According to the ancient Chinese book I Ching (The Book of Changes),
in about the year 2800 BC the Emporer Yu was boarding a royal barge to cross
the River Shu. He was a tortoise hiding in the reeds. On the back of the
animal were some curious markings and the Emporer had a scribe copy them.
The result of this, according to tradition, was the discovery of the first
magic square, the 'Lo Shu'. It is often called the 'River Writing' because
of this story.
In this drawing of the 'Lo Shu', numbers are represented by dots - even
numbers are black and odd numbers are white.
Look at the diagram below, can you see a connection? Try adding up the
numbers in the rows, columns and diagonals.
The 'Lo Shu' is a magic square of the order 3.
Magic squares were thought to have protective powers against illness and
evil spirits. From the earliest of times their construction has fascinated
mathematicians all around the world.
What makes a square magic?
The numbers in a magic square sould be the same as the number of cells (for
examples in a square of 3 x 3 the numbers are 1 - 9); and they are placed so
that each row, column and the two main diagonals all sum to the same total.
The smallest magic square is the 'Lo Shu' which is of the order 3. Any other
arrangements of order 3 are only reflections and rotations of 'Lo Shu'.
The order of a magic square
The word 'order' refers to the number of squares, or cells, on one side of
the square. A magic square of order 4 has 16 cells (4 x 4), a magic square
of the order 9 has 81 cells (9 x 9) and so on.
The magic summation
The numbers in the rwos, columns and diagonals of the 'Lo Shu' all sum to
fifteen. This constant sum is called the 'magic summation'.
Making other magic squares
The making of magic squares of different sizes has been enjoyed for
centuries by mathematicians as a pastime. What most interests them is
working out ways that always work for filling in a particular order of magic
square. Rules have been found that work for all squares that are odd (5 x 5,
7 x 7 etc.) but there isn't just one rule that will work for all even
Can you find any other combinations where the rows, columns and diagonals all add to the same number?