I don't believe the article defines a magic square (nor was I aware). A magic square is a n by n grid of numbers where each individual row, column, and diagonal add up to the same value.
I recently posted a small example showing how to construct a magic square of odd degree using a surprisingly simple algorithm called De la Loubère (or sometimes also the Siamese method), with Python:
If you're interested in why they're called "magic" squares, search for "Agrippa" on the Wiki page. The history of mathematics is chock full of such curiosities.