Beautiful and hardest puzzles I've ever seen!

The beauty of this problems are not the answers, it is the process of solving them!

Correct

These puzzles have no real solution.

The reason why is because they are not real puzzles.

You have left out information in the puzzle.

Which means the solutions can be any solution we want them to be.

As long as it is under the assumption of legal moves.

 

For example:

People have said the answer to problem 1 is en-passant.

Well I disagree.

I think white is a terrible chess player and instead of taking the free rook on b5.

They decided to do a rook move.

The white king in my diagram is on h4.

I was thinking of putting my king on h8, but I didn't because my king feels lonely in the corner.

---

For position number 2.

Since information was left out of it as well.

I have an easy solution for it.

Each side needs to account for 8 pawns + 4 minor pieces + 3 major pieces

 

First look at the material count for black.

Black has 7 pawns left.

Black has 2 minor pieces left.

Black has 0 major pieces left.

Which means black is missing 1 pawn + 2 minor pieces + 3 major pieces.

 

 

Now look at the material count for white.

White has 7 pawns left.

White has 0 minor pieces left.

White has 2 major pieces left.

Which means white is missing 1 pawn + 4 minor pieces + 1 major piece.

 

We have to keep this in the back of our mind.

Now look at the spots were the pawns are located at!

Black is missing a B pawn + F pawn + H pawn.

White is missing a F pawn

 

All black needs to account for is the pawn captures.

The black B pawn turned into the a6 pawn.

The black F pawn turned into the c4 pawn.

It turned into those other pawns by capturing.

It is a total of 4 captures.

The piece on h4 can be:

A white pawn, rook, bishop, knight, queen

or

A black pawn, rook, bishop, queen.

Black is missing a piece on c7.

@Xplayer: You are sometimes too quick to put the blame on others. There is no missing information in the puzzles. The puzzle position, along with the text provided, is sufficient to logically deduce the answer required. In the first puzzle, it is stated the white king is knocked off the board (implicitly: only the white king is missing), and not any other piece, like the white rook you would arbitrarily put on h4. And in the second puzzle, it is possible to identify the piece on h4 solely from the position and the fact that it was a legal game of chess. Only the piece on h4 is unknown; no other piece is missing off the board, nor is any information missing.

I or someone else might have said this in the past, but retros are not your thing.

@OP: These 2 problems are indeed beautiful (especially the first), but also just the tip of the iceberg. (I have seen them before, many times in fact.) I can point you to many, many other problems of this type if you want. (And I don't mean those by Smullyan.)

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Never seen no.2 but love it! Took me ages then finally got it seconds after asking for help.

I can confirm that there are no hidden tricks and it is solvable with the information given.

Regarding puzzle 2: I haven't solved it but expanding the thought of the others we have the following:

As @SJFG pointed out, the black king got in check after a white c-pawn captured a black piece at d8. This means that in the previous move the pawn was at c7 so white had no promotions until that point.

Could the piece at h4 be a black piece? Apparently it can't be a rook or queen (white king would be in check), it can't be a light-squared bishop and it can't be the dark-squared bishop because g and e pawns haven't moved (the bishop has no legal way to travel there). So the dark-squared bishop must have been captured at its starting position. This leaves only the case of a black knight to eliminate.

Black has both of its knights so for a black knight to be at h4 that would mean that the h-pawn got promoted to a knight.

(here comes the part I am not 100% sure about): what piece did the white c pawn captured at d8? I think it could not have been a queen/rook/bishop (for the same reasons as for the case of h4) so it could only be the h-pawn promoted to a knight!

If the above is correct, then the piece at h4 is certainly a white piece but this is as far as I came up with

 

P.S. Very fun puzzles. I think we are ready for the next tip

 

You are correct so far if I am right also! Next step: work out how many captures black has made!?

I'm afraid the only thing with no real solution is your hopeless case.

Oh!!! White's king is on b3 and black's last move was b4xc3because white before that played c4 to block the check. Now white played Kxc3 and it's blacks move.

no idea

@hsa23f, one thing I've been thinking about is that Black could have moved the h-pawn to h3, captured on g2, and promoted to a dark-squared bishop on g1, which then moved to f2, e1, a5, and then d8 (before some of the pawn moves). But you're right that it could be a black knight too. I don't know that it matters, though, because in either case it cannot be a Black piece on h4.

