This comes as a sequel to the post here about the 4×4 edition of what I am calling a “Barron Square” until someone corrects me with a better name. What is a Barron Square? Well, it is a:

Nonzero square matrix where the product of the first column(s) and the last column(s) equals the center rows and the product of the first row(s) and the last row(s) equals the middle rows.

An example of this would be the matrix

6 4 2 7

4 2 0 5

8 4 8 6

8 6 4 8

Because looking at it row wise: 6 * 7 is 42, 4 * 5 is 20 etc etc and column wise 6 * 8 is 48, 4 * 6 is 24 etc etc

The logical next step is to find the next size matrix: In this case the 8×8 (if one exists) where the first two rows times the last two rows equal the middle four rows and the first two columns times the last two columns equal the middle four. After working on the 4×4 style I realized that by setting the corners of each matrix you can solve for the rest of the matrix and see if there are any conflicts (i.e. set the four corners of the above matrix, solve for the edges and then the middle. In an 8×8 you would set the 4 2×2 corners). I wrote matlab code that searches for such a matrix in this manner. Yet after several million iterations I haven’t been able to find one. This is where you come in.

Can you help me pave this quirky thought process into a tangible matrix? Is there a better way to do this than simply running more and more interations? Does one even exist?