This one asks: White to play and mate in how many?
The obvious reply is three: 1.Rxb8 hxg2 (or 1...Kxb8 2.Kb6 followed by 3.h8=Q) 2.Rb7+ Kc8 (or d8) 3.h8=Q mate. However, with Loyd the obvious is usually wrong. It's time for a little detective work:
Black has clearly made two pawn captures: hxg6 and exf6. So his pawn on h3 must have arrived there from d7, making captures on e6, f5, g4 and h3. That's six black pawn captures accounted for.
A white bishop must have perished on f1, since the pawns have never moved from e2 and g2. So we have now accounted for every missing white man.
White must also have made at least six pawn captures, with the pawn on h7 having travelled from the d-file at least.
The missing white pawn must have promoted, for it is among the six white men captured, and can never have drifted far enough to the K-side to have lost its life while still a pawn. And since the pawn on b5 must have started life on b7 (we've accounted for all the white pieces captured), no white pawn can have promoted without making at least one capture.
We are now in a position to work out Black's last move.
Well it wasn't with the h-pawn, which we know arrived there from g4. It can't have been Ba7-b8 or b6-b5, because White would have been in check. And it can't have been exf6 because that would mean the bishop on b8 is a promoted piece (it could never have emerged from f8) and no black pawn can have promoted without exceeding the number of possible captures by one side or the other. Black's king cannot have just moved out of check from c8 or d8, for the white rook could not legally have moved to h8 to deliver the check. Finally, if the king moved from d7 or b7, then White's last move must have been d5xc6, which, bearing in mind the pawn on h7, makes too many captures.
There remains only one possibility: Black's last move was b7-b5. White mates by capturing en passant, axb6 mate!