Black's b-pawn made one capture. Black's f-pawn made three captures. Black's h-pawn must have made one capture. So that's five captures. White starts the game out with seven pieces (aside from the king), and the rook on d7 could not have been captured. That means that the piece Black did not capture is the one on h4. The thing to realize is that all of Black's pawns must have captured on light squares. The only piece that couldn't get to a light square is White's dark-squared bishop, which is why that's what I said must have been on h4. I wrote that in white earlier. Am I wrong?

headache please give answwer

I have seen this puzzle before and I know the solution, but I think yours is even better. Well played.

with puzzle 2, the only thing i know about it is that last move was c7xd8=R

Well obviously you are to quick to blame me and put me down.

When I am not blaming anyone!

I am simply relaying the information which was provided by the OP.

He said there are missing pieces.

People gave a solved to problem number 1.

However, I disagreed because it required pawns.

If the above diagram is the solution to the problem.

Than why did the OP say pieces were missing?

In this solution no pieces were used.

Only pawns were used.

I don't call my pawns pieces.

I have 8 pawns, 7 pieces, and 1 king when I play chess.

I don't have 15 pieces and 1 king.

For puzzle #2, I tried my best to solve the puzzle, but the complications are definitely entertaining me. I will have to check this puzzle at a later date, however, because I can not go past here. 

For solvers unfamiliar with problems like these, the current position must be achieved by legal yet nonsensical moves. In this case, our mission is to figure out what piece belongs on the h4-square. This includes color and denomination, which proves to be challenging. Let's proceed to the solving! 

Before delving into the puzzle too much, I recognize that the Black king is currently in check. This means that it was White's turn. Another helpful observation is that the d7 rook had no legal moves to achieve its current position. So, if the White rook could not physically obtain its post by moving, then something, interfered with its range, did. The only possible square for this phenomenon to hold true is on c7. Suddenly, after extracting some more or less obvious information about this puzzle, we have an immediate, unanswered question. What piece was on c7? Well, we can begin with the process of elimination. Can not be a rook or a queen since that would renew check to the Black king, and, therefore, the position would be impossible.

What about a bishop? We have only 7 pawns as White, so, at first glance, it looks likely, but how can a dark-square bishop maneuver to that square? H8? No, because once it promotes, it must capture g7 to get to the ideal square. F8? A similar story occurs; it is imprisoned, and can not leave without permanently damaging the current position. The same can be said for all other available promotion squares. If a bishop can not be, then what about a knight? Can that be the answer. Unfortunately, the answer is still no. This is actually quite simple; there is nothing a knight can do to help the current situation.

So, what about a pawn? Instinctively, the pawn is trapped, so where would the pawn move? However, we seem to have it a roadblock since we have eliminated all the other options. Now, we mustn't forget that pawns are special for its movement. It is special for two reasons:

1. When capturing, it moves diagonally
2. When it reaches the back rank, it transforms into another piece. 
3. en passant can occur

Can these specialties about a pawn be utilized? En passant can not be performed unless a White pawn is on the 5th rank. Finally, it strikes me. A pawn from c7 captured a piece on d8! So, what piece did it capture? Our options are the following:

1. Queen
2. Rook
3. Knight
4. Dark-square Bishop

Let's investigate the queen. Note that this piece would leave White in check. This would mean that Black must have made a legal move that placed White in check. Is there a square that would allow for this? No. Therefore, it could not be the queen. Likewise, the rook has the same problem. Without much thinking at all, we have managed to eliminate a few choices. Of course, recalling from our earlier list, we still have a dark-square bishop and a knight to consider.

We have condensed our options to the following:

1. Knight
2. Dark-square Bishop

If you have read until this point, I am proud, but there is more to come:

If a knight was captured, the first yet obvious thing to notice is that there are three knights on the board. For this to be possible, there must have been a promotion by Black. Simple addition of the pawns would indicate that there are only seven, so this is possible. Now, we know that this is a possibility, but we must consider the other option just in case that is possible, too. In logic puzzles, all options must be thoroughly examined. If not, you may run into a contradiction later, and you will incorrectly say that no solution exists.

What about a bishop? 

 

In all the diagrams except the last one, I am missing a knight on the a1-square. Please forgive me.

 

English reading practice for you: The OP posted for the first position "White's king has fallen off the board." The OP did not mention that pieces were missing, nor did I say that pieces were missing.

Furthermore, look at the diagram you posted. In the final position, the only extra unit on the board is the white king on c3. No pawn there that I can see. That is the answer: White king on c3. I will also note that you include "king" in your definition of piece, so what exactly is your problem?

@SJFG: You aren't wrong.

@yourChess: If it was a black bishop captured on d8, how did it get there from the start of the game?