Henry Ernest Dudeney/Modern Puzzles/172 - A Magic Square Delusion/Solution

Modern Puzzles by Henry Ernest Dudeney: $172$

A Magic Square Delusion

Here is a magic square of the fifth order.

$\qquad \begin{array}{|c|c|c|c|c|} \hline 17 & 24 & 1 & 8 & 15 \\ \hline 23 & 5 & 7 & 14 & 16 \\ \hline 4 & 6 & 13 & 20 & 22 \\ \hline 10 & 12 & 19 & 21 & 3 \\ \hline 11 & 18 & 25 & 2 & 9 \\ \hline \end{array}$

I have found that a great many people who have not gone very profoundly into these things believe that the central number in all squares of this order must be $13$.

One correspondent who had devoted years to amusing himself with this particular square was astounded when I told him that any number from $1$ to $25$ might be in the centre.

I will show that this is so.

Try to form such a magic square with $1$ in the central cell.

Solution

Make $9$ squares like this one and put them together to form a larger square:

$\qquad \begin{array}{|c|c|c|c|c|} \hline 9 & 11 & 18 & 5 & 22 \\ \hline 3 & 25 & 7 & 14 & 16 \\ \hline 12 & 19 & 1 & 23 & 10 \\ \hline 21 & 8 & 15 & 17 & 4 \\ \hline 20 & 2 & 24 & 6 & 13 \\ \hline \end{array}$

Then you can pick out a $5 \times 5$ square in any position and it will be a magic square.

Hence it is seen that you can arrange for any number you like to be in the centre.

This is what is known as a Nasik square (named by Andrew Hollingworth Frost after the place in India where he was living).

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $172$. -- A Magic Square Delusion
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $385$. A Magic Square Delusion