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  ; )

You mean, after white played a4, black was the last to move. Just because it's a retrograde analysis doesn't make time go backwards. The position is listed as white to move (not to mention white is in check), so obviously the last move made was by black. Before black's last move, white had to have made a move. I don't think anyone has been confused about this, yet you keep "correcting" us for some reason.

White is in check from the knight on h5. Therefore black must have played the knight to h5 on his last move. I don't see any way around that. On the otherhand, it also isn't particularly important to the rest of the analysis.

Now, about white's last move (which was obviously before black's last move). Let's answer the question: could it have been a king move? The king would have to move from f8 or f7. The king could not have been on f7 with the black bishop on g8 and black knight on h8 -- there's no way for black to give this double check. The king could not have been on f8 and triple checked by the queen on d6 and the knights on h7 and g6. Therefore, there are no squares for the white king to have come from on white's last move.

White's last move must have been either a4 or b3.

What I pointed out in my first post is that it takes black a lot of promotions and therefore a lot of captures to get to this position. There are a couple different ways to go about it, but the upshot is that you can't move both the a and b pawn because one of these needs to be held in reserve. 

When I said, "before White played a4," I was referring to one of the two positions below:

In these positions, it is White to move (he's about to play a4), so Black was the last to move, right?

 Getting back to the final position (where White is in checkmate), there IS another possible last move that Black could have made, and it IS significant to the analysis once you find it. There's also Position 2 (from the original puzzle in post #116) to consider... although any hour now einstein will have the whole thing solved and post the full analysis here, so I'm not even sure why I keep bothering to point these things out

Loomis, that game fails to reach either of the two positions from the puzzle - check post #116 again... or was that just a joke? 

Eh, I realized right after posting it the queen was on the wrong square. But the position of the queen isn't so important, so I left it. I'll go back and fix it just to make you happy.

Omg, either you're really getting a kick out of this, or you still haven't checked post #116. Einstein_69101, where are you? :P jk

No, I actually did overlook the knight on g1. Oops.

I was actually trying to demonstrate that there aren't enough captures available. I was trying the line where white promotes twice by capturing on a2, but since black has made no pawn moves other than a3, this took both knights to accomplish.

I think I'll go back to my original conclusion that there simply aren't enough captures.

 It seems you're getting close to solving it, now. Looking at that "proof game" more carefully, I noticed that you had found the other possibility for Black's last move: N4xh5# (instead of N4h5#). But Position 2 has not yet been addressed - at all. The original puzzle's question involves the legality of both positions.

You can stop bolding and italicizing that point. I'm not missing it.

I probably should have mentioned that regardless of what black played before white's a4 move, white didn't have any other legal move.  Black has made eight captures already so this means that when black checkmates white, then white didn't have another piece to move.

There is the possibility that black's b-pawn promotes with just one capture (bxa3) and promotes on the a1 square after white has played a4.  In this case, black can promote six times with just seven captures (five captures by the a-, b-, c- and d-pawns, one capture on f1 and another to capture white's h-pawn.)  This requires that black plays axb2 before white plays a4.  But this also means that white has to play a4 and b3 early.  And the only possible move that white could have made was Rh5+ right before black plays N4xh5#.  The continuation of the proof is similiar to the proof for post #114.

Position 2 is nearly identical to Position 1.  The difference is that the rook is on f5 instead of c4, the black knight is on f4 instead of h5 and it is black's turn to move.  This means that white just made a move.  White didn't just move his king because of reasons stated earlier.  The white knight is trapped, and white's last move was not b3.  Black needs white to play b3 so his c-pawn can promote with just one capture.  This means that white has just played a4.  And black is required to have a minimum capture count of eight as shown from the analysis of Position 1.  It is impossible that white's a-pawn has moved from a2 to a4 because white had to play a3 early so black can get a promotion on a1 and meet the maximum of eight captures.  Regardless of what black played before white played a4, white had no previous legal move.

Edit: I should say that black can get all of his promotions with just seven captures in Position 2, but this requires white to play a4 and b3 early.

I say they are both illegal.

Legal or Illegal?

Legal or Illegal?

Hmm looks like a Bongcloud variation to me...