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Magic Squares

A closeup of the magic square from Albrecht Durer's famous engraving, MelancholiaA 3 by 3 magic square with the numbers on each row, column and diagonal adding to 15Almost from the time that people added numbers, someone would have noticed a curious pattern … that 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6.  As one number grows, the other decreases by the same amount, so the total remains constant.  It would not have taken long before similar sums were drawn across each other, for example, 9 + 5 + 1 vertically and 3 + 5 + 7 horizontally, sharing the five (see the square to the right).  From this it would be a short step to a pattern with a fascinating property: a magic square, like the one at right, in which the totals of all columns and rows and both diagonals are the same.

This page will explore these patterns and why they ‘work.’  We will examine different ways of designing such squares, and the surprising variety among those that exist.  We will construct squares of increasing size and ask ourselves whether there is a limit to their size.  We will see squares that are built from prime numbers, others that look the same when rotated 180°, and magic squares with other strange properties.

We will also learn some of the history of these fascinating squares and how Albrecht Dürer (1471-1528) managed to construct a magic square for his engraving, Melancholia, in which the middle numbers at the bottom give 1514, the year of the engraving!

I attend university in the US and am starting the more difficult math courses. I have never studied and managed to stumble along for a very long time and now I have no idea what I am doing. I can follow along with lecture and understand what they are saying but when left on my own I have no clue what to do and where to start. So I’ve recommitted to start over and build a solid foundation. Looking around youtube alot I have found some good quick fix stuff but man I am converted to Graeme Hendersen. I really appreciate your very thorough approach to learning and can’t get enough of your explaining mathematical concepts. I am eating up everything and am excited to master math. Man, I sound super nerdy but it’s true. Thanks
Garett M (on a CCM YouTube video about How to Find the Equation of a Parallel Line in 4-5 Lines of Work)

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