How It Works

Principally, we take advantage of the property
Magic_Sum = 3 x Middle_Square.

Then, using the 3 given numbers, we can derive the others.

Here are some examples:

With this pattern, since the diagonal sums to (3 * Middle_Square), Middle_Square = 1/2 * (Sum of other diagonal elements). From that, we can get the bottom right square ( = 2 * Middle - Upper_Left ), and all the others in similar fashion.

In this example, we have to work a little harder, but not too hard. Let the squares be designated

a b c
d e f
g h k ...  with a, b, and f given.

Then a + b = f + k (since they both, together with c, add up to the Magic Sum). And, since a, b, and f are all given, we have k = a+b-f. Then, having a and k, we can compute e (the Middle_Square) as in the previous example, and the rest follow.

This last example works as follows:
a, b, d are given. a+b+c = 3e ; g + c = 2e ; c = 2e - g ; a + b + 2e - g = 3e; g = a + b - e;
And, a + d + g = 3e; g = 3e - a - d; So,
a + b - e = 3e - a - d ; 4e = 2a + b + d; and e = (2a + b + d)/4
From there the sum is known, and the rest follows.

More Math

The sum of each row, called the Magic Sum, is the same, so the sum of all the numbers equals 3 times that.

There are four pairs of opposite numbers, comprising eight of the numbers, each with the same sum, which is Magic Sum - Middle Number. Hence,
Total Sum = 4 * (Magic Sum - Middle Number) + Middle Number.
3 * Magic Sum = 4 * Magic Sum - 3 * Middle Number, and
Magic Sum = 3 * Middle Number.
Then each opposite pair sums to twice the middle number, and the opposite pairs all consist of two numbers of the same parity, equidistant from the middle number.

Every corner is half the sum of the two squares which are "knight's move" away (the middle squares of the opposite sides), and falls halfway between them.

From all that, we see that the nine numbers always form eight arithmetic progressions:
The middle row and column(2), the main diagonals(2),
and 4 are "broken diagonals", consisting of each corner square and the two opposite middle edge squares, just mentioned above.

If all 9 numbers form a single arithmetic progression, then the magic square can be derived from the basic 816-357-492 square by a linear transformation: A * x + B, where A and B are constants, and x is value in a square.

For example: 5 8 11 14 17 20 23 26 29, which becomes:

26  5 20             8 1 6
11 17 23  = (3x+2) * 3 5 7
14 29  8             4 9 2

Others are not quite so simple to reduce, but there is always some regularity. Take 4 7 8 10 11 12 14 15 18 for example:

This forms a set of arithmetic progressions:
4 7 10 ... 8 11 14 ... 12 15 18
all with constant difference 3.
Also 4 8 12 ... 7 11 15 ... 10 14 18
all with constant different 4.
The remaining ones are:
10 11 12 ... 4 11 18 (the middle row/column);

Note the differences between this one and the previous:

26  5 20     15  4 14    11  1   6   which is itself
11 17 23  -  10 11 12  =  1  6  11   a magic square!
14 29  8      8 18  7     6  11  1

It is always the case that the sum or difference of two magic squares is another magic square.

Or start with (a progression of) arithmetic progressions:
0 3 6 ... 11 14 17 ... 22 25 28

 3 .. ..     .. 28 ..     .. .. 11     3 28 11
.. ..  6  +  22 .. ..  +  .. 14 ..  = 22 14  6
..  0 ..     .. .. 25     17 .. ..    17  0 25

Starting with 0 11 22 ... 3 14 25 ... 6 17 28 also works. The two sets of progressions are complementary.