Almost 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 have been watching your “Integration technique” videos with high expectation and excitement. They never fail to give me a lot of joy and much pleasure. You must have been one hell of teacher at your time! … just continue making highly enjoyable and intellectually stimulating videos. … Unfortunately, what you learn at the university, you hardly ever see again in your office job. That is why I try to repeat some thing I learned: to enjoy it a bit more, to marvel upon beauty, and to keep my brain sharp. There is where your videos come in perfectly: interesting, amusing and thought provoking problems neatly presented. … You do your best to share your passion, knowledge and experience. The least I can do is to say an honest “Thank you!” As I said, your videos give me much pleasure. You may freely let other people know that. … Rest assured that I shall be watching your videos in the future. With joy and excitement. And I shall comment on them from time to time. You just continue making them.

Zoran V (in private correspondence via YouTube, quoted with permission)

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