Maths: Magic squares

Artist Albrecht Dürer knew about magic squares, and showed one in the top right corner of his engraving Melencolia. A magic square is one in which all of the rows and all of the columns add up to the same number. 

Two magic squares, one of which you saw.

A magic square is one in which all of the rows and all of the columns add up to the same number. In truly excellent magic squares, the diagonals sum to the same value, like the example on the right, above. There’s more about this one at the very end of the chapter, because it appeared in an engraving by Dürer.

It is possible to make much larger magic squares. The 3 x 3 example is not very elegant: can you do better, and produce one with those numbers and with diagonals summing to the same value?

An 8 x 8 magic square.

Benjamin Franklin created the 8 x 8 square above. Each row adds to 260, and stopping halfway along each row makes 130. Going diagonally up four, across one, and down four also adds to 260 (that is, 9 + 58 + 59 + 12 + 21 + 38 + 39 + 24 = 260). As well, the four corners and the four middle numbers add to 260, and the sum of the numbers in any four-box square is 130.

Any four numbers lying diametrically equidistant from the centre also add to 130: try this with the four corner numbers (52, 45, 17 and 16) or with the four centre numbers (54, 43, 23 and 10). If you examine the patterns and balances in his solution, maybe you can work out how he did it.

* Design a 4 x 4 magic square, different from the example shown above. (If you count rotations and reflections as one, there are 880 different 4 x 4 magic squares.)

* Design a 3x3 magic square with a 5 in the middle, leaving out 3 and 7, but using 10 and 0 instead. (It can be done!) That is, you use 0, 1, 2, 4, 5, 6, 8, 9 and 10. These add up to 45, which tells you that your target sum is 15, and that’s all the help you get!

* Is it possible to construct a magic square of any sort, using only prime numbers? I suspect that it is not possible, but you may be able to prove me wrong—or you may be able to prove me right. I really didn’t know the answer, but a quick internet search in April 2025 reveals a whole mob of solutions, and Wolfram knows them (that's a hint).

* In what year did Dürer engrave Melencolia? Use the internet to find out, then find that year in the engraving. Old Albrecht was a cunning chap, and well worth an hour or two of your time.

Notes

Here is 3 x 3 square where both the diagonals add to 15.

4	9	2
3	5	7
8	1	6

 Something to explore

* Look at the pattern formed by the 1, 2 and 3 in the 3 x 3 example, and at the pattern formed by the 7, 8 and 9. This should tell you something.

* Can yopu find a similar pattern in larger magic squares?

To search this blog, use this link and then use the search box.

Another way: use the index!