White’s queen is under attack. The obvious impulse would be to move it to prevent RxQ, but that would be premature. First take stock of your offensive possibilities. Here you have a knight to work with; it's on a dark square, and so are Black’s king, queen, and rook—and the queen and rook can be forked by Nxf7. But the needed square is guarded by Black’s king. Ask whether the king is protecting anything else White can take, and see that it guards the Black rook. So: 1. QxR+ (extinguishing the threat with a temporary sacrifice), KxQ; 2. Nxf7+ and when the smoke clears White will have gained a rook.

Here as in the previous problem, the exchange at the beginning both made the fork possible and improved it by turning the guardian of the forking square into a target. Notice that Black really has no choice but to take White’s queen with his king; if Black’s king merely moves out of the way, White plays QxQ—a formation known as a skewer (you attack two pieces on the same line; the more valuable gets out of the way, leaving the less valuable to be taken).

There are other little trains of thought that might have brought you to the solution here. You could have just examined any checks you are able to give and then realized that 1. QxR+, KxQ leaves Black vulnerable to a fork; or you might have seen that White can attack the Black queen via Nxf7, and wondered whether there was a way to draw Black’s king onto another dark square that also could be attacked from f7. Equivalently, you might have noticed that the Black king and queen are both on dark squares, but on the same diagonal one square apart—an arrangement that cannot be forked. Dragging the royal couple apart by an additional square makes the fork work.