This is an interesting problem I found.
Let there be a 2-digit sequence that can start with 0, like 04 or 93. Let a "nudge" be defined as exactly one of the following operations:
1) Increasing one of the digits by 1.
2) Decreasing one of the digits by 1.
3) Changing a 0 to a 9.
4) Changing a 9 to a 0.
For example, possible nudges of 19 are 09, 29, 18, and 10.
Say that $S$ is some set of 2-digit sequences so that it takes 3 or more nudges to transform any element of $S$ into some other element of $S$. What is the maximum possible number of elements of $S$?
I thought of a nudge as a knight moves on a chessboard, but with a 10x10 board numbered starting with 1 on the top left and 100 in the bottom right. Clearly the maximum is $100-1=99$ nudges, but that is without considering the extra condition in the problem statement. Any idea how to continue?