Hello all, the Knight’s tour is an interesting issue of calculation, visualization and memory. It’s one I have wondered about; but I will throw it open to the master mnemonics Mavens and magnificent mathematical Moguls who can help out!
Does anyway know of a mnemonic way to get around the board? I have tried using a Dominc PAO to encode squares; but it gets pretty confusing pretty fast.
Are you trying to memorize a knight’s tour path? If you use a notation that is just numbers (1-8 for columns and 1-8 for rows), then each square would be a 2-digit number. Then you would just have a 128-digit number to memorize (kept in order with a memory palace).
I used to do this ‘trick’ at times. I used the story-telling approach (rather than loci/memory palace) with a variety of characters/actions en route. Depending on what numbered square someone chose as the starting point, I simply jumped into the story at that point. Once I’d done it a few times, it became very easy to remember.
The sillier the story the better!
The only downside is that it does take time to get round the board - so you need a committed/interested audience! 
I’m not sure if memorizing the path is the way to go here. But if anybody insists on that approach, you can cut @Josh’s approach in half by considering the fact that the knight cannot randomly move to any of the other 63 squares.
In fact, there is only 8 moves, so every two moves gives you a 2-digit number for a total of a 64-digit number rather than Josh’s 128-digit number. Just number them like on a clock face. You can’t move to 12, 3, 6, and 9 and then one to the right and two up is 1 and one up and two to the right is 2, etc.
Either way, it’s overkill… just cut the board in half vertically and horizontally to get four 4x4 boards and memorize the patterns for:
…and, you’re D-O-N-E done!!! That’s not a joke. Just see which kind of system your piece is on and complete the moves in your 4x4 and go into the next 4x4. Decide on clockwise or counterclockwise – one gets you stuck the other let’s you move into the next 4x4. Rinse and repeat twice more. After completing the last 4x4 you randomly choose your next system. Diamond always follow square systems and vice versa.
Say you start on a left square system, your next system could be either left diamond or right diamond, followed by right square. If you picked left diamond before, the last one will be right diamond or vice versa. Either way, you’ll complete all four systems in the end.
Nice (<10 mins) video here: https://youtu.be/ChiOwpOjH_4 explaining the four systems and moves described above. Equally nice video here: https://youtu.be/dWM5pKYZCHw building on the first one solving the advanced knight’s problem where not just the starting square but also the end square is given.
In summary, you need to rememeber:
I assume that choosing to move clockwise or counterclockwise within a 4x4 is straight forward and needs no memorizing because the rule is: one gets you stuck, so pick the other one that doesn’t.
Much better than memorizing a path… esspecially since there are 26 trillion different ways of solving the knight’s problem. 
That seems pretty cool. As James says below someone can choose any of the 64 squares, so how would one account for that. I suppose it would be sixty-four 128 number stories. Sounds doable.
Yahoo. Very clear and very elegant.
Nelson Dellis just did a tutorial on this puzzle today, if anybody is interested…
