Magic Squares

Each month the Math Forum sponsors a student activity that can be done locally and shared globally. There are:

• discussion suggestions for the teacher or student activity leaders,
• a detailed lesson plan for the classroom,
• ways to share what you do and learn,
• and here's a link to a Java Applet that will allow you to experiment with this puzzle.

What is a magic square?

In a magic square, the rows, columns, and diagonals all add up to the same number.

Let's begin with the first three numbers:

Can you fill in the empty cells?

Need help?

Definition:

Some mathematicians define a magic square formally as an arrangement of the numbers from 1 to n^2 (n-squared) in an nxn matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same.

Discussion suggestions:

1. What is the sum of each row, column, diagonal of the magic square? Can a square be made that results in a different sum? [Advanced question: Why or why not?]
2. How did you figure out what numbers to use and where to put them in the square?
3. What number did you place in the middle? Is this the only number that can go in the middle? [Advanced question: Why or why not?]
4. If we start with a puzzle like the one above, is there only one solution? [Advanced question: Why or why not?]
5. [Advanced question: Can you show that the sum of the entries of any row, any column, or any main diagonal must be n(n^2+1)/2?]

Interactive exploration: a Magic Square Java Applet

A lesson plan for the classroom, by Suzanne Alejandre.

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