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 S3 = 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 S2 , 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 NSq(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).