diff --git a/Advanced/Retrograde Analysis/main.tex b/Advanced/Retrograde Analysis/main.tex new file mode 100755 index 0000000..92e86fb --- /dev/null +++ b/Advanced/Retrograde Analysis/main.tex @@ -0,0 +1,130 @@ +% 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} \ No newline at end of file diff --git a/Advanced/Retrograde Analysis/parts/easy.tex b/Advanced/Retrograde Analysis/parts/easy.tex new file mode 100644 index 0000000..0964925 --- /dev/null +++ b/Advanced/Retrograde Analysis/parts/easy.tex @@ -0,0 +1,317 @@ +% 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 \ No newline at end of file diff --git a/Advanced/Retrograde Analysis/parts/hard.tex b/Advanced/Retrograde Analysis/parts/hard.tex new file mode 100644 index 0000000..6c0bff2 --- /dev/null +++ b/Advanced/Retrograde Analysis/parts/hard.tex @@ -0,0 +1,137 @@ +% 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} +\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 \ No newline at end of file diff --git a/Advanced/Retrograde Analysis/parts/intro.tex b/Advanced/Retrograde Analysis/parts/intro.tex new file mode 100644 index 0000000..6f8c7e4 --- /dev/null +++ b/Advanced/Retrograde Analysis/parts/intro.tex @@ -0,0 +1,140 @@ +\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 \ No newline at end of file diff --git a/Advanced/Retrograde Analysis/parts/medium.tex b/Advanced/Retrograde Analysis/parts/medium.tex new file mode 100644 index 0000000..aa58847 --- /dev/null +++ b/Advanced/Retrograde Analysis/parts/medium.tex @@ -0,0 +1,237 @@ + +% 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} + +\vfill +\pagebreak + +\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} + +\vfill +\pagebreak \ No newline at end of file