A Math Forum Web Unit

Review Unit

Squaring Magic Squares

Suzanne's Magic Squares || Multiplying Magic Squares: Contents || Exploring the Math

Objectives: [NCTM Standards: Number and Operations, Communication, and Connections]

  1. Students will work with arrays of numbers.
  2. Students will calculate and compare sums of numbers in the context of magic squares.
  3. Students will apply a method for multiplying two magic squares.


  1. Overhead transparencies or handouts:

      Large empty 3x3 grid
      The nine cells in numerical order
      Ordering the cells to make a magic square
      The completed 9-cell magic square

  2. Blank paper
  3. Blank overhead transparency and pens
  4. Rulers
  5. Calculators


To square a number means to multiply it by itself.

[Just as we can talk about squaring a number, we can talk about squaring something with any notion of multiplication. We can square an nxn matrix by multiplying it by itself under matrix multiplication, and we can square a magic square by multiplying it by itself.]

Considering Square A:

Remember the method we followed to multiply Square A by Square B? (To review, return to Classroom Activities.) If this method can be used to 'multiply' two magic squares, let's use it to square a magic square.

To square Square A, we'll follow the same steps, but instead of using Squares A and B, we'll use Square A twice.


Square A           Square A

We'll again need a large grid. It will have as many cells as there are cells in Square A.  

    Check for Understanding:

    1. Verify that Square A is a magic square.
    2. How many cells does Square A have?
    3. How many cells will we need in the large grid?

Generating the second square:

Following the multiplication method, we will fill the nine cells in the empty 3x3 grid. We will begin by adding the same number to each entry in Square A to make a new magic square.

Square A is made up of nine entries. When we add 9 to each of them, we get

We will continue this process by adding 9 to each entry to make each successive cell.

Continued next page

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