legal AND illegal positions

Legal - preceding moves were Kg8+ Ka1.

or something like Bb2-a1+ Kxa1

i really enjoyed them all thank you

It's also possible that the rook on b8 was a pawn that was just promoted to a rook giving check.

Legal or illegal?

Nice and subtle: It's illegal.  Pawn capture count is OK on White's side, but Black's a pawn can't have gotten over to the b-file nor could it have promoted to get there without the pawn on b5 first having changed files (kind of a chicken and egg problem).  Even if one of the other pawns promoted (which is possible) it can't address this problem.

 Actually, I don't see what the big problem is. The position in post #110 is legal:

Right, second promotion.  Of course....

Here's one of mine: legal or illegal?

That looks illegal.  Black's last move was N4g5#.  It is possible that that knight captured a white rook.  If no piece was captured, then white had no possible move before that.  The moves axb4 and cxb4 are not possible because black still has all of his pieces on the board.  If N4g5# captured a piece, then it could of only captured a rook.  That is the only piece that white could have moved previous to the checkmate.

 Correct, the position is indeed illegal. I once posted this puzzle in another thread, and though everyone was "certain" it was illegal, none gave any analysis to prove it. So I'd say you're the first to actually "solve" this one - nice!

A more complicated version is given below; let's see who can work it out:

Position 1:

Position 2:

Which of the above positions is legal?

A.) Position 1

B.) Position 2

C.) Both are legal

D.) Neither is legal

Post #116, Position 1 I think is illegal.

1. White's last move had to be the a or b pawn.

2. Black has to promote 6 pawns

3. Black can only make 8 captures.

4. The a and c-pawns can get to b1 with just one capture. The b-pawn takes 2 captures to get to b1 (because we have to hold a pawn move for white until the end). The d-pawn will take 2 captures (one of which can be the f2 pawn). This is already 6 capturs.

5. The h-pawn needs to be captured and a pawn will need to capture on the back rank to promote on a dark square (either a1 or g1). So this brings us to all 8 available captures.

We haven't answered what happened to the black h-pawn and who captured the white h-pawn. This seems to uncover the impossibility.

When it is black to move, then it is legal.

 That's a good start, but there's more to it than that.

(1)  In Position 1 it's White to move; doesn't that mean the last move was made by Black, not White?

(2) Well, there are three possibilities: either the pawn promoted, was captured, or is now on g4. Can all three of these be ruled out?

Sorry for being vague, but those are the only things I can address without giving any hints :)

I don't have anything to contribute to this post, as I do not have the skill (or the will) to solve any of these puzzles, but I can say this. After only reading a few pages, my mind is blown. 

1. Yes, black's last move was N4h5+. And since the white king and knight could not  move to the squares they are on, white's last move must have been with the a or b pawn.

2. Black's h-pawn can not have been captured. Black has made 6 promotions and still has two pawns. We've already counted up 8 captures, for black to get all these promotions, so the h-pawn capturing to get to the g-file isn't possible. In order for the black h-pawn to have promoted, it needs to make captures, but we've run out of possibilities.

Missionary is a legal position.

I think you are on the right track.  :)  Another thing to point out is that black had to capture white's bishop on f1.  This capture does not help with pawn promotions.  You pointed out a variation where black promotes on the a1 (one time), b1, f1 and h1 squares.  I agree that that variation is not possible.  There is another variation where black's a-pawn captures twice to go around white's a-pawn so that it can promote on a1.  In order to do this, white needs to play a3 early so black can play bxa2 twice (once for the a-pawn and another time for the b-pawn.)  The first promotion on a1 must be a knight so it can get out by playing Nb3.  The second promotion can be a bishop.  The bishop needs to wait for white to play b3 so it can get out as well.  The black pawns need to make at least six captures as you pointed out.  White's h-pawn needs to be captured by a black piece so that the black h-pawn can go straight for promotion.  With black capturing white's bishop on c1, the total capture count is eight.  White would have pawns on a3 and b3.  This gives white only one available move before N4h5# was played.  But white had no legal move before he played a4 in this variation.

 Before White played a4, Black was the last to move, not White.  :)  Furthermore, Black's last move was not necessarily N4h5#, nor was White's last move necessarily a4. More importantly, has anyone given Position 2 a look? The original question (from post #116) cannot be answered without doing so  ; )