Finding Magic Squares

Date: 05/12/98 at 22:11:54
From: Chris Holaday
Subject: Magic squares - how to find them

I've looked at magic squares, many of them actually, and just couldn't 
find a pattern in the numbers. Is there a pattern for each type of 
magic square?  A universal pattern that would work on all magic 
squares?  Also, are there similar patterns in magic cubes as well?

Date: 05/14/98 at 11:27:05
From: Doctor Jeremiah
Subject: Re: Magic squares - how to find them

Hi Chris:

I really like magic squares. I have found that any size magic square 
with an odd number of squares on a side (3x3 9x9 101x101) is easy to 
do. I do not know of an easy way to do magic squares with an even 
number of squares on a side (4x4 10x10). So the way I would create a 
4x4 magic square is probably by trial and error.

A magic square has a certain number of smaller squares placed together 
to form a larger square. The numbers 1 through however many smaller 
squares exist are placed each exactly one time into one of the smaller 
squares so that all the rows, columns and both diagonals add to the 
same number:

     +---+---+---+       +---+---+---+
     |   |   |   |       | 8 | 1 | 6 |
     +---+---+---+       +---+---+---+
     |   |   |   |  ==>  | 3 | 5 | 7 |
     +---+---+---+       +---+---+---+
     |   |   |   |       | 4 | 9 | 2 |
     +---+---+---+       +---+---+---+

Here is the method that I use for doing a magic square with an odd 
number of squares on one side (5x5):

If things are not lining up very well, change to a monospaced font 
like Courier.

Step 1: Always put the number "1" in the center bottom square.

     +---+---+---+---+---+
     |   |   |   |   |   |
     +---+---+---+---+---+
     |   |   |   |   |   |
     +---+---+---+---+---+  
     |   |   |   |   |   |  
     +---+---+---+---+---+  
     |   |   |   |   |   |
     +---+---+---+---+---+
     |   |   | 1 |   |   |
     +---+---+---+---+---+

Step 2: Place successively bigger numbers in the squares in a downward
        diagonal direction. If you run off the end, wrap around as if
        the sides were attached to each other.

     +---+---+---+---+---+
     |   |   |   | 2 |   |
     +---+---+---+---+---+  
     |   |   |   |   |   |  
     +---+---+---+---+---+  
     |   |   |   |   |   |  
     +---+---+---+---+---+  
     |   |   |   |   |   |  
     +---+---+---+---+---+  
     |   |   | 1 |   |   |
     +---+---+---+---+---+

Step 3: Continuing with Step 2 . . .  

     +---+---+---+---+---+
     |   |   |   | 2 |   |
     +---+---+---+---+---+  
     |   |   |   |   | 3 |
     +---+---+---+---+---+  
     | 4 |   |   |   |   |  
     +---+---+---+---+---+  
     |   | 5 |   |   |   |  
     +---+---+---+---+---+ 
     |   |   | 1 |   |   |
     +---+---+---+---+---+

Step 4: When you get to a square that is already filled, move one
        square up instead.
 
     +---+---+---+---+---+
     |   |   |   | 2 |   |
     +---+---+---+---+---+
     |   |   |   |   | 3 |
     +---+---+---+---+---+  
     | 4 | 6 |   |   |   |  
     +---+---+---+---+---+  
     |   | 5 |   |   |   |
     +---+---+---+---+---+
     |   |   | 1 |   |   |
     +---+---+---+---+---+

Step 5: Repeat this over and over.

     +---+---+---+---+---+
     |11 |   |   | 2 | 9 |
     +---+---+---+---+---+
     |10 |   |   |   | 3 |
     +---+---+---+---+---+
     | 4 | 6 |   |   |   |  
     +---+---+---+---+---+
     |   | 5 | 7 |   |   |
     +---+---+---+---+---+
     |   |   | 1 | 8 |   |
     +---+---+---+---+---+

Step 6: Repeat this over and over. 

     +---+---+---+---+---+
     |11 |   |   | 2 | 9 |
     +---+---+---+---+---+
     |10 |12 |   |   | 3 |
     +---+---+---+---+---+
     | 4 | 6 |13 |   |   |  
     +---+---+---+---+---+
     |   | 5 | 7 |14 |16 |
     +---+---+---+---+---+
     |   |   | 1 | 8 |15 |
     +---+---+---+---+---+

Step 7: Repeat this over and over.

     +---+---+---+---+---+
     |11 |18 |   | 2 | 9 |
     +---+---+---+---+---+
     |10 |12 |19 |21 | 3 |
     +---+---+---+---+---+
     | 4 | 6 |13 |20 |   |  
     +---+---+---+---+---+
     |   | 5 | 7 |14 |16 |
     +---+---+---+---+---+
     |17 |   | 1 | 8 |15 |
     +---+---+---+---+---+

Step 8: Stop when all the squares are filled. 
     
     +---+---+---+---+---+
     |11 |18 |25 | 2 | 9 |
     +---+---+---+---+---+
     |10 |12 |19 |21 | 3 |
     +---+---+---+---+---+
     | 4 | 6 |13 |20 |22 |  
     +---+---+---+---+---+ 
     |23 | 5 | 7 |14 |16 |
     +---+---+---+---+---+
     |17 |24 | 1 | 8 |15 |
     +---+---+---+---+---+

This particular square adds up to 65 in each direction.

As far as a 4x4 magic square is concerned, check out:

  http://www.astro.virginia.edu/~eww6n/math/MagicSquare.html   

because it has a solved 4x4 magic square and a somewhat complicated 
method for solving it. And for a lot of information about magic 
squares see "How to construct magic squares" at

  http://mathforum.org/alejandre/magic.square/adler/   

I hope this helps. 

-Doctors Jeremiah and Sarah,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/