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Books about Magic Squares at Amazon.com Books about Magic Squares at Amazon.Canada and at Amazon.co.UK Magic Squares A Magic Square is a square grid filled with numbers, arranged in such a way that every row, every column, and both diagonals add up to the same Magic Sum.

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Regular Odd-Size Magic Squares

Click these links to see actual magic squares, and
to see directions for how to make your own magic square
(any odd-number size)

3-by-3 magic square
5-by-5 magic square
7-by-7 magic square
9-by-9 magic square
11-by-11 magic square
13-by-13 magic square
15-by-15 magic square
17-by-17 magic square
19-by-19 magic square

And if a 19x19 square isn't big enough for you,
I prepared these much larger magic squares,
some of which are larger than you'd ever want to see:

21x21 * 23x23 * 25x25 * 27x27 * 29x29 * 31x31 * 33x33 * 35x35 * 37x37 * 39x39 
41x41 * 43x43 * 45x45 * 47x47 * 49x49 * 51x51 * 53x53 * 55x55 * 57x57 * 59x59 
61x61 * 63x63 * 65x65 * 67x67 * 69x69 * 71x71 * 73x73 * 75x75 * 77x77 * 79x79 
81x81 * 83x83 * 85x85 * 87x87 * 89x89 * 91x91 * 93x93 * 95x95 * 97x97 * 99x99

I see no reason to make a magic square larger than this, do you?

Instructions for making these magic squares can be found at the bottom of each page.

Regular 4N+0 Magic Squares

Click these links to see actual magic squares, and
to see directions for how to make your own magic square
(width equal to any positive multiple of four)

4-by-4 magic square
8-by-8 magic square
12-by-12 magic square
16-by-16 magic square
20-by-20 magic square
24-by-24 magic square
* 28x28 * 32x32 * 36x36 * 40x40 * 44x44 * 48x48 * 52x52 * 56x56 * 60x60 *
* 64x64 * 68x68 * 72x72 * 76x76 * 80x80 * 84x84 * 88x88 * 92x92 * 96x96 *

Instructions for making these (multiple-of-4) magic squares can be found at the bottom of each page.

Regular 4N+2 Magic Squares

Click these links to see actual magic squares, and
to see directions for how to make your own magic square
(even width not a multiple of four)

6-by-6 magic square
10-by-10 magic square
14-by-14 magic square
18-by-18 magic square
22-by-22 magic square
26-by-26 magic square
* 30x30 * 34x34 * 38x38 * 42x42 * 46x46 * 50x50 * 54x54 * 58x58 *
* 62x62 * 66x66 * 70x70 * 74x74 * 78x78 * 82x82 * 86x86 * 90x90 * 94x94 * 98x98 *

Instructions for making these (multiple-of-4) magic squares can be found at the bottom of each page.

More About Magic Squares

If you'd like to read more about magic squares,
click one of these links and look for books about Magic Squares
at Amazon.com, at Amazon.canada, and at Amazon.co.UK.

Irregular Magic Squares

Yes, there are irregular 3x3 magic squares, ones that cannot be converted to a regular magic square by any sets of mathematical operations. The simplest irregular magic square is made with the numbers 0, 2, 3, 4, 5, 6, 7, 8, and 10. Can you see some major differences between these two magic squares, the regular 3-by-3 magic square that I teach you to make here (click link) and the irregular one that I teach you how to make further below?

Here is one sequence of steps to make this irregular square:

• Put zero in the middle of the top row
• Put two and three in the corners furthest from the zero
• Put four above the three, with five and six to its right
• Come up from the six to write the seven
• Put eight in the other corner
• and finish with the ten in the last spot.

If you just want to have an easier way to remember this irregular square, just put zero, five and ten down the middle verticle of the square, and complete an "L" by putting two next to the ten. Then just use your math to calculate all the other squares. You don't even need to remember that the magic sum is 15 because you can add up the middle column to remind you of that number..

Do you need to remember the zero?

Actually, you only need to remember the 5, 10 and 2 to recreate this square. Why? Because the middle value ALWAYS must be one-third of the sum in a 3x3 Magic Square. So, if you know the middle number, just multiply it by three to get the magic sum.

The fact that the middle number is one third of the sum can be proved algebraically.

Let's PROVE that the middle number will
ALWAYS BE
one-third of the Magic Sum

Look at a magic square of letters?

