Thursday, December 29, 2016
Replacing Rotations With More Primitive Operations
Since the rotations of the magic square can be defined in terms of the reflections they are all we need in order to find new arrangements. The group can be reduced to just the three primitive operations of multiplying a magic square M and its transpose MT by the arrays F and G either on the left or right side. So each magic square belongs to a set or class of 32.
Even though we can exchange the order in which we multiply by F and G the order above needs to be fixed so we don't double count the number of magic squares. Using reflections and rotations in our group of operators risks double counting magic squares.
Why can we exchange the order of F and G and the order of the transpose? F and G only exchange rows with rows and columns with columns. Transpose exchanges rows and columns. There appears to be something fundamentally different about doing this. Rotations exchange rows and columns too.