Tuesday 6 June 2023

A Unique approach for Odd Order Magic Squares

Lo Shu Magic Square

I have been interested in Math History and Recreation Math for a really long time,  (yes, I'm that old), so when I came across a new approach on twitter that I had never seen, I was a little surprised.  When I read that it was about 400 years old, I was even more surprised (and no, I'm not THAT old).

I've written about Magic Squares over the years, from the earliest known 3x3 supposedly found on the back of a turtle in Chinese Mythology, called the Lo Shu Square literally: Luo (River) Book/Scroll)  and about the magic square on the Passion Facade on the Sagrada Familia Cathedral in Barcelona and then about  a magic square relationship to Matrices I just learned this year (2018) from John D. Cook's blog.

I usually not surprised in finding out new relationships in magic squares, but part of what surprised me this time, was that it was a method created by Claude Gaspard Bachet de Méziriac, Who I've read a lot about, and written a little about, and was aware that he worked with recreational math and number theory. He published a Latin translation of the Greek text of Diophantus’s Arithmetica in 1621. This is the translation that Fermat made his famous margin note that became the famous Fermat's Last Theorem. He asked the first ferrying problem: Three jealous husbands and their wives wish to cross a river in a boat that will only hold two persons, in such a manner as to never leave a woman in the company of a man unless her husband is present.

So, anyway, if I'm not the only person in the world who never saw it before, here is a really unique method of constructing nXn magic Squares when n is odd. which I found on a animated tweet created in Geogebra by Jason-Automaths@palajsn, and thanks to Vincent Pantaloni for sharing.

You start by constructing a Diamond stack of squares with 1 square in the first row, three squares in the second, etc. until you get to 2n-1 squares for the desired nXn desired, then descending back down to one. Here is the example for the 7x7 square.

Then you start at the 1'st diagonal down the right side and write the numbers in order, 1 to 7. Skip to the third diagonal and do the next 7 digits. Continue in like fashion and you get something like this.

Now here is the slickest little move imaginable, you take the pyramid of six numbers above the the top row of seven squares, and move it down until the number one is just below the center square (25 in this case)...

Make a similar translation of the pyramids on the other three sides across to the similar position on the other side of the center square, and you have a magic square.

The first known magic square was the Lo Shu, shown above. If you learned the quick method I did for odd squares, you start at the bottom center with one (apparently the Chinese put North on their calendars at the bottom, and that was an influence on the future evolution of magic squares). Then you just number up and right (or down and right) on the diagonal (as if the edges were connected right and left, top and bottom like a torus). Each time you come to a multiple of n, you drop down one and continue. Notice this works the same way, except that the diagonals go down and right, and at ever multiple of n, you drop down two rows, instead of one.

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