VULPES:
Is this position legal?

i am pretty sure this one is illegal.
Obviously. Black is to move as he is in check, and what was White's last move before which the check did not exist ? Bf8-f1 ?

#175:
1st one is legal, as the number of required captures is fewer than the missing pieces. Quite the mess though.
2nd one is illegal, as the bBh8 can't have gotten there. If the board was flipped, it is legal as 8 captures are needed and 16 men missing.
3rd one legal (almost trivially, but it's just slightly more interesting than that.) I give a simple (but slightly tricky) puzzlette based on it:
Is this legal with white to move? What about black to move?

#179, White to move:
Black to move is illegal. The reason is that whichever knight pair starts, they will both need either an even or an odd number of moves. If knights of the same colour start, they must both reach a knight of the opposite colour; the reverse is also true. Since it's impossible for a knight to reach a particular square in both an even and an odd number of moves, and because the rooks can only move one square backwards and forwards (ie always an even nummber of moves), White can't lose or gain a tempo and will always be first to move once the position is reached.

#179, White to move:
Black to move is illegal. The reason is that whichever knight pair starts, they will both need either an even or an odd number of moves. If knights of the same colour start, they must both reach a knight of the opposite colour; the reverse is also true. Since it's impossible for a knight to reach a particular square in both an even and an odd number of moves, and because the rooks can only move one square backwards and forwards (ie always an even nummber of moves), White can't lose or gain a tempo and will always be first to move once the position is reached.
And you fell into the trap here. Black to move IS legal. What did you overlook?
Hint: Obviously white and black did not mirror each others' moves. How to break the parity - what is the square at which the mirror breaks?

Short explanation of Remellion's latest : (in white)
I already mentioned this sooner. The key is you can win a tempo by choosing the square on which you capture the knight correctly.
Of course #175 is much simpler, as there is ...d6-d5 vs ...d5 or queen triangulation possibilities.
yes, no illegal moves were made that I could see.... silly, yes, legal, yes.
Edit that was odd, no responses when I typed, 10 pages after, something glitched.

1 and 2 trivially legal. 3 and 4 illegal because of the a4- and a8-queens both checking the black king. 5 and 6 trivially legal. 7 illegal because of the bishops.
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I finally managed to uncook an older puzzle. This is a triple bill: Is the position legal 1) as is; 2) with the g2-pawn on g3; 3) with the g2-pawn on h3?
Seems trivially legal.
Last move was dxe6+