2011 Cayley Problem 23

A *Fano table* is a table with three columns where

- each entry is an integer taken from the list 1, 2, 3, \ldots, n, and - each row contains three different integers, and - for each possible pair of distinct integers from the list 1, 2, 3, \ldots, n, there is exactly one row that contains both of these integers.

\begin{array}{|c|c|c|}\hline 1 & 2 & 4 \\ \hline 2 & 3 & 5 \\ \hline 3 & 4 & 6 \\ \hline 4 & 5 & 7 \\ \hline 5 & 6 & 1 \\ \hline 6 & 7 & 2 \\ \hline 7 & 1 & 3 \\ \hline \end{array}

The number of rows in the table will depend on the value of n. For example, the table shown is a Fano table with n = 7. (Notice that 2 and 6 appear in the same row only once, as does every other possible pair of the numbers 1, 2, 3, 4, 5, 6, 7.) For how many values of n with 3 \le n \le 12 can a Fano table be created?

\textbf{(A)}\ 2\quad \textbf{(B)}\ 3\quad \textbf{(C)}\ 5\quad \textbf{(D)}\ 6\quad \textbf{(E)}\ 7

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.