Thanks to Suzanne Alejandre for this activity and its accompanying lesson plan, and Mike Morton for the nifty Applet.

## Magic Squares## Activity of the Month## October 1996 |

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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.

we'll discover how to make a 3x3 magic square.

## 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?

## 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:

- 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?]
- How did you figure out what numbers to use and where to put them in the square?
- 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?]
- If we start with a puzzle like the one above, is there only one solution? [Advanced question: Why or why not?]
- [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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