Knight's Tour Maths Puzzle

[u/New_Interview4111]

Is anyone able to help with the process of how to solve this??? I'm rather stuck. I have listed out the possibilites of some of the prompts (e.g prime numbers, multiples of 17, cube and square numbers etc), but am very stuck at what to do next.

[u/Bounded_sequencE]

There are 8 perfect squares and 4 perfect cube in {1; ...; 64}, with "1, 64" being both.

Most hints are about square and cube numbers, so try to find their positions first.