Correct.  Only one capture is required to PROMOTE, but two captures required for the knight to be on g8.

Here's a similar position.

^Illegal. White's h pawn must have captured on g7 to promote, but then the knight would not be able to get to h8, because both g6 and f7 were blocked by black's pawns.

 Incorrect (assuming you're referring to post #234). The most recently-posted position is legal.

Wait, why is the fact that both kings can castle relevant? Thx :)

 It means that neither of the kings or rooks could have moved. This makes the position harder to reach, since the final pawn captures must be synchronized so that neither side runs out of moves before the final position.

Right, I see.

BTW, where do you get all these problems from?

I see now that I should have probably posted this here in the first place. It's not too hard, made by me:

 Legal or illegal?

AAARGAHRAGHARHRAHRAHRGAHRAGHARGHRAGHAHAHGGHH

How do you people come up with these?! They're ridiculously amazing!

 Kind of easy for a thread like this one. It's legal, of course:

Yes, I said it's not really hard - but I think the easier ones are also nice to have here. Maybe a piece of cake for you, but some players may struggle a bit with this!

Also, I composed this, so sorry for it not being super-hard

P.S. did you check out my "Can white castle" posts? They're a bit more challenging, albeit if this was a walk at the park for you, I suspect you'll make short work of them as well 

Keep the puzzles rolling guys! I'm enjoying these very much so!

@shoopi - Your puzzle was an interesting one.  :)  Feel free to post more.  I might have to compose more puzzles.  I've only composed one so far.  :)

if it is legal mate in 1

 This position is legal, as well:

Well, the first position is definitely legal.

Black is missing two pieces - a light bishop, and a pawn. Just enough to make the two captures by white's pawns possible.

Because of that, the d5 pawn could have only got via 2 captures. Likewise, the h5 pawn could have only got there through capture of white's g and h pawns. All of these captures are made on light squares, so white's dark bishop is pretty useless. Thankfully he has just enough pieces (must promote his f pawn though) for these 4 captures. The rest is just a matter of positioning.

Nice try, shoopi - but that proof game fails to reach either of the positions (wrong side to move).

 What's even more embarrassing is, I was trying to reach the second position!

Heh, I can't believe the side to move slipped by me like that. Working on it.

I'm pretty sure the last sequence is O-O+ followed by Ka2.

On second thought, that's not possible.

The last move by black couldn't have been e6 because then the black bishop could have never got out (you also can't promote to a bishop for lack of captures afterwards). Then it must have been Nxg4 - which means that the white dark-square bishop was captured.

Black had played dxe6, then white got his d pawn to d7, then black played exd6 (the dark-square bishop) followed by d5 to allow white's c pawn to play cxd6.

But now we have a new problem, since d6 is a dark square, so black couldn't have used his light-square bishop. This means that the position is in fact illegal.

Edit: Also worthwhile to mention that Kxa2 fails for the same reason mentioned above.

The second position is also illegal.

Black has absolutely no moves he could have just played, other than Kxa2. But then white must have promoted his g pawn - in return, there must have been atleast 4 captures by white. However black is only missing 3 pieces.

It also couldn't have been Ba2+  Kxa2, since black will then have no moves prior to that.

Position 2 of post #249 by cobra91 is illegal if it is White to move. Black's only previous moves could have been: -

1) A capture by a black pawn on f7 to e6 - not possible as the black f pawn is now on h5.

2) Advancement of a black pawn from h6 to h5 - not possible as the h5 pawn must have traversed diagonally from e6 where a white knight now resides.

3)  A move by the black king from a1 or b1 to a2 - not possible. The reason for this is that the move preceding this must have been a castling move by White due to the position of the White rook on f1. But this implies that the White king would have been on e1 all the time prior to this and there is no way that the Black king could have got to the first rank if this was the case (the White king on e1 would have protected the d2 square - the only avenue of approach for the Black king, given White's pawn structure).