In an earlier section we have created a 3-by-3 magic square using a guess-and-check method. In this section we will create a 4-by-4 magic square using the de la Loubere's method. We will also look at some interesting patters that evolve from the Franklin square and conjecture some geometrical concepts. Finally, we will see how Lo Shu's magic square can be combined with different symmetries.

# Antoine de la Loubere

This Frenchman created a method for constructing magic squares using consecutive numbers starting with 1; an n-by-n square would contain the numbers 1, 2, 3, ..., n2. We will use this method to construct a 5-by-5 magic square.

## Steps

1. Open the following Excel document to use as a template: Loubere's Method (xls, 10kb)
2. Draw a 5-by-5 square and divide it into 25 cells (already done).
3. Add a border of cells in top and right edges (already done).
4. Shade the cell in the top-right corner; think of this cell as occupied (already done).
5. Write a 1 in middle-top cell of the original square.
6. As a general rule, fill in cells diagonally upward and to the right with numbers that increase by 1.

Note: This rule has two exceptions. First, if you land in a cell that is outside the original square, then you can get back into the original square by shifting completely across the square, either from top to bottom or from right to left, and continuing with the general rule. Second, if you land in a cell that is already occupied, then you must write the number in the cell immediately beneath the one last filled, then continue with the general rule.

# Franklin Square

You have been given a copy of Franklin's 8-by-8 magic square. Complete the following:

## Steps

1. Draw a line connecting the numbers from 1 to 64.
2. What do you see?
3. What types of symmetry do these shapes have?
4. Do you notice any other interesting patterns within the magic square?

# Lo-Shu Magic Square

Consider the Lo-Shu magic square you created earlier (you can use the 3-by-3 Form to help you). On your sheet titled "Lo-Shu Squares", create a different square by reflecting the original Lo-Shu by its center row. Is this new square a magic square?

If so, find other examples of magic squares using reflections of the Lo-Shu magic square.

Can you find other Lo-Shu magic squares using another type of transformation?