Given a $2$-digit sequence (which can start with $0$), let us define a nudge as the operation of either increasing or decreasing one of the digits by $1$, or changing a $0$ to a $9$ or a $9$ to a $0$. For example, we can nudge $19$ to get $09$, $29$, $18$, or $10$.
Suppose $S$ is a set of $2$-digit sequences such that it takes at least $3$ nudges to transform any element of $S$ into another element of $S$. What is the maximum possible number of elements of $S$?
I know I have to use either One-to-one Correspondences or Pigeonholes to solve this problem, but I really have no idea.