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This problem by Hans-Peter Rehm won first prize in the composing competition at the 1960 Chess Olympics. It is a fine example of the sort of effects a good problemist can create. It is White to play and mate in seven.

Black's king is caught in a tight net, but the only spare white man with which to create mating threats is the f-pawn. But how do we get the king out of its way? Let's try 1.Kg8, threatening f8=Q mate. Black must play 1...Rg2+, which seems to get him out of trouble since 2.Kh8 or Kh7 takes the protection away from f7. Ah, but what about the clever 2.Ng4 Rxg4+ 3.Kf8? White is now threatening Bxe7 mate, and if the rook returns to the e-file with 3...Re4, then it obstructs the bishop on c2 and allows Rxf5 mate. That's fine as far as it goes, but what if Black defends e7 with the other rook? After 3...Re3! White has no good continuation.

So let's try the other direction, starting with 1.Ke8. Now the threat of promoting the pawn forces 1...Ba4+. White could try 2.Bd7 Bxd7+ 3.Rxd7, but now the e5 square is unguarded and Black's king can make a bolt for it and survive past move seven.

The only other idea is 1.Ke8 Ba4+ 2.b5, when after 2...Bxb5+ we can return with 3.Kf8 having created the threat of 4.Rxf5 mate. And 4...Re5 is mated after 5.Bxe7+ Rxe7 6.Rxf5. The trouble is that Black's bishop can now return to d3 defending f5 just as effectively as it did from c2.

You now have all components needed for the answer. There is one important respect in which the bishop on d3 differs from the bishop on c2, and we have already seen the variation in which that difference is important. Are you with me? Here's the full answer: 1.Ke8 Ba4+ 2.b5 Bxb5+ 3.Kf8 Bd3 4.Kg8! Rg2+ 5.Ng4! Rxg4+ 6.Kf8. Now the rook on a3 cannot reach e3 and 6...Re4 (to stop Bxe7 mate) allows 7.Rxf5 mate. Bravo!