Thursday, June 16, 2011

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.

 

 

 

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