INVESTIGATING MAGIC SQUARES

A magic square consists of a square grid of size n x n. The numbers from 1 to are entered into the cells in such a way that each row, each column and each diagonal add up to the same total. Each number is only entered once. It can be shown that the total to which the rows and columns sum is . You do not need to prove this. A magic square puzzle is created by missing out numbers from the cells of a magic square, and then asking someone to complete it.

The first task is for each member of your group to create a magic square puzzle different from the ones on the web sites listed below. You can use the web sites to help you to design a square (of size at least 5 x 5) and then eliminate some entries to create the puzzle. Investigate how many entries can be eliminated while still allowing the puzzle to have a unique solution.

The second task is to investigate some mathematical properties of magic squares. The second website below defines a method for “multiplying” two magic squares. Read through the classroom activities on that web site to learn how to “multiply” two magic squares A and B to get the product A*B. Illustrate the method using two 3x3 magic squares of your choice. Once you understand the multiplication method, your group should investigate the following questions, using examples of 3x3 magic squares.

• If A and B are magic squares, is their product A*B a magic square?
• If you add a constant to each entry in a magic square do you get a magic square?
• If you multiply each entry in a magic square by a constant do you get a magic square?
• An identity magic square would be a magic square that acts like the number 1 in multiplication. If ID was the identity magic square, then for all other magic squares A, we would have A*ID = ID*A = A. Is there an identity magic square?
• Magic square multiplication would be associative if for any three magic squares A, B and C, we had (A*B)*C = A*(B*C). Is magic square multiplication associative?
• Magic square multiplication would be commutative if for any pair of magic squares A and B, we had A*B =B*A. Is magic square multiplication commutative?

You may find a computer package such as Excel helpful in carrying out magic square multiplication.

Magic Square web sites:

http://www.dr-mikes-math-games-for-kids.com/magic-squares.html

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

Developed by Susan Worsley and Geoff Martin.

http://www.maths.uq.edu.au