My son was very proud of himself yesterday evening after completing the knight tour without any assistance. I had shown this to him a few months back as a fun puzzle but it’s only recently that he got really interested. And in my view it’s a great improvement tool.
How does it help? I believe that it develops perception of the board as a whole as well as expert handling of the trickiest of all the pieces. Even very strong players can have difficulty seeing backward knight moves, though I don’t think they’d have this problem if they’d done knight tours as children.
There are several other knight puzzles that I think are helpful, the first being to manoeuvre a knight round to visit every square of the chess board but allowing multiple visits to each square and doing the squares sequentially. So you bring it from a1 to b1 and then c1, d1, e1, f1, g1 and h1. After that you go to h2 and then all the squares on the second rank until you get to a2.
This can be enhanced by placing Black pawns on f3, f6, c3 and c6 and then disallowing the knight to visit any square that is either occupied or attacked by one of the pawns. They say that a young and talented GM could probably do this in about 2 minutes, not that I’ve ever managed it at such speed!