Fastest solution (found by 
, but congrats also to 
for mentioning the winning move 1. Rh5! first):
1. Rxh5! gxh5 (Other moves lose as well.) 2. Bh6! Nf1 (2. ... h4, which nobody suggested, extends the check mate a little bit.) 3. h4! Ng3 4. e4 Nf1 5. b5 Ng3 6. c5! dxc5 7. b6 cxb6 8. d6 c4+ 9. Kd4 Ne2+ 10. Ke3 Nc3 11. d7 Nd1+ 12. Kd4 c3 13. d8R#
Final position:
RE: Chess problem 37 / Schachaufgabe 37