# Magic Squares

January 6th, 2010 by Math Tricks | Filed under Magic Squares.

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Long before Sudoku began showing up in newspapers, ancient peoples had their own math tricks up their sleeves.  What were these math tricks?  They are called magic squares.  Magic squares were known by the Chinese as far back as 650 BC.

What are magic squares?  They are squares of order n with n2 numbers within the square’s n x n matrix such that the numbers in each column, row and diagonal add up the same number.   For example, the magic square for order n=3 looks like this:

Look carefully at the rows, columns and diagonals – they each add up to the same number – 15!  Now this magic square is a special case; it is the only magic square for order 3.  Sure you can rotate it and make reflections, but this is the only arrangement for an n=3 magic square.

Magic squares exist for n >= 3.  There exists a trivial magic square for n=1, and for n=2, there is no magic square.  The number for which the columns, rows, and diagonals add up to is called the magic constant.  The value of the magic constant for a square of order n is determined by the formula:

So for n=3, the magic constant (Cm) = 15.  For n=4, Cm = 34.  For n=5, Cm = 65, etc.

How many solutions are there for magic squares for n>3?  There are many!  For n=4, there are 880.  For n=5, there are 275305224.  How many are there for n=6?  I have seen an estimate for over 1.7745×1019!

### 5 Responses to “Magic Squares”

1. vishurthi | 3/08/10

2. dafid | 3/08/10

superb
no words to say

3. deepak | 25/10/10

awesome and interesting

4. dan | 15/03/13

I have a 3×3. Upper left is 11, upper middle is 2 and right column middle is 4. I think the bottom right should be 9. Also as I throw in numbers in the upper right I always end up with a 18 in the lower middle. This seems like it should be easy but is kicking my buttowski. Without giving me the answer is there a trick to this other than brute force and filling in numbers?

5. Math Tricks | 15/03/13

Dan,

Wish I had the time to help. There may be sites out there where you can input the data you have and it will solve it for you. Also, you can use Excel to make solving it yourself easier if you are comfortable with using that program.

Good Luck!

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