\section{Very difficult problems}

% 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 separately.

	\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}
\difficulty{5}{7}

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