Look at the Lo Shu, the smallest magic square (if order m=1 is excluded). Its order m is 3.
The numbers from 1 to 9 were placed in a 3x3-grid.
Add all numbers: 1+2+3+4+5+6+7+8+9 = 45
Divide the result by 3 to get the magic constant S_{3} = 15.
The three numbers in each row, column and diagonal sum up to the magic constant.
Thus the conditions for a magic square were fulfilled.
In this simple case 8 different sums of 3 numbers equal 15.
Such a sum like 4 + 9 + 2 is called a magic series because its result equals the magic constant.
We speak about the same magic series when the order of the numbers is changed: 9 + 2 + 4
Generally we write the numbers in an increasing order: 2 + 4 + 9
How many magic series of order 3 are there?
In other words: How many sums of 3 distinct positive integers equal the magic constant 15?
(1) 1 + 5 + 9
(2) 1 + 6 + 8
(3) 2 + 4 + 9
(4) 2 + 5 + 8
(5) 2 + 6 + 7
(6) 3 + 4 + 8
(7) 3 + 5 + 7
(8) 4 + 5 + 6
Surprisingly there are equally as many magic series as we need to build a magic square of order 3.
We can't exspect many magic order-3 squares. And indeed there is only one.
(As usual we don't count rotated or reflected variants of the square.)
Magic Series of Order 2
A magic square of order 2 should consist of the numbers 1, 2, 3, 4.
Calculate the magic constant S_{2} , find all magic series of order 2
and determine how many magic series are neccessary to build a magic order-2 square.
What is your conclusion?
Magic Series of Order 4
A magic order-4 square has 4 rows, 4 columns and 2 diagonals.
Thus 10 magic series would be sufficient. But there are 86 magic series of order 4.
This enables us to construct 880 different magic squares of order 4.
Magic order-4 series can be found without computer. Find them all and confirm the number 86.
Magic Series of Order m
The numbers of magic series form a monotone increasing sequence.
But more magic series for a certain order m enable more magic squares of this order.
There is no doubt that the sequence N_{Sq}(m) of the numbers of magic squares is monotone increasing.
It should be possible to estimate the number of magic order-m squares from the number of order-m series.
See also: Eric W. Weisstein's article in MathWorld: "Magic Series." (external link).