Magic Squares

Karyn Carson

This is a Ômagic squareÕ.

 8 1 6 3 5 7 4 9 2

What do you notice about it?  What makes it ÔmagicÕ?

Find the sum of each row, column and diagonal.  What pattern do you see?

 8 1 6 15 3 5 7 15 4 9 2 15 15 15 15 15 15

The magic sum of this square is 15.  Every row, column and diagonal all sum to 15.

Now, this is not all thatÕs cool about a magic square.  There are several things IÕd like you to investigate:

á      Do you think this is the only 3x3 magic square?

á      What happens if you add a number to each of the amounts?  Is it still a magic square?

LetÕs add 5 to each number:

 13 6 11 8 10 12 9 14 7

What is the new magic sum? Why is this the new sum?

What amount did you add? Was your result a magic square?  Why?

a.  Can you multiply each of the numbers by a particular number and still have a magic square?

LetÕs multiply by 2:

 16 2 12 6 10 14 8 18 4

Is this still a magic square?  Why?  WhatÕs the new sum?

Now you try it.  Why is it still a magic square?  What is the new magic sum?  Can you give an algebraic reason?

b. Must the numbers in the magic square be whole numbers?  Divide each of the numbers by a constant.

LetÕs divide by 2:

 4 1/2 3 3/2 5/2 7/2 2 9/2 1

If you believe itÕs still a magic square, what do you think will happen to the sum?

Is it still a magic square?  What is the new magic sum?  Give an algebraic sentence that proves that it will be a magic square.

c.  Can the numbers be negative or positive?

LetÕs subtract 10 from each number:

 -2 -9 -4 -3 -5 -7 -6 -1 -8

WhatÕs the new magic sum?  Why?  Now you try it.  Subtract the numbers by a constant.  Prove it using an equation.

d.   Can you change the order of the numbers.  Use what we came up with in ÔeÕ to help you answer this.  Play with the magic square to see if you can come up with another magic square using the original nine numbers.  Is there more than one?

Do you think you can use multiplication in a magic square?  Try it!

Actually, you can.  One easy way to make a multiplicative magic square is to use 2 as a base and raise it to the power of the original addition magic square:

 8 1 6 3 5 7 4 9 2

 28 21 26 23 25 27 24 29 22

What is the magic product of this square?

What is the middle number of this square?  What is the relationship between the middle square and the magic product?  What do you notice about the relationship between the lower left-hand corner and the middle square?

Use this information to solve the following multiplication magic square:

 2 1 3 4