If we add up all the numbers in the grid we have
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
and this total would have to be the same as adding the totals of 3 rows. This
shows that each row (and column and diagonal) must add up to 15. Now let’s
look at the middle cell – we’ll call this c. Two diagonals involve c as does the
middle row and the middle column. If we add the numbers in these four lines
together we get 15 +15 +15 +15 = 60 and this must equal all the numbers
added together plus 3 extra lots of c. From the equation 3c + 45 = 60, we see
that c must be 5. Other facts can also be learned such as not being able to place
a 1 in a corner cell. Some clues gathered, we are in a good position to use the
trial and test method. Try it!
A solution for the 3×3 square by the Siamese method
Of course we’d like a totally systematic method for constructing magic
squares. One was found by Simon de la Loubère, the French ambassador to the
King of Siam in the late 17th century. Loubère took an interest in Chinese
mathematics and wrote down a method for constructing magic squares that have
an odd number of rows and columns. This method starts by placing a 1 in the
middle of the first row and ‘going up and across and rotating if necessary’ to
place the 2 and subsequent numbers. If blocked the next number beneath the
current number is used.
Remarkably this normal magic square is essentially the only one with 3 rows
and 3 columns. Every other 3×3 magic square can be obtained from this one by
rotating numbers about the middle and/or reflecting numbers of the square in
the middle column or middle row. It is called the ‘Lo Shu’ square and was known
in China around 3000BC. Legend says that it was first seen on the back of a turtle
emerging from the Lo river. The local people took this as a sign from the gods
that they would not be freed of pestilence unless they increased their offerings.
If there is one 3×3 magic square, how many distinct 4×4 magic squares are
there? The staggering answer is that there are 880 different ones (and be
prepared, there are 2,202,441,792 magic squares of order 5). We don’t know
how many squares there are for general values of n.