In many cases it's more simple and less frustrating just to write from scratch instead of figure out what the problem is.

Well, ... not sure I agree. But to make everyone happy, I've done some renaming.

there is nothing to replace because None gets returned if the path is not valid

This is most commonly used in electrical engineering.
Instead of incrementing bits (representing switches) by their integer value, you want to increment them so that only one bit/switch changes.
That way you prevent two switches to change at the "same" time, resulting in inaccurate measurements.

This is called "single-distance codes".

The Gray Code is one possibility to arrange your bit range exactly like that and can be used in accurate position encoding.
In this kata we wrote a function to give the integer value of the numth position of the gray code.
Though I'm not sure what that integer would be used for.. the inverse does give the current position during incrementation.

Assume num has n bits. Then mystery(num) is the number whose binary representation is the numth entry in the table T(n), where T(n) is defined recursively as follows:

The pre defined function takes n as an argument, not num. The descrepancy between the description and the pre written code is a bit confusing.

very clear solution, thanks

eliminate all bugs from the supplied code

Just writing new code is not the intention of this kata I think.
Most pythonic solution to the task itself, though.