Hi, I think there is a likelihood that this is similar to proving the Collatz conjecture and would be very hard to prove. Any time numbers and divisibility are involved, you enter a realm of dragons and monster spiders where you can easily choose to spend your entire lifetime trying to solve something impossible.

As for “maximum steps”, I think in general for knights that jump a large n and then only “1” in the other direction, the amount of steps will tend to grow larger and larger — but there will always be exceptions. Again, I have no proof. I would love to converse more about this, and hope maybe Sloane reads the article once and your comment to come up with any insight.