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Before the emergence of the modern chess problem in the mid-19th century conditional problems were common, and this is one of the best examples. The mating check must be delivered by the pawn.
My apologies to solvers. I took the problem from a source which gave an
incomplete stipulation. The proper stipulation should read “Mate in 5 with
the pawn, without capturing the rook”.
Solution: With Black ready to capture the pawn it would appear that a sequence of checks is called for, so after 1.Qc8+ Rb8 the quiet continuation 2.Ke7 is not so easy to spot. There follows 2...Rxc8 3.Bg2+ Kb8 4.Ba7+ Kxc7 5.b6 mate. Without the condition forbidding capture of the rook there is a mate in 4 by 1.Qa4+ Kb8 2.Rc8+ Kb7 3.Qa6+ Rxa6 4.bxa6 (or 1...Ra6 2.Rc8+ Kb7 3.bxa6.) My thanks to the solvers who pointed this out.
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