1. The question's wording is weird. Just say "Is it possible that it's White to move" or something; I almost misread "have made" to "make" (which completely negates the question's meaning). The answer is yes.
2. If it's White to move, yes (later when Black has the move); if it's Black to move, no.
Interesting.
In position 2, Black may have queenside castling rights, but would need to move the bishop first.
For 2, if the king and rook have not moved, and it is black's turn, than yes. If he has, he can play Bb7 and do it.
There is a whole book with these puzzles:
"The Chess Mysteries of Sherlock Holmes"
The title suggests that it is for kids, but the puzzles are actually quite hard. Both 1) and 2) are in there.
Could somebody explain these puzzles? 
#2 is unknowable. There is a legal sequence of moves where black can castle and also where black cannot castle.
rtr1129: According to the Codex of Chess Compositions, if a side's castling ability is unable to be ruled out, it is assumed that they can castle. (In other words, the question is "Is it possible that Black still has castling rights?".)
Question 1: Yes, the position is legal (with White to move).
Question 2: If it's Black to move, no, it's impossible for Black to castle kingside.
Since it's Black to move, White has made a last move; it must necessarily be a2-a3. Before this, Black must have captured one White unit, otherwise White cannot have another last move (White cannot retract). This unit must be either rook or knight. A rook is impossible, since White's rooks cannot have possibly got out from their starting points (Ra1 is bound by Pa2, Pb2, and Bc1, and Rh1 is bound by Bf1, Pg2, and Ph3.) So Black's last move was to capture White's knight somehow.
Certainly the capture is not with any of the pawns. If the capture was by either the king or the kingside rook, then clearly Black can no longer castle. If the capture was by the queenside rook, the knight has no previous move. If the capture was by the bishop, the knight's only previous move is Nd6-c8, but this means White moves while giving check (knight on d6 is check). So there is no way to retain Black's kingside castling right.
You guys are great, both puzzles have been solved correctly.
1.: Yes, 2.: No (the first perfect answer given by chaotic_iak)
By the way:
"Raymond M. Smullyan wrote two well-received retrograde analysis riddle books:
(Wikipedia quote)
For more on retros, do check out the Retro Corner. Smullyan's compositions are mostly a tad contrived (conditions like "no promoted knights" that feel artificial even for retro problems); normal retros usually just have a position and question like "what were the last 6 moves?"; sometimes not even the side to move is given (that has to be worked out.)
Also you can check out some of my threads, or chaotic_iak's, or BigDoggProblem's, or shoopi's, and the group here on chess.com; we like retros, and have made some original puzzles.
Hi,
I just stumbled upon an article about Smullyans (famous mathematician) chess compositions which you have to solve using logical conclusions. No matter how unrealistic the following positions are, can you answer the question to each diagram with a sound an coherent logical proof?
Diagram 1: Can Black have made a legal turn in this position?
Diagram 2: Can Black castle in this position?
Please enjoy !!
Yours, ubuntux