, we now see that the white king must
be on D5, E4, G4, or H5 if he isn't on C7.
\linehack{}
The white queen has to have moved from F8 to capture a piece on G8 to put Black in check. What was Black's move before this?
It couldn't have been the king from G7, since the white queen wouldn't have been able to enter F8.
It couldn't have been any other piece on the board, since they are all trapped.
So, Black's last move must have been with the mystery piece on G8.
\vspace{2mm}
Where did it come from? This piece can't be a bishop (how would it get in?), so it must be a queen, rook, or knight.
If it is a queen or rook, it must have come from G7, which is impossible---the white queen wouldn't be able to get in.
The mystery piece must therefore be a knight. It couldn't have come from H6 (again, the queen couldn't have gotten in to deliver a check),
so it must have come from F6.
\linehack{}
We now know that the white king is not on D5, E4, G4, or H5, since all those were in check when the black knight was on F6.
However, the white king must be on one of those four squares if he isn't on C7. This is a contradiction --- therefore the king must be hiding on C7.
\end{solution}
\vfill
\pagebreak
% Arabian Knights 3
\problem{The king in disguise once more}
\difficultynote{2}{5}{(Assuming you've solved \ref{kingdisguise})}
The white king is again exploring his kingdom, now under a different disguise. Where is he? \par
\hint{\say{different disguise} implies that the white king looks like a different piece!}
\manyboards{
nb8,be8,Qg8,kh8,
pa7,Pb7,pc7,Nd7,pe7,Pf7,ph7,
pa6,Pc6,Pg6,
ra5,pb5,Rd5,Ph5,
Pa4,Nc4,Pe4,Bg4
}
\begin{solution}
Use the same arguments as before, but now assume that the king isn't a black pawn.
\vspace{2mm}
Again, the king is disguised as a white piece, and must be on D5, E4, G4, H5, or B7. \par
For the same reasons as above, he can't be on D5, E4, G4, or H5, so he must be on B7.
\end{solution}
\vfill
\pagebreak

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\section{Introduction}
To solve the problems in this handout, you mustn't be a chess master---you just need to know how the pieces move.
I'm sure you're all familiar with the basic rules of chess. The odd ones are listed below.
\generic{Board orientation:}
The bottom-left square of a chessboard is \textbf{always} black.
\generic{Starting pawns \& en passant:}
A pawn may move two squares on its first turn. \par
An opposing pawn may capture this pawn as it does this. \par
This is called an \textit{en passant} capture (Which means \say{in passing} in French)
\begin{minipage}{0.3\textwidth}
\begin{center}
\chessboard[
smallboard,
maxfield=b4,
setpieces = {
pa4,pb3,
Pa1,Pb1
},
addpgf={
\tikz[overlay]
\draw[red,line width=0.1em,->]
(a1)--(a3);
},
]
White moves two squares
\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\begin{center}
\chessboard[
smallboard,
maxfield=b4,
setpieces = {
pa4,pb3,
Pa3,Pb1
},
addpgf={
\tikz[overlay]
\draw[red,line width=0.1em,->]
(b3)--(a2);
},
]
Black captures en passant
\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\begin{center}
\chessboard[
smallboard,
maxfield=b4,
setpieces = {
pa4,pa2,
Pb1
},
]
Result
\end{center}
\end{minipage}
\vfill
\generic{Promotion:}
When a pawn reaches the last row of the board, it may be promoted to \textbf{any} other piece.\par
(Except a king or a pawn, of course.)
\generic{Castling:}
A king and rook can \textit{castle} under the following conditions:
\begin{itemize}
\item No pieces are in the way
\item The king has not yet moved
\item The rook has not yet moved
\item The king is not in check
\item The king does not move through check while castling
\end{itemize}
\begin{minipage}{0.3\textwidth}
\begin{center}
\chessboard[
smallboard,
maxfield=h2,
setpieces = {
Ra1,Ke1,Rh1
},
addpgf={
\tikz[overlay]
\draw[red,line width=0.1em,->]
(e1)--(g1);
\tikz[overlay]
\draw[red,line width=0.1em,->]
(e1)--(c1);
},
hmarginwidth=0mm
]
Possible castle directions
\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\begin{center}
\chessboard[
smallboard,
maxfield=h2,
setpieces = {
Rd1,Kc1,Rh1
},
hmarginwidth=0mm
]
Queenside castle result
\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\begin{center}
\chessboard[
smallboard,
maxfield=h2,
setpieces = {
Ra1,Kg1,Rf1
},
hmarginwidth=0mm,
]
Kingside castle result
\end{center}
\end{minipage}
\par
\vfill
\pagebreak

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% Sherlock, A matter of direction
\problem{A matter of direction}
\difficulty{3}{5}
The results of a game of chess are shown below. \par
Did White start on the north or south side of the board? \par
\manyboards{
ka8,Kc8,
Qe7,
Bc5,Pe5,
Pd4,
Ph3,
Bh1
}
\begin{solution}
Let us first find White's last move. It wasn't with the pawns on D4 and E5, since Black wouldn't have a move before that.
(Note the double-check on A7).
\vspace{2mm}
How, then, did White put Black in check? There are no pieces that could've uncovered this check, and the bishop on H1 couldn't
have moved from anywhere. We thus see that that bishop must be a promoted pawn, proving that White started on the north side of the board.
\end{solution}
\vfill
\pagebreak
% Arabian Knights 1
\problem{Where is the king?}
\difficulty{3}{5}
The white king has turned himself invisible. Find him. \par
\hint{White started on the bottom. En passant.} \par
\manyboards{
rb5,bd5,
Ba4,
kd1
}
\begin{solution}
Looking at the board, we see that the white king is on B3 or Black is in check.
