As I was watching The Story of Math with my class after finals, I thought I'd try a little experiment with the magic square shown in the video.
Each row, column, and diagonal sums to 15. Incidentally, if you add an amount n to each value, the square retains its magic and the sum changes to 15+3n.
But what if each unit square were comprised of nine smaller squares, like a sudoku board? Could you follow the same overall pattern to generate a fractal magic square?
I made this one by adding successive multiples of nine to each mini-square, following the same "path" as in the original:
Now taking this square as the unit and adding multiples of 81 to it, we have (with the magic of spreadsheets):
In this way you can make an arbitrarily huge magic square! Here's my Excel file if you want to play with it.