Minor cleanup
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356
Advanced/Retrograde Analysis/parts/03 medium.tex
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356
Advanced/Retrograde Analysis/parts/03 medium.tex
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\section{Slightly harder problems}
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% Sherlock, A matter of direction
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\problem{A matter of direction}
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%\difficulty{3}{5}
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\starvalue{5}
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The results of a game of chess are shown below. \par
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Did White start on the north or south side of the board? \par
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\manyboards{
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ka8,Kc8,
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Qe7,
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Bc5,Pe5,
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Pd4,
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Ph3,
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Bh1
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}
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\begin{hintlist}
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Hint 1:
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\hintcontent{2}{5}{The bishop on H1 is important. How did White deliver this check?}
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\end{hintlist}
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\begin{solution}
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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.
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(Note the double-check on A7).
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\vspace{2mm}
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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
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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.
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\end{solution}
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\vfill
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\pagebreak
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% Arabian Knights 1
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\problem{Where is the king?}
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%\difficulty{3}{5}
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\startimes{8}
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The white king has again become invisible. Find him. \par
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\hint{White started on the bottom. En passant.} \par
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\manyboards{
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rb5,bd5,
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Ba4,
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kd1
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}
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\begin{hintlist}
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Hint 1:
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\hintcontent{3}{8}{
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Either the white king is on B3, or Black is in check. \par
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First, show that the latter implies the former.
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\vspace{2mm}
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}
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Hint 2:
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\hintcontent{2}{5}{
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Moving back in time, you'll need to add two pieces to the board (not counting the king). \par
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They have been captured!
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\vspace{2mm}
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}
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\end{hintlist}
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\makeatletter
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\if@solutions
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\vfill
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\pagebreak
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\fi
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\makeatother
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\begin{solution}
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Looking at the board, we see that the white king is on B3 or Black is in check.
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\vspace{2mm}
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First, we show that the latter implies the former: assume the black king is not on B3. \par
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How did White deliver this check?
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Not by moving the bishop, so this check must have been discovered by the white king moving from B3.
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Therefore, if the white king isn't on B3 now, he was there on the previous move.
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\vspace{4mm}
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How did the white king end up on B3? That seems to be an impossible double-check from both the rook and bishop!
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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.
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The sequence of moves is now as follows:
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\begin{minipage}{0.5\linewidth}
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\begin{center}
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\chessboard[
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setpieces = {
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rb5,
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Ba4,pb4,be4,
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Kb3,
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Pc2,
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kd1
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}
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]
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}{0.48\linewidth}
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Black: E4 $\to$ D5 \par
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White: C2 $\to$ C4 \par
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Black: B4 $\to$ C3 (en passant capture) \par
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White: B3 $\to$ C3 \par
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So, the white king must be on C3.
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\end{minipage}
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\end{solution}
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\vfill
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\pagebreak
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% Arabian Knights, intro (given with solution)
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\problem{Double-checks}
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%\difficulty{3}{5}
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\startimes{10}
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White to move. Which side of the board did each color start on? \par
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\hint{What was Black's last move? }
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\manyboards{
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Re3,
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Nc2,Rd2,
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Nd1,kf1,Kh1
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}
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\begin{hintlist}
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Hint 1:
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\hintcontent{4}{10}{
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Double-checks make all positions seem impossible... \par
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Try E1 and F2 anyway. Can you add pieces to make it make sense? \par
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Don't forget about promotion.
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}
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\end{hintlist}
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\begin{solution}
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Black's last move was from F2, where his king was in double-check from both a rook and a knight.
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How did this happen?
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\vspace{2mm}
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White started on the north side of the board, and put Black in check by capturing a piece on D1 with
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a pawn and then promoting that pawn to a knight.
