Have you ever tried the knight’s tour on a chessboard? Simply place your knight on one of the four corner squares (the easiest place to start) and try to visit each square on the board only once. An interactive board can be found here and it will color the squares you’ve already visited. A perfectly-completed knight tour will take sixty-three moves, assuming you allow yourself to finish on any square. On my first try I hit fifty-eight squares and, interestingly, on another attempt I hit every square except for one. The universe can be quite a cruel prankster. The knight’s tour hasn’t yet improved my chess, but it has actually made me think about my own chess thought process more critically.
When in doubt, try thinking inductively. What needs to be left near the end for the problem to be complete?
What is your organizing principle? What are your limitations?
What are the rules of the game? If I start in the corner of a square chess board will I automatically be able to solve the knight’s tour for any smaller square? The answer to this is quite interesting and complex.
When do you have your first opportunity to go wrong? Perhaps you can make an inference about what would render your task would impossible, and can avoid computing whole variations.
When are moves that look identical choices of pure aesthetics and when is one actually better than the other? Think of how many opportunities there are to find interrupting moves and tiny finesses in chess. Each of them may or may have a particular merit, but unless you want a nasty surprise sometime you’ll at least have to take heed of them.
What can you learn from a given exercise? When you’re doing a chess problem or looking at a particular chess position what are you actually seeing? If you don’t know what you’re looking for you will not actually learn anything. Are you paying attention to the endless series of triangles you’re forming or are you focused on calculating your way through the tour?
I only solved the knight’s tour once I figured out the rules governing it. So are you just going to strand your poor horse over and over again or are you going to come up with a plan?