...without moving to the same square twice.
When you found this did you try it? and if so, what was your score/results? Or did you work out the equation first?
I first attempted this challenge about 40 years ago, using a chessboard and pistachio shell halves to mark the squares I had moved to. I guess pawns would have been easier to use but I didn't have 63 pawns handy at the time.
I gave it a shot from time to time over a year or two and never managed to get to the end without having to double dip on at least one square. I recall having read in books that it is possible, though, so I never forgot about it.
A few months ago, I noticed this feature on Maths Is Fun and didn't try it because I had been frustrated with it long ago (like in 40 years ago). But just the other day, I had a little time and gave it a shot. The computer option is far superior to keeping track using a physical board because the latter practically necessitates using two hands, one to move the knight and the other to place a marker on the square evacuated. It's really distracting to do that. But with a computer program you get a lot of continuity, and the memory of what pattern you tried last time and the time before and the time before, etc., up to twenty or thirty attempts, is far more easily retained than when you spend 5 or more minutes doing it manually. With the computer, 30 seconds is often enough, and if you find yourself stuck halfway, 15 seconds may suffice. So you get a lot of practice in a short time, and you learn faster that way.
In answer to your question, when I first tried it a couple weeks ago, I got very close to success right away, but it did take about a day or two (2 hrs per day) before I finally succeeded, and it was a great feeling of victory when I did. I was so thrilled I had to share it somewhere, so CathInfo was my victim!
I was getting scores of 65 to 70 quite often, but a few 64. Finally I got 63, and then using the principles I outlined above, I was able to get 63 again and again, about half the time, along with 64, 65, 66 and 67.
As for "equation," I never really discovered what the mathematical formula is for the solution, although I expect there must be one, somehow. It's a very logical and structured sequence so I'm sure it's possible to set up an equation of some sort. But I have no clue whatsoever how to do that.
BTW -- the color of the squares might make a difference for some players; so that you know, the accepted norm for chess boards is: square a1 is black and square h1 is white.