Magic Square Variations

Date: 08/29/2001 at 21:20:35
From: Anonymous
Subject: Magic Squares

A magic square consists of numbers arranged in a square so that all 
rows, columns, and usually the two diagonals will add up to the same 
sum. Try to create a magic square by arranging the first nine counting 
numbers in the nine square cells. 

There is only one possible way. Can you please help me?

Thank you,
Anonymous

Date: 08/30/2001 at 06:58:00
From: Doctor Jeremiah
Subject: Re: Magic Squares

Hi there, and thanks for writing.

There are actually 8 different ways, but they are all rotations and 
mirror images of the same one.

First consider that if you have 3 cells wide by 3 cells high, you will 
have to put the numbers 1 through 9 in these nine cells. The sum of 
the diagonals, rows, and columns will be the same.

That means that the sum of all three columns must be the same as the 
sum of all nine numbers because the nine numbers fit into the three 
columns.  (Let's call the sum of a column, row, or diagonal S):

 sum of 3 columns =      sum of all nine numbers

       3S         = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

       3S         = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
       3S         = 45
       3S / 3     = 45 / 3
        S         = 15

So the "magic sum" is 15.

Let's think a bit about what number must be in the middle. It's an 
important number because it is used in every sum (all the rows, 
columns and diagonals).

If that middle cell holds 6, then what? In which cell can you put the 
9? You can't put it anywhere, because 6+9 = 15 with only two numbers, 
and we need to make _three_ numbers add to 15. So 6 and above cannot 
be in the middle cell.

If that middle cell holds 4, then what? In which cell can you put the 
1? You can't put it anywhere, because 4+1 = 5, and to make 15 you need 
to put 10 into a cell, but 10 isn't a choice because it's not one of 
the 9 counting numbers. So 4 and below cannot be in the middle cell.

That leaves 5. With 5 in the middle cell the solution is easy 
(especially if you know the magic sum is 15).

Try that and let me know if you get stuck again. And for more 
information, see Suzanne Alejandre's Web unit:

   Magic Squares
   http://mathforum.org/alejandre/magic.square.html   

   How to Construct Magic Squares
   http://mathforum.org/alejandre/magic.square/adler/adler4.html   

or search the Dr. Math archives for the keywords  magic square (that 
exact phrase):

   http://mathforum.org/mathgrepform.html   

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