The skill called for in that last problem—the ability, when you examine a check or other forcing move, to keep the resulting appearance of the board clear in your mind’s eye—is one of the keys to good chess. It gets even more important as we turn to positions that require you to follow up on a first check with a second one before the fork is ready. Gradually your ability to recognize an emerging forking pattern will kick in as you are examining the checks and follow-up checks available to you. By working back and forth between forcing moves and glimpses of patterns, you build a combination.
Begin with the leftward position. Anytime you have a battery of rooks on an open file like this, consider what would happen if you drove them both through to your opponent’s back rank. Sometimes the result may be a mating sequence there; even if not, though, the threat is powerful enough to force results—and forced results always have to be inspected in search of forks or other tactical opportunities they may create. Thus White imagines Rc8+, to which Black would reply NxR; then comes RxN+, and Black’s king is forced to f7. If you were looking only for checkmate you would have to consider the sequence a failure, since the king escapes. But if you're looking for a tactic the sequence is a spectacular success, as it leaves Black’s king and queen both on light squares and ready to be forked with Ng5+. White wins a queen and a knight for a rook and ends up with two attacking pieces on the board against none.
Notice how remote the chances for a knight fork by White appear to be on the face of this position; the knight on f3 just seems too far away from Black’s king. It’s a study in the importance of reevaluating such possibilities whenever you can make the enemy king move. A useful rule of thumb is to ask every time the king moves whether you have any new checks against it. Here that would turn up Ng5 on White’s third move.