\vspace{2mm}
First, we show that the latter implies the former: assume the black king is not on B3. \par
How did White deliver this check?
Not by moving the bishop, so this check must have been discovered by the white king moving from B3.
Therefore, if the white king isn't on B3 now, he was there on the previous move.
\vspace{4mm}
How did the white king end up on B3? That seems to be an impossible double-check from both the rook and bishop!
Looking at the hint, we place a black pawn on B4 to block check from the rook, and a white pawn on C2 that this black pawn will capture.
The sequence of moves is now as follows:
\begin{minipage}{0.5\linewidth}
\begin{center}
\chessboard[
setpieces = {
rb5,
Ba4,pb4,be4,
Kb3,
Pc2,
kd1
}
]
\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
Black: E4 $\to$ D5 \par
White: C2 $\to$ C4 \par
Black: B4 $\to$ C3 (en passant capture) \par
White: B3 $\to$ C3 \par
So, the white king must be on C3.
\end{minipage}
\end{solution}
\vfill
\pagebreak
% Arabian Knights, intro (given with solution)
\problem{Double-checks}
\difficulty{3}{5}
White to move. Which side of the board did each color start on? \par
\hint{What was Black's last move? }
\manyboards{
Re3,
Nc2,Rd2,
Nd1,kf1,Kh1
}
\begin{instructornote}
\textbf{Hints to give:}\par
Clearly, Black just moved his king. From where? \par
All possible positions may seem impossible (thanks to double-checks), but E1 and F2 are a bit more reasonable than others.
Move the king to both and try to add (or un-promote) pieces to make the position make sense.
\end{instructornote}
\begin{solution}
Black's last move was from F2, where his king was in double-check from both a rook and a knight.
How did this happen?
\vspace{2mm}
White started on the north side of the board, and put Black in check by capturing a piece on D1 with
a pawn and then promoting that pawn to a knight.
\begin{center}
\chessboard[
smallboard,
setpieces = {
Re3,
Nc2,Rd2,Pe2,
bd1,kf2,Kh1
}
]
\end{center}
\end{solution}
\vfill
\pagebreak
% Arabian Knights 4
\problem{A missing piece}
\difficulty{4}{5}
There is a piece at G4, marked with a $\odot$. \par
What is it, and what is its color? \par
\manyboards{
ra8,ke8,rh8,
pc7,pd7,
pb6,
pa5,
Ug4,
Pb3,Pg3,Ph3,
ba2,Pb2,Pc2,Pd2,Pf2,qg2,rh2,
Kc1,Rd1,nf1,Bh1
}
\begin{instructornote}
\textbf{Hints to give:} (in this order)
\begin{itemize}
\item How did the black bishop on A2 get there? \note{(Part 1)}
\item How many captures has Black made? \note{(Part 1)}
\item What color is the missing piece? \note{(Part 1)}
\item What was White's last move? What does this imply? \note{(Part 2)}
\item Which white pieces were captured? \par
How did they move from their starting positions? \note{(Part 3)}
\item How did the bishop on H2 get to where it is now? \note{(Part 3)}
\item Which pawn was promoted to the bishop now on H2? \note{(Part 4)}
\item Which black pieces are still missing? \note{(Conclusion)}
\item Remember that White cannot castle through check. \note{(Conclusion)}
\end{itemize}
\end{instructornote}
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\begin{solution}
\textbf{Part 1:}
The black bishop on A2 cannot be original, since the white pawn on B3 would have prevented its getting there.
That bishop is a promoted bishop. \par
The black pawn it was promoted from must have come from E7,
captured four pieces to get to A3, then moved to A2, and then made a capture on B1, where it was promoted. \par
Thus, the pawn from E7 has made five captures.
\vspace{2mm}
The white bishop from from C1 never left its home square
(since neither of the pawns on B2 or D2 have moved), and hence was captured on C1. This makes six captures of
white pieces, which tells us that the mystery piece is black.
\vspace{2mm}
\textbf{Part 2:}
White's last move could not have been with the rook from E1, which would have checked Black,
nor with the king (which could only come from B1, an impossible check),
nor could it have been with any piece other than the rook or king.
Therefore, White just castled, and thus the white king never moved before that.
\vspace{2mm}
\textbf{Part 3:}
Among the white pieces captured by the black pawn from E7 was the white rook from H1. Since White has just castled,
and the white king never moved before that, how did the rook from H1 get onto the board to be captured?
The only possible explanation is that the pawns on G3 and H3 cross-captured to let out the rook:
the pawn on G3 really came from H2 and vice-versa. Since the pawn on G3 comes from H2, the black bishop
on H2 has always been confined to G1 and H2. How did the bishop get there? It must have been promoted.
\vspace{2mm}
\textbf{Part 4:}
The promoted black bishop on H2 must have been promoted on G1. The pawn which was promoted must have come from G7,
since neither of the pawns from F6 or H6 could make a capture to get to the G-file (all six missing white pieces have been accouted for).
The Pawn from E7 has promoted to the bishop on A2.
What happened was this: the white pawn from G2 made its capture on H3 while the pawn on G3 was still on H2. This allowed the black pawn
to come down and be promoted (after the white rook from H1 got out), and then the pawn on H2 made its capture on G3.
\vspace{2mm}
\textbf{Conclusion:}
We already know the mystery piece is black. It can't be a pawn, because we've accounted for all missing black pawns.
It can't be a queen or a rook, since there couldn't have been any more promotions by Black. It is therefore a bishop or a knight.
However, White has just castled and moved his king over D1, so the mystery piece cannot be a bishop (the king may not cross through
check while castling). Therefore, the mystery piece must be \textbf{a black knight}.
\end{solution}
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