An Ancient Magic Square

Place the numbers $1$ to $16$ in a four-by-four grid so that every row, every column and both main diagonals add to the same total. The old Indian square below does this and a good deal more besides. What is the total, and what are the “and more” properties?

Solution

The numbers $1$ to $16$ add up to $\tfrac{16 \times 17}{2} = 136$. Four rows share that sum equally, so each must total $136 / 4 = 34$. The square $$\begin{array}{|c|c|c|c|} \hline 7 & 12 & 1 & 14\\\hline 2 & 13 & 8 & 11\\\hline 16 & 3 & 10 & 5\\\hline 9 & 6 & 15 & 4\\\hline \end{array}$$ has every row, every column and both diagonals summing to $34$. But more is true, and all of it can be checked by eye:

• the four corner cells, $7 + 14 + 9 + 4 = 34$;

• the central two-by-two block, $13 + 8 + 3 + 10 = 34$, and likewise each two-by-two block at the four corners;

• the “broken” diagonals that wrap around the edges, such as $12 + 8 + 5 + 9 = 34$ and $1 + 11 + 16 + 6 = 34$.

A square whose broken diagonals also share the magic total is called pandiagonal, and this one is the most celebrated example, carved on a temple wall in Khajuraho. Many four-by-four magic squares exist, but only the pandiagonal ones carry all these extra balances at once.