Fractal Magic Square
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.
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.