Additive and Multiplicative 3X3 Magic Squares - Construction and Some Not So Well Known Properties

I had discussed a couple of party games with 4x4 magic squares that have proved really popular among friends and in gatherings. Magic squares have fascinated people for thousands of years - they have an aura of mysticism and intrigue that is irresistible.

Wiki's article on magic squares comes with a lot of historic background etc. (also see 1, 2).  In this blog, I wish to concentrate on the additive 3x3 magic squares - particularly on their construction and its less well known relative - the multiplicative magic square (MMS), and describe some other interesting variations. I shall end this blog with a discussion of the majestic 16x16 magic square constructed by Benjamin Franklin more than 200 years ago.

Let us look at magic squares with sequential numbers.

 A general way to find the number in the central cell is described in the next slide

 I shall now discuss two variations of the magic squares that are not well known, but have really interesting properties.

The Multiplicative Magic Square:  In the magic squares that we have considered so far, the numbers in each row, column and diagonal add up to the same value. In a multiplicative magic square, the product of the numbers in each row, column and diagonal has the same value.

This is from Wiki:

Sum of Products of Rows and Columns in a Magic Square:  This is a property of magic squares that is relatively unknown.  It was analysed by Professor Hahn in 1975.  Essentially, in an additive magic square, the sum the products of numbers in each row is equal to the sum of products of numbers in each column. 
Hahn shows, in a rather formal looking paper, that this property is always true for a 3x3 magic square but only holds for some (balanced) 4x4 and higher order magic squares. I refer you to the paper that is available online to read.

A second important point here is that the sum of products of numbers in rows or columns is not equal to the sum of products of numbers in the diagonals. 

I could algebraically prove these results for a general 3x3 magic square but the calculation is too long and tedious to present here.
I give some examples in the following:

The Majestic 16x16 Magic Square:  Professor Bill Richardson has described this magic square that was constructed by Franklin more than 200 years ago - without the advantage of computers!!  I refer you to the 1991 publication for all the details.  In the following is a brief summary:
The 16x16 magic square is shown in the slide:

https://en.wikipedia.org/wiki/Magic_square

https://ektalks.blogspot.com/2016/03/variations-on-magic-squares-interesting.html