## Aug 31 Benjamin Franklin's Magic Square

Benjamin Franklin was brilliant at basically everything. While he may be more famous as a founding father of the USA, a slavery abolitionist or his exploits with a kite (which may or may not have happened), he was also very accomplished with mathematics.

During his work as a clerk of the courts he would entertain himself by constructing magic squares and the following was his masterpiece:

There is a lot going on here. Firstly this 16x16 square contains all of the numbers from 1 to 256 without repeats and has all rows and columns add to 2056. Unfortunately the diagonals don't add to the same constant which technically makes this only a semi-magic square.

However it surpasses this shortcoming by having loads of extra properties. Any half row or column adds to 1028. Pick any 4x4 square from anywhere on the grid and it adds to 2056.

But most impressive property is that any broken diagonal adds to the same magic constant of 2056. Here's one collection of them:

Summing each of the cells of the same colour gives 2058. I've provided one such colouring, but by rotating the pattern by 90, 180 or 270 degrees gives 3 other patterns which are also true.

Writing on the subject of magic square in 1793 Franklin said: "I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so unentertaining that I was induc'd to amuse myself with making magic squares or circles."

There are many techniques for constructing magic squares, but surprisingly Franklin's method seems to have died with him.

I'll leave you with a link to my favourite magic square on one of my favourite websites, The Futility Closet. It is truly majestic.