Friday, December 16, 2016

More on Magic Squares

The lack of recent posts is due to a fatal computer crash which prevented me from going online. However I had the opportunity to look at magic squares more closely. There is an Indian magic square from the 11th or 12th century in which there are more sums that turn out to be the same and only the integers 1 through 16 are found in the square.  Dürer's Melancolia from 1514 contains a similar magic square. Following some simple rules I was able to deduce another one that is similar to Dürer's along the bottom row but with some of the other numbers rearranged.

There are at least 16 ways in these magic squares in which one can arrive at a sum of 34.

If one looks at the list of sums one will notice that a pair of sums occurs in more than one equation so one can therefore subtract these equations to find two pairs whose sum is equal.

I found 18 pairs with identical sums.

These pairs of sums add structure to the magic squares and limit the choices for filling a magic square.

Supplemental (Dec 18): Corrected misstatement about my magic square being Dürer's flipped.