MAGIC SQUARES

* Note: This is an abbreviated project idea, without notes to start your background research or a procedure for how to do the experiment. You can identify abbreviated project ideas by the asterisk at the end of the title. If you want a project idea with full instructions, please pick one without an asterisk.

Abstract

A magic square is an arrangement of numbers from 1 to n² in an n x n matrix. In a magic square each number occurs exactly once such that the sum of the entries of any row, column, or main diagonal is the same. You can make several magic squares and investigate the different properties of the square. Can you make an algorithm for constructing a Magic Square? Can you show that the sum of the entries of any row, column, or main diagonal must be n(n²+1)/2? Are there any other hidden properties of a Magic Square? Show the differences between special instances of the Magic Square, like the Lo Shu, Durer, Ben Franklin, or Sator Magic Squares. Can magic squares be constructed in 3 dimensions? You can also investigate other shapes, like magic circles and stars (Alejandre, 2006; Pickover, 2002). Or test the question, "Is there really no math in Sudoku?" (Hayes, 2006).

Materials:

Method 1: If Internet display capabilities are available or students have access to the Web at individual computer stations, the activity can be structured by viewing the story of the Emperor Yu interactively and then continuing with the activity.

Method 2: Prepare overhead transparencies and/or handouts before presenting the activity, including:

1. Emperor Yu's Story
2. Lo Shu
3. Magic Square
4. Magic Square with lines
5. 3x3 grid

For both Methods 1 and 2:

1. Blank paper
2. Blank overhead transparency and pens
3. Rulers

Procedure:

Relate "Emperor Yu's Story."

Story Questions:

1. Using dots, draw what you think the turtle would look like in the night sky.
2. What is it about the Huang-He (Yellow River) and water that relaxes the emperor?
3. Write a short description of how the turtle got its dots.
4. Emperor Yu didn't understand the number pattern on the turtle's back. Explain in your own words how the position of the numbers makes the square "magic."

Number Questions:

1. What is magic about the arrangement of the numbers in the 3x3 cell square?
2. What is the first number?
3. What is the last number?
4. How many numbers are there?
5. Is any number repeated?
6. What is the sum of the numbers in the1st row? 2nd row? 3rd row?
7. What is the sum of the numbers in the1st column? 2nd column? 3rd column?
8. What is the sum of the numbers on one diagonal? the other diagonal?

With the teacher modeling on the overhead projector, have the students construct a 3x3 grid. Use the drawing of the turtle and write in the grid the numerals that correspond to the configurations of dots on the turtle's back. [The numerals should be positioned in the center of each cell of the grid.]

Draw a dot in the center of each numeral for reference. Using a ruler, connect the dots starting at 1, going to 2, 3... to 9 [refer to Magic Square with lines].

Symmetry Questions:

1. What patterns do you see?
2. If you draw a reference line from the 8 through the 5 to the 2, what symmetrical relationships do you see?
3. Are there any other symmetrical relationships?
4. 5 is in the center of the square. Calculate the difference of each of the other numbers and 5.

This is the result:

3 2 1
4 5 4
1 2 3

Does that make the symmetrical relationships clearer?