The Questions are as important as the Answers
The right question to ask here would be, 'What would my opponent play if it were his move' is a good place to start your thought process and with a little bit of imagination, you might be able to arrive at the solution.
The position is deceptively simple and the sheer number of possibilities/ candidates make it difficult for white to arrive at the right move. Especially, as only one move achieves the goal of mating in three moves.
Instead of forcing mate on the opponent, it helps to get into black's head and see 'Where the black king would try to escape if it were his move?'
During a practical game a strong player would think only when something is bothering him, only when he is trying to find an answer to a question in his mind.
Carlsen Magnus-Hikaru Nakamura Tata steel 2011.
White aims to play Nd4 here and the question to ask is 'What does black intend against Nd4?'
If you can’t find the solution of a puzzle after prolonged effort and thought, don't fret as you can still find the question that arrives at the answer.
2.Black to play the best way for him to avoid quick mate would be take the king to d4.So white in preparation for that play 1.Rf1!! Kd4 2.Qd3 Ke5 3.f4#.A truly extraordinary study.
3.Black achieves some counterplay after 1. Nd4 with 1...Nbc4 2.Qe1 Qb6 3.Bc1 Na3 4.Ka1 Qxd4! White avoids all of these complications with 1.Ka1 !