Question 173
There are several identical caro papers (square-grid papers) of size $5 \times 5$. Someone uses $n$ colors to fill in each paper such that two cells at the same position on two sides share the same color. Two papers are considered congruent if they can be stacked together in such a way that the pairs of squares at the same position have the same color. Prove that, by the definition of congruence, one can obtain at most
$$
\frac{1}{8}(n^{25} + 4n^{15} + n^{13} + 2n^{7})
$$
distinct colored caro papers.
$$
\frac{1}{8}(n^{25} + 4n^{15} + n^{13} + 2n^{7})
$$
distinct colored caro papers.