Associated Topics || Dr. Math Home || Search Dr. Math

### Largest Magic Square Ever Known

```
Date: 09/18/2001 at 03:46:49
From: Jayson Javellana
Subject: Largest Magic Square Ever Known

Dr. Math,

I am eager to know what was the largest magic square ever constructed
and known in the world of mathematics. What is the easiest way to find
the sum of each row, column, and main diagonals of magic squares?

Thank you.
Jayson
```

```
Date: 09/18/2001 at 11:35:02
From: Doctor Rob
Subject: Re: Largest Magic Square Ever Known

Thanks for writing to Ask Dr. Math, Jayson.

There are magic squares of every odd order, so there is no maximum.
Here is a 9-by-9 magic square that shows the pattern for the
construction of odd orders (see where the numbers 1, 2, ..., 9 are
placed; then 10, 11, ..., 18; then 19, 20, ..., 27; and so on):

37 28 19 10  1 73 64 55 46
47 38 29 20 11  2 74 65 56
57 48 39 30 21 12  3 75 66
67 58 49 40 31 22 13  4 86
77 68 59 50 41 32 23 14  5
6 78 69 60 51 42 33 24 15
16  7 79 70 61 52 43 34 25
26 17  8 80 71 62 53 44 35
36 27 18  9 81 72 63 54 45

There are also known constructions for large even orders, but they are
more complicated.

If an n-by-n magic square contains the numbers from 1 to n^2, then
the sum of any row, column, or main diagonal must be n*(n^2+1)/2.
For example, if n = 9, as above, then the sum would be 9*82/2 = 369.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 09/18/2001 at 11:48:07
From: Doctor Jubal
Subject: Re: Largest Magic Square Ever Known

Hi Jayson,

Thanks for writing to Dr. Math.

I searched the Dr. Math archives on the keyword:

magic square (exact phrase)

and found this link

Finding Magic Squares

which gives instructions for creating a magic square as large as you
want, so long as the square has an odd number of squares on a side.
Actually, however, you can create magic squares as large as you want,
even if they have an even number of squares on a side.

If the number of squares on each side is divisible by four, start off
by just numbering the squares in order across each row:  (I'll
illustrate the method for 8x8.)

1   2   3   4   5   6   7   8

9  10  11  12  13  14  15  16

17  18  19  20  21  22  23  24

25  26  27  28  29  30  31  32

33  34  35  36  37  38  39  40

41  42  43  44  45  46  47  48

49  50  51  52  53  54  55  56

57  58  59  60  61  62  63  64

Now, divide the square into smaller 4x4 squares, and draw in the
diagonals of those four-by-four squares. Some of the numbers will be
crossed out in this process

\   2   3   /   \   6   7   /

9   \   /  12  14    \   /  16

17   /   \  20  21    /   \  24

/  26  27   \   /  30  31   \

\  34  35   /   \  38  39   /

41   \   /  44  45   \   /  48

49   /   \  52  53   /   \  56

/  58  59   \   /   62 63  \

Finally, replace all the numbers that got crossed out with
n^2 + 1 - a, where n^2 is the number of cells in the square (64 in
this case) and a is what the number originally was. This gives you

64   2   3  61  60   6   7  57

9  55  54  12  14  51  50  16

17  47  46  20  21  43  42  24

40  26  27  37  36  30  31  33

32  34  35  29  28  38  39  25

41  23  22  44  45  19  18  48

49  15  14  52  53  11  10  56

8  58  59   5   4  62  63   1

which is a magic square. This will hold for any square of side n
divisible by 4.

There is also a way to generate magic squares as large as you want
with sides of 4n+2 (even, but not divisible by 4), but it's more
complicated.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Puzzles
Middle School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search