A Math Forum Web Unit
Allan Adler and Suzanne Alejandre's

How to Construct Magic Squares



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Suzanne's Magic Squares || Multiplying Magic Squares: Contents || Exploring the Math
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It doesn't matter how you select your magic squares, because no matter which two magic squares you use, you will be able to produce a new magic square if you follow the 'multiplication' method.

After following along and understanding the classroom activities, if you wish to find other magic squares on which to try the method, here are some examples:
  1. Lo Shu Magic Square
  2. Dürer Magic Square
  3. Mutsumi Suzuki's Magic Square Pages
Another option is to learn to generate your own magic squares:

3x3 Magic Squares

    Consider the 3x3 magic square previously given (square A):

        8 1 6
        3 5 7
        4 9 2

    Can you see how it is related to the following?

        6 7 2
        1 5 9
        8 3 4

    What about this one?

        2 9 4
        7 5 3
        6 1 8

    How is this one related to the other three?

        4 3 8
        9 5 1
        2 7 6

    Can you find examples of

     

3x3 Magic Squares: Exploring the Math

A more detailed discussion of the math underlying constructing 3x3 magic squares.
 

Generating 4x4 magic squares

This way of constructing a 4x4 magic square can be found in a book by Jerome S. Meyer, Fun With Mathematics (Cleveland: World Publishing Company, 1952).
  1. Draw a 4x4 square and go through the boxes one row at a time, left to right, top to bottom, counting from 1 to 16, but writing down the number of the box only when it falls on the diagonal.

  2. Then count down from 16 to 1, and using only the numbers not yet in the square, fill in the boxes that are left (see numbers in red in (2), below).

    (1)      (2) 


  3. Extension:

    Notice that the four entries in the upper left-hand corner of (2) (1,15,12,6) add up to 34, which is the same as the sum for each row, column and main diagonal, and that the four in the upper right-hand corner (14,4,7,9), the four in the lower left (8,10,13,3), and the four in the lower right (11,5,2,16) also add up to 34.

    Let's see what happens if we try to use these quadruples for the columns of a 4x4 square. We get:

        (3) 


    If we treat this square the same way to make a new 4x4 square, we arrive at another magic square variant of the original. Example 4 is also obtained from Example 2 in the following way: interchange the middle two columns of 2, then interchange the middle two rows, then flip over the main diagonal.

        (4) 


    Students can try many more such experiments.

5x5 and other odd-numbered magic squares

 
Review Unit: Squaring a 3x3 Magic Square



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