  The exact formula for the average hitting time (HT, as an abbreviation) of
simple random walks from one vertex to any other vertex on the square $C^2_N$
of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of
Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author
gives the expression for the even $N$ case and the expression for the odd $N$
case separately. In this paper, by using an elementary method different from
Chair (2014), we give a much simpler single formula for the HT's of simple
random walks on $C^2_N$. Our proof is considerably short and fully
combinatorial, in particular, has no-need of any spectral graph theoretical
arguments. Not only the formula itself but also intermediate results through
the process of our proof describe clear relations between the HT's of simple
random walks on $C^2_N$ and the Fibonacci numbers.