First, remember that, by definition, every row, every column, and both diagonals must add up to the same number. we know that this particular magic sum is equal to 15, but for purposes of this proof, let's say that the magic sum is simply Sum.

by definition: Every sum must be equal to the same number
(otherwise, it is not a magic square)

Second, list what we know about the rows of the puzzle:

A + B + C = Sum
D + E + F = Sum
G + H + J = Sum

Third, add these up and we get:

A + B + C + D + E + F + G + H + J = 3 x Sum

Fourth, let's list everything that we know about this square that includes the middle square "E" (row, column and two diagonals)

A + E + J = Sum
D + E + F = Sum
C + E + G = Sum
B + E + H = Sum

Fifth, add all four of these equations together, and we get:

A + B + C + D + 4E + F + G + H + J = 4 x Sum

Finally, subtract the Third equation from the Fifth equation and what do you get?

A + B + C + D + 4E + F + G + H + J = 4 x Sum
  A + B + C + D + 1E + F + G + H + J = 3 x Sum  
0 + 0 + 0 + 0 + 3E + 0 + 0 + 0 + 0 = 1 x Sum

3E = Sum

So, what must E be equal to? Divide both sides by 3, and we get:

3E / 3 = Sum / 3

E = Sum / 3

We have proved that the middle square must be equal to one-third of the magic sum, so if you know the middle value, you can multiply it by three to get the magic sum.

So, what does this mean? It means if you have three values, and they include the middle value, you should be able to figure out the entire magic square.

Let's solve this incomplete Magic Square together

a	b	c
d	12	f
14	15	j

What is the magic sum equal to? We have proved that the magic sum must be equal to three times the middle square, so all we have to do is triple that number and we have our magic sum.

Magic Sum = 3 x 12
Magic Sum = 36

Next, you are ready to calculate the values for B and J

B + 12 + 15 = 36 ---> B = 36 - 12 - 15 ---> B = 9
14 + 15 + J = 36 ---> J = 36 - 14 - 15 ---> J = 7

Let's fill those numbers into the square and see what we see?

Now, you can easily calculate values for A and C:

A + 12 + 7 = 36 ---> A = 36 - 12 - 7 ---> A = 17
C + 12 + 14 = 36 ---> C = 36 - 12 - 14 ---> C = 10

We're almost done ... figure out what D and F must be equal to and we'll have our square?

17 + D + 14 = 36 ---> D = 36 - 17 - 14 ---> D = 5
10 + F + 7 = 36 ---> F = 36 - 10 - 7 ---> F = 19

Put those values into the magic square, and be sure to double-check all the sums to make sure they all add up to 36.

The numbers in this particular magic square are, in order, 5, 7, 9, 10, 12, 14, 15, 17 and 19.

At first glance, these appear to be an irregular set of number, but they are a regular sequence of numbers ... each group of three numbers is counted by twos, and each group of three is separated from the other groups by ones. This magic square is the same as our regular 3x3 magic square with the numbers 1-9. 

Here's how to change this magic square into the one that you probably saw first in math class:

1. Subtract 1 from each of the numbers 5, 7 and 9
2. Add 1 to each of the numbers 15, 17 and 19.
    (we haven't changed the row sums because if you look, we've added 1 and subtracted 1 from every row, every column, and both diagonals)
3. Subtract 2 from every number in the square
    (now we're changing the magic sum, from 36 down to 30)
4. Divide every number in the square by 2, and
5. Flip the puzzle over so that the numbers match from one square to the other.

An Algebraic Magic Square

We could use Algebra to represent a Magic Square.
1. Let's decide to make a magic square that has a magic sum of 3E (3 times the value of E) so let's figure out what our nine numbers will be first
2. The middle number in the list, the fifth number, must be E.
         ?, ?, ?, ?, E, ?, ?, ?, ?
3. Let's now decide that the numbers just before and just after that number will be E-A and E+A
         ?, ?, ?, E-A, E, E+A, ?, ?, ?
4. Let's have a difference of C between the fifth number and the second number and with the eighth number. In other words, E-C and E+C
         ?, E-C, ?, E-A, E, E+A, ?, E+C, ?
5. And let's do the same -A and +A that we did with the middle three numbers.
         E-C-A, E-C, E-C+A, E-A, E, E+A, E+C-A, E+C, E+C+A
6. If we put those expressions into the regular 8-1-6-3-5-7-4-9-2 magic square shown near the top of the page, we get this Algebraic Magic Square:

Let E = 5 and A = 1 and C = 3 and you have our first Magic Square, the one that your math teacher probably showed you first ... try it, you'll see. Try any other numbers for A, C and E, and you'll prove you are an ACE, and can create many such squares. Have fun with Magic Squares.