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\begin{center}
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\chessboard[
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smallboard,
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setpieces = {
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Re3,
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Nc2,Rd2,Pe2,
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bd1,kf2,Kh1
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}
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]
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\end{center}
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\end{solution}
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\vfill
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\pagebreak
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% Arabian Knights 4
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\problem{A missing piece}
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%\difficulty{4}{5}
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\startimes{20}
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There is a piece at G4, marked with a $\odot$. \par
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What is it, and what is its color? \par
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\manyboards{
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ra8,ke8,rh8,
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pc7,pd7,
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pb6,
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pa5,
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Ug4,
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Pb3,Pg3,Ph3,
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ba2,Pb2,Pc2,Pd2,Pf2,qg2,bh2,
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Kc1,Rd1,nf1,Bh1
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}
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\begin{hintlist}
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Hint 1:
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\hintcontent{1}{20}{
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What color is the missing piece? Count captures.\par
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Look at the region bounded by A1 and B3. How did the bishop get there? \par
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\note[Note]{This is Part 1}
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\vspace{2mm}
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}
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Hint 2:
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\hintcontent{1}{19}{
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What was White's last move? \par
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What does this tell us about White's king? \par
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\note[Note]{This is Part 2}
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\vspace{2mm}
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}
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Hint 3:
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\hintcontent{1}{18}{
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Now, look at the region bounded by G1 and H3. \par
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In Part 1, we found that all of White's pieces were captured---including the H1 rook. \par
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How did it get off its home square to be captured? \par
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What does this tell us about the bishop on H1? \par
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\note[Note]{This is Part 3}
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\vspace{2mm}
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}
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Hint 4:
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\hintcontent{1}{17}{
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The black bishop on H2 must have been promoted on G1. \par
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Which pawn was it, and how did it get there? (Remember, we counted captures in Part 1). \par
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In what order did the cross capture by the G1 and H1 pawns occur? \par
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\note[Note]{This is Part 4}
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\vspace{2mm}
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}
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Hint 5:
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\hintcontent{5}{16}{
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Which black pieces are still missing? \par
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Remember that White cannot castle through check.
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}
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\end{hintlist}
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%\begin{instructornote}
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% \textbf{Hints to give:} (in this order)
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% \begin{itemize}
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% \item How did the black bishop on A2 get there? \note{(Part 1)}
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% \item How many captures has Black made? \note{(Part 1)}
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% \item What color is the missing piece? \note{(Part 1)}
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% \item What was White's last move? What does this imply? \note{(Part 2)}
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% \item Which white pieces were captured? \par
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% How did they move from their starting positions? \note{(Part 3)}
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% \item How did the bishop on H2 get to where it is now? \note{(Part 3)}
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% \item Which pawn was promoted to the bishop now on H2? \note{(Part 4)}
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% \item Which black pieces are still missing? \note{(Conclusion)}
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% \item Remember that White cannot castle through check. \note{(Conclusion)}
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% \end{itemize}
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%\end{instructornote}
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\vfill
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\pagebreak
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\begin{solution}
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\textbf{Part 1:}
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The black bishop on A2 cannot be original, since the white pawn on B3 would have prevented it from getting there.
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That bishop is a promoted bishop. \par
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The black pawn it was promoted from must have come from E7,
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captured four pieces to get to A3, then moved to A2, and then made a capture on B1, where it was promoted. \par
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Thus, the pawn from E7 has made five captures.
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\vspace{2mm}
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The white bishop from from C1 never left its home square
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(since neither of the pawns on B2 or D2 have moved), and hence was captured on C1. This makes six captures of
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white pieces, which tells us that the mystery piece is black.
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\vspace{2mm}
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\textbf{Part 2:}
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White's last move could not have been with the rook from E1, which would have checked Black,
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nor with the king (which could only come from B1, an impossible check),
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nor could it have been with any piece other than the rook or king.
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Therefore, White just castled, and thus the white king never moved before that.
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\vspace{2mm}
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\textbf{Part 3:}
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Among the white pieces captured by the black pawn from E7 was the white rook from H1. Since White has just castled,
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and the white king never moved before that, how did the rook from H1 get onto the board to be captured?
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The only possible explanation is that the pawns on G3 and H3 cross-captured to let out the rook:
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the pawn on G3 really came from H2 and vice-versa. Since the pawn on G3 comes from H2, the black bishop
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on H2 has always been confined to G1 and H2. How did the bishop get there? It must have been promoted.
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\vspace{2mm}
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\textbf{Part 4:}
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The promoted black bishop on H2 must have been promoted on G1. The pawn which was promoted must have come from G7,
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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).
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The Pawn from E7 has promoted to the bishop on A2.
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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
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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.
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\vspace{2mm}
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\textbf{Conclusion:}
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We already know the mystery piece is black. It can't be a pawn, because we've accounted for all missing black pawns.
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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.
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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
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check while castling). Therefore, the mystery piece must be \textbf{a black knight}.
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\end{solution}
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\vfill
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\pagebreak
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