The following game was included in Attacking Strategies for Club Players by GM Michael Prusikin, which I'm currently working through during lunch hours at work (when I can take them, that is.) While calculation obviously played a large part in this tactical masterpiece, the final winning idea by Black (GM Eduard Gufeld) is something that I certainly never would even have begun to calculate when trying to generate candidate moves. This highlighted the insight that calculation is not enough - one has to first have the necessary ideas in mind, then recognize that the position in front of you may allow them to work. (The related training quote of the day #47 is relevant as well.)
In this case the sequence beginning on move 24, which uses the concepts of clearance and deflection sacrifices, is particularly striking. The necessary move of simply hanging your bishop (move 25) is highly unlikely to occur to anyone employing a "brute force" type of thought process. Understanding that this move will both clear the 8th rank for the queen and divert the protection of the c4 square are both necessary ideas. Clearance sacrifices in particular I think are a sign of mastery, and I expect will be a separate topic in the future.
[Event "URS-ch41 Semifinal"]
[Site "Kirovabad"]
[Date "1973.06.??"]
[Round "?"]
[White "Bagirov, Vladimir"]
[Black "Gufeld, Eduard"]
[Result "0-1"]
[ECO "E84"]
[PlyCount "64"]
[GameId "270424022216"]
[EventDate "1973.06.06"]
[EventType "tourn"]
[EventRounds "15"]
[EventCountry "URS"]
[SourceTitle "EXT 2009"]
[Source "ChessBase"]
[SourceDate "2008.11.26"]
[SourceVersion "1"]
[SourceVersionDate "2008.11.26"]
[SourceQuality "1"]
1. d4 g6 2. c4 Bg7 3. Nc3 d6 4. e4 Nf6 5. f3 O-O 6. Be3 Nc6 7. Nge2 Rb8 8. Qd2 a6 9. Bh6 b5 10. h4 e5 11. Bxg7 Kxg7 12. h5 Kh8 13. Nd5 bxc4 14. hxg6 fxg6 15. Qh6 Nh5 16. g4 Rxb2 17. gxh5 g5 18. Rg1 g4 19. O-O-O Rxa2 20. Nef4 exf4 21. Nxf4 Rxf4 22. Qxf4 c3 23. Bc4 Ra3 24. fxg4 Nb4 25. Kb1 Be6 26. Bxe6 Nd3 27. Qf7 Qb8+ 28. Bb3 Rxb3+ 29. Kc2 Nb4+ 30. Kxb3 Nd5+ 31. Kc2 Qb2+ 32. Kd3 Qb5+ 0-1