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. » Read more

The Magnificent Magic Cube

If you are fascinated by magic squares, you will be more astonished that magic cubes also exist, one of which is shown in the first figure.


Magic Cubes are cubical arrangements of numbers from 1 to n^3 such that the sums of the rows and the sum of he columns are equal. In the magic cube above, the numbers from 1 to 3^3=27 are arranged in a cube and the sum of the numbers on each column and each row is 42. » Read more

The Magic Circle Puzzle

I was browsing a copy of the  The Mathematical Palette, the book where I got the title of my other blog and I saw an interesting puzzle about magic circles.

magic circle

Magic circles are quite similar to magic squares.  In the magic circle above, we have to place the numbers 1 through 9 in the interior of the red circles.  If we add the numbers on any circle (blue or yellow) and the number at the center, the sum should be equal to the sum of each “diagonal” of the circle.

For a little trivia, magic circles are said to have been invented by Japanese mathematician Kittoku Isomura (c. 1660).