Impossible-to-reach position with the least amount of pieces?

That pieces must occupy  a single square in the board layout of a legal position follows from art 2.3 in the FIDE handbook (the initial board layout), together with the definitions of "legal move" and "illegal position" in art. 3. In particular art. 3.10.3 "A position is illegal when it cannot have been reached by any series of legal moves".

The term "position" has no explicit definition so would take the usual English meaning of "situation occurring", the board layout existing in such a situation being only one aspect of the position. Other relevant aspects would include (but not be limited to) which player has the move, whether the last completed move was an initial double square pawn move, which pieces the player having the move has touched and in which order since his opponent last made a move, whether the board layout contains rooks and if so whether they have previously moved and whether the kings of the same colours have previously moved, which, if any, players have resigned, whether a draw has been agreed, if the game is being played under competition rules, how many times a position with the same board layout as the position after completion of the last move, the same player having the move and the same legal moves available has previously occurred, how many moves by each player have been made since the last pawn move or capture, what times remain on the clocks.

So @Rocky64 is correct in saying that I haven't given positions in #33, I've given only board layouts. The second board layout on the other hand could be the current board layout of a position occurring in a legitimate game if the player having the move is Black and he is moving his queen. Such a position, though legitimate, would be "illegal" with the FIDE definition in art. 3.10.3 because it is not reached by a series of "legal moves". (The position at the completion of the previous move would be so reached.)

Legal moves from board layouts in which pieces occupy single squares result in board layouts in which the pieces also occupy single squares. This fact combined with the fact that all the pieces occupy single squares in the initial board layout defined in art. 2.3 means that no position in which the associated board layout contains pieces which do no not occupy single squares can be legal (though such positions regularly occur in legitimate games).

Similar reasoning applies to positions in which the board layout has squares occupied by multiple pieces (but these, in common with most of the positions with board layouts posted in this thread, do not occur in legitimate games).

White to move

Na its this ^

*enter Dab here*

But not quite in accordance with OP's stated requirements.

Least possible pieces for impossible position.

@bbmaxwell

Certainly that, but OP stated two kings. The first position is not specifically excluded by OP's question. 

This is a reference to how that's legal, however, it is not in the rules for the challenge, it just saying that is a key component for it to be legal.

A moot point, but I don't expect anybody to do it with fewer pieces at any rate.

It can, of course, be reached by a series of legal moves from the starting position if those result in a win or draw and the players then put the pieces back in the box.

How about a white pawn like a1 as you cant move a pawn backwards unless promoted (in case you didnt know)

A chess position is a discrete state where each piece occupies one whole square. Each move will generate only one such position. Just because in an OTB game a player has to physically move a piece across the boundary of two squares, that doesn't mean a move generates intermediate "positions" where the piece is sitting on neither square. Such an idea implies that every move generates not one but an arbitrary (infinite?) number of positions, making the term meaningless.

Legal and illegal positions have something in common, namely they are not nonsensical. A piece standing between two squares is nonsensical. 

If you say so. The FIDE laws don't. Are the positions before and after a player touches a piece on his move the same? The possible continuations of the game aren't.

Solved lmfao the bishop is checking the king however it had no way to get to that spot without checking the king on the previous move

but thats just a theory

A GAME THEORY

The definition of "position" is important in the rules because the concept of "three-fold repetition" needs to be unambiguously defined.

As you note, two positions being "the same" entails more than just the board being visually the same. (From the perspective of that rule, "position" also includes the player to move, castling rights and en-passant right. If any of that changes, it's not the "same position" anymore.)

There is no definition of "position" in the rules. The 3(5) move repetition rule defines only an equivalence between positions under which it calls the positions the same (though they obviously can't be). The scope of that definition is only arts. 9.2.2 and 9.6.1.

In the absence of a definition the normal English meaning of position should be assumed, which, I would suggest, is simply a situation occurring.

FIDE doesn't need to define "position". There are obviously many different ideas of the term among chess players, but the purpose of the laws is not to resolve such issues, merely to describe how to play chess.

Nevertheless 9.2 could be better formulated. As it stands, if you draw White against Carlsen and you're not very good, your best course could be to wait two seconds, call the arbiter and point out that the position is the same for the purposes of art. 9.2 as it was at the start of the game and one second later, so you wish to claim a draw.   

Which FIDE laws say that such a Q being moved between two squares counts as a "position"?

None. That's the point. The FIDE laws don't say what counts as a position.

You have to understand "position" in the laws in standard English, which I would suggest means simply state or situation occurring during a game, which clearly would apply.

In fact in #33 the queen wasn't intended to be moving between squares. It was meant to represent the situation after moving the black king to a different square in your board layout and adding a black queen to the board at the centre. Obviously that would be a board layout that would not occur in any legal position (as FIDE defined) but that's what you asked for.

Right, the meaning of position is open to interpretation and a matter of (unspoken) convention. And your interpretation of the word is so loose that it includes what most people would consider nonsensical states of a piece standing between two squares.

Your argument that such a state counts as an illegal position is based on the rule that an illegal position is one that cannot be reached via a series of legal moves. I agree with this rule, but note that it does not imply that anything that cannot be reached via legal moves is automatically an illegal position. For instance, would adding a white piece of cheese to my problem position work as a solution? After all, no series of legal moves could ever result in a "position" with a piece of cheese on the board. The answer is of course not, because it's nonsensical. The point is that asking for an illegal position does not give the solver free rein to say "anything goes". Even an illegal position has certain boundaries that make them count as positions in the first place.  

I took your problem part (a) to mean a white chess piece should be added (I didn't, in any case, have a symbol for cheese) but certainly if the white bishop in my first diagram were replaced by a white piece of cheese that could not occur in a legitimate game of chess. 

I wouldn't say it's nonsensical. You could set it up on a chess board and that's what it would mean. 

Whether it's an illegal position is, as you point out, open to interpretation. The FIDE rules don't help here because they don't define "position". If you accept the suggested English interpretation of the term in my previous post then "position" would mean only "situation occurring in a game", in which case your solutions wouldn't actually be the board layout of any position and neither would my first diagram in #33 (with or without cheese).

The FIDE laws do define what it means for a "position" to be illegal:

art. 3.10.3 A position is illegal when it cannot have been reached by any series of legal moves.

Such illegal positions routinely occur in a legitimate chess game whenever events or actions referred to in the laws other than a move being "made" according to art. 4 occur. (By defining the term "illegal" in this context the FIDE laws remove its natural meaning.)

My second diagram in #33 is a different matter. It could be the board layout in a situation arising in a chess game. The position would necessarily be "illegal" according to FIDE's definition because some event referred to in the laws (in particular touching the queen with the intention to move) must have occurred since the completion of the last series of legal moves. So I think this would qualify as a solution to (b) with my suggested interpretation of "position". Again I see nothing nonsensical about such positions - they occur all the time.

It's quite clear that the answer to my original question is "three pieces" (as a minimum amount of pieces in a position that cannot be reached): Two kings and either a bishop or a rook. In addition the third piece can be a pawn as well:

But maybe this is "too trivial" in a sense.

Now I'm thinking if there would be an impossible-to-reach position (with not a huge amount of pieces) where it's far from trivial to see why it can't be reached, ie. it's not immediately obvious. At first glance it looks like a rather normal position that could happen, yet there's something about it that makes it impossible to reach.