2006 Gauss 8 Problem 25

In the diagram, the numbers from 1 to 25 are to be arranged in the 5 by 5 grid so that each number, except 1 and 2, is the sum of two of its neighbours. (Numbers in the grid are *neighbours* if their squares touch along a side or at a corner. For example, the "1" has 8 neighbours.) Some of the numbers have already been filled in. Which number must replace the "?" when the grid is completed?

\begin{array}{|c|c|c|c|c|} \hline & & & 20 & 21 \\ \hline & 6 & 5 & 4 & \\ \hline 23 & 7 & 1 & 3 & ? \\ \hline & 9 & 8 & 2 & \\ \hline 25 & 24 & & & 22 \\ \hline \end{array}

\textbf{(A)}\ 15\quad \textbf{(B)}\ 12\quad \textbf{(C)}\ 14\quad \textbf{(D)}\ 11\quad \textbf{(E)}\ 13

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 Gauss contests here.