Magic Squares

As an educator of both pre- and in-service mathematics teachers I frequently urge my students to use activities in their teaching which require mathematical thinking, perseverance, and problem solving.  In addition, teachers are always searching for activities which can be used by all the students in a class rather than choosing different activities for the different groups of students.

One of our favorite activities is Magic Squares.

I’m sure all of you are familiar with Magic Squares.  They are square arrangements of numbers so that the sum of the numbers is the same for each row, column, and the two primary diagonals.  Some magic squares have additional properties, but we’ll ‘stick’ to the ‘basic’ magic square for now.

Take a moment to notice that each row, column, and diagonal sums to 15.  Also, the numbers in the cells are the positive integers from 1 to n; where n is the total number of cells.  A 5 x 5 magic square of this type would contain the numbers from 1 to 25 (note: there are more than two hundred seventy-five million distinct 5 x 5 magic squares).

So, how can we use these when we choose instructional activities for our students?  Simply said, we can have students build or solve (more about that below) magic squares using positive and negative integers, decimals, fractions, etc.  We can, therefore, use the same ‘activity’ to challenge the diverse set of students found in our classrooms.

Some suggestions I have for teachers are:

  • For some students you’ll ask them to find the Magic Sum (the sum of each row, column, or diagonal).  Here are a few examples which my students use in a single 5th grade classroom. Please notice that the 3×3 example does not use integers from 1 to n; notice the 4×4 example uses decimals (which you could change to fractions), and the 5×5 example uses both positive and negative integers.
  • By using a digital tool, students can explore, report, and explain the effect of changing the size of the square, the Magic Sum, and other parameters.
  • Another activity is to solve Magic Squares.  Solving requires students to find the values of the ‘blank’ squares.  You’ll find that most students use guess-and-check, but this tedious activity can lead to deeper thinking and the development of strategies.  A simple teacher strategy is to take one of your complete magic squares and eliminate one or more values.  The more values you remove the more difficult the task (in general).  I suggest you put your ‘incomplete’ magic square into the Solver portion of the website I identified above.  It’ll prevent you from asking students to solve an unsolvable magic square. 

Another example below uses a 3×3 magic square (using Excel). Then each cell is multiplied by 5.  Notice that multiplication of each cell by the same number results in a ‘new’ magic square.

Of course, dividing each cell by the same number results in a new magic square.  Excel makes multiplication or division accurate and easy. On the left side below the Magic Square is ‘divided’ by 4, and then (using the equation editor in Excel) changed some of the decimal numbers to fraction format.

By using the last Magic Square, students will need to add whole numbers, decimals, and fractions to determine the Magic Number. Similarly, of course, you can leave one or more squares blank and ask students to find the missing number(s).

As the teacher you can assign different Magic Squares to each student or group of students.  In this way, everyone is working with Magic Squares, using addition and subtraction in non-trivial ways, problem solving, etc.

I suggest you visit the website: which should be on your list of favorite/useful websites which you should use to support your teaching/learning. You’ll find useful classroom activities using Magic Squares.

Perhaps you’re more familiar with the website: which is another source of rich problems/activities including Magic Squares.

In summary, Magic Squares can fulfil many of the requirements you have for activities which challenge your students. Feel free to send me your own examples.

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