Question 173
Jimmy the grasshopper is sitting on a corner square of a rectangular $9 \times n$ board, where $n \ge 8$. To describe the hops Jimmy can make, we label the squares of the board by pairs of positive integers $(a, b)$, where $1 \le a \le 9$ and $1 \le b \le n$.
Jimmy can hop in both directions between squares $(a, b)$ and $(a', b')$ if either
$$
\begin{gathered}
b' \equiv b + 7 \pmod{n} \quad \text{and} \quad a' = a + 2, \quad \text{or} \\
b' \equiv b + 2 \pmod{n} \quad \text{and} \quad a' = a + 7.
\end{gathered}
$$
How many squares of the board can Jimmy reach by hopping repeatedly?
Jimmy can hop in both directions between squares $(a, b)$ and $(a', b')$ if either
$$
\begin{gathered}
b' \equiv b + 7 \pmod{n} \quad \text{and} \quad a' = a + 2, \quad \text{or} \\
b' \equiv b + 2 \pmod{n} \quad \text{and} \quad a' = a + 7.
\end{gathered}
$$
How many squares of the board can Jimmy reach by hopping repeatedly?