Magic Squares: magic sum

Magic Squares
history

1	15	22	8	19
23	9	16	5	12
20	2	13	24	6
14	21	10	17	3
7	18	4	11	25

history

4	10	16	22	13
21	12	3	9	20
8	19	25	11	2
15	1	7	18	24
17	23	14	5	6

definition

3	6	12	25	19
15	24	18	1	7
16	2	10	14	23
9	13	21	17	5
22	20	4	8	11

magic sum

1	15	24	18	7
23	17	6	5	14
10	4	13	22	16
12	21	20	9	3
19	8	2	11	25

transformations

2	14	21	8	20
23	10	17	4	11
19	1	13	25	7
15	22	9	16	3
6	18	5	12	24

count

8	2	16	25	14
20	24	13	7	1
12	6	5	19	23
4	18	22	11	10
21	15	9	3	17

number

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Magic sum

This constant sum, the so-called magic sum is given by

This means that n² numbers are placed in a grid with n rows and n columns. There is a wellknown formula for the sum of first n consecutive numbers:

If we now add up to n², this formula will change to

With this result it is easy to calculate the magic sum

The following table shows some magic sums:

n	M(n)
3	15
4	34
5	65
6	111

For a magic square of order n=8 we will get therefore M(8)=260.