Simultaneous Equations with Integral Solutions
Date: 11/29/96 at 23:58:31 From: F. Klasic Subject: Magic squares Hello! 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 Hi, 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 Check out our web site! http://mathforum.org/dr.math/
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