Question 165
Let $S$ be the set of two-digit numbers that do not contain the digit $0$. Two numbers of $S$ are called friends if their largest digits are equal and the difference between their smallest digits equals $1$. For example, $68$ and $85$ are friends, $78$ and $88$ are friends, but $58$ and $75$ are not friends.
Determine the largest integer $m$ such that there exists a subset $T$ of $S$ with $m$ elements, such that no two elements of $T$ are friends.
Determine the largest integer $m$ such that there exists a subset $T$ of $S$ with $m$ elements, such that no two elements of $T$ are friends.