22. Two numbers in the $4 \times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).
What is the sum of these two numbers?
A 12
B 15
C 22
D 26
E 28
\begin{tabular}{|c|c|c|c|}
\hline 9 & 6 & 3 & 16 \\
\hline 4 & 13 & 10 & 5 \\
\hline 14 & 1 & 8 & 11 \\
\hline 7 & 12 & 15 & 2 \\
\hline
\end{tabular}
Solution: $\mathbf{E}$
We have included, in italics, the row and column totals. The circled numbers are the totals of the numbers in the two main diagonals.

The sum of all the numbers in the square is $1+2+\ldots+15+16$. This is the 16th triangular number, $\frac{1}{2}(16 \times 17)=136$. So the total of each row, column and main diagonal should be $\frac{1}{4}(136)=34$.
We thus see that we need to increase the totals of row 2 and column 2 by 2 , and decrease the totals of row 4 and column 3 by 2 . Clearly, the only way to achieve this by swapping just two numbers is to swap 13, in row 2 and column 2, with 15 in row 4 and column 3 . When we swap 13 and 15 the total on the main diagonal from top left to bottom right is increased by 2 to 34 , while the total on the other main diagonal is unchanged. So by swapping 13 and 15 we create a magic square. $13+15=28$. The discussion above makes it clear that this is the only swap that makes the total of each row and column, and the main diagonals, equal to 34 .