The Mystery and Mystique of Magic Squares

Magic squares are square arrays of numbers that have the same sum horizontally, vertically, and diagonally. It is the arrangement of n^2 numbers in an n \times n square. They were known to mathematicians since antiquity and they were believed to have mystical powers.  The most popular magic square is the 3 \times 3 Lo Shu square dated about 650 BC as shown in the first figure.

Magic Square

Magic squares are of different sizes. The magic squares with 16, 25, and 36 numbers are shown below. Of course there are still larger magic squares. In fact, in 1663, Muramatsu, a Japanese mathematician constructed a 19 \times 19 magic square. Some mathematicians even found ways to generate magic squares. The 3 \times 3 square above can be generated using elementary school algebra.


There are also modifications of magic squares. For instance, there are magic squares that are all primes. It was also proven that there are arbitrarily large magic squares consist of prime numbers.


The great mathematician Leonhard Euler, created an interesting an 8 by 8 magic square puzzle where each 4 by 4 corner squares are also magic squares. What is brilliant about Euler’s magic square puzzle is that the numbers are arranged such that the Chess knight can move from square 1, 2, 3, all the way up to 64.


However, Euler’s square is quite different. The sum of the diagonals are different from those of horizontal’s and vertical’s.

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