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Simultaneous Equations with Integral Solutions

Date: 11/29/96 at 23:58:31
From: F. Klasic
Subject: Magic squares


I am a seventh grade student and I love math. We have a science fair
coming up and I would really like to enter the math category, but I am
really having a hard time finding a project. We need a hypothesis 
that can be researched and proved or disproved. Do you have any ideas 
at all or can you tell me someone who might be able to help me?  Math 
is the least covered topic in all the science fair project books.

I really like magic squares (I already wrote to the magic square 
person) and algebra. One idea is to see if a magic square can become 
a magic cube - but that doesn't really give me much to research other 
than to construct a model with cubes and see if it works. It's hard 
to think of something that I can graph the results of. I sure would 
appreciate ANY ideas at all. Thank you very very much. My teacher 
really wants me to come up with a project. I talked about a 
correlation between measurement of body parts (like arm and foot) but 
she said someone else is already doing that. My biggest problem is 
that I'm running out of time to come up with an idea and I'm afraid 
I'll end up doing something about which wood burns hottest which is 
OK, but I really like math!

Thank you very very much, 

Your friend, Meghan Klasic 

Date: 12/06/96 at 14:32:58
From: Doctor Ceeks
Subject: Re: Magic squares


Your idea of trying to find magic cubes is a very natural question
to ask after thinking about magic squares. I'd encourage you to think 
about them. You might even conclude that they don't always exist.

If you're going to try to think about magic cubes, I suggest you start
with very small cubes.  For 1 x 1 x 1, the answer is pretty simple, 
but what about for 2 x 2 x 2?  Can you understand that case?

Do you know the book _An Invitation to Number Theory_, by Oystein Ore?  
It begins with magic squares, but it doesn't consider magic cubes.

On a more abstract level, a related question which fits better with 
the construction of a whole theory is to try to figure out conditions 
under which simultaneous equations have integral solutions.

Let me explain.

When you make a magic square, you can think of the problem as trying 
to solve a bunch of equations in the following way.  Suppose your 
square is 3 x 3.  We can assign labels to the numbers which go in each 
box of the magic square like this:

          A   B   C

          D   E   F

          G   H   I

If you can fill in the magic square with numbers, you are, in effect,
finding a solution to all these equations (where S is the number which 
all rows, columns, and diagonals add up to):

          A+B+C = S
          D+E+F = S
          G+H+I = S
          A+D+G = S
          B+E+H = S
          C+F+I = S
          A+E+I = S
          C+E+G = S

But even more importantly, your solution is a solution where all the 
numbers are integers!

Now, forget about magic squares for a moment and think of just a 
simple equation like this where a and b are integers and "ax" means 
the number "a" times the number "x": ax = b

What condition must a and b satisfy so that the value of x which makes 
the equation true is an integer?  Maybe that's too easy for you, since 
the best you can say is that a must divide b.

But now, suppose we have two equations where a, b, c, d, m, and n are 
all integers:

          ax+by = m
          cx+dy = n

Can you find a general condition under which integer values of x and y  
make both equations true?

Or suppose you just have the equation where a,b, and m are integers:

          ax+by = m

You want to know if you can find integers x and y which make the 
statement true.  When can you do that?  For instance, no matter what, 
it turns out that you cannot find integers x and y so that the 
equation 3x+6y = 20 holds.  But you CAN find integers x and y so that 
3x+7y=20 holds!  (Try x=2 and y=2.)

Well, this problem of trying to determine when a family of equations
has integer solutions is a real challenge.

Feel free to write back with any other questions!

-Doctor Ceeks,  The Math Forum
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