The solution is 1 Qd5+ Kh8 2 Nf7+ Kg8 3 Nh6+ Kh8 4 Qg8+ Rxg8 5 Nf7#.
The Point of Chess Problems Although chess problems are interesting in their own right, the main reason I do them is to get tactical ideas for actual play. Solving a chess problem is quite different from making the next move in a game because we know that the chess problem has a solution, a clever tactic that wins quickly, either by gaining material or mating. Occasionally, situations like this arise in actual games - the trick is to recognise them. Even better is to plan for and create them. This position illustrates a classic tactical idea, the smothered mate. If you are unfamiliar with it then see whether you can show how White wins.
The solution is 1 Qd5+ Kh8 2 Nf7+ Kg8 3 Nh6+ Kh8 4 Qg8+ Rxg8 5 Nf7#.
Note that Black is better off giving up the exchange by playing 2… Rxf7 rather than being mated. The point of doing problems like this is to recognize patterns that win. In this case it is that White’s queen can check on the white diagonal, that Black cannot interpose a piece without losing material, and that the white Knight covers the square f7.
So as White we would aim for such a position when Black has no pawn on f7.