Magic Squares Odd Ordered (3x3, 5x5, 7x7, .....)

We have been working with magic squares in which each column, row, and both diagonals sum to the same value. However, the only magic square children every get to solve is the 3x3 which is relatively easy. We are now going beyond 3x3 inspired by Ben Franklin, a founding father and mathematician who explored the properties of magic squares.

We are going to use his algorithm for solving odd-ordered magic squares. These are 3x3, 5x5, 7x7, and so on. The procedure is difficult to describe in words without a diagram so look on with page one of the attached pdf while you read these steps.

PROCEDURE:

1. Always start with the number one in the middle upper row.
2. Every move starts with one cell up and to the right.
3. If you end up above a column, go all the way down to the bottom of the column.
4. If you end up to the right of a row, go all the way to the left.
5. If you hit a cell with a number entered or move from the upper right corner, place the next number directly under the LAST number entered.

A few observations that the children made this week:

1. The middle square will always be the median or middle number between 1 and the number of squares in the grid. For example, between 1 and 9 in a 3x3 square, 5 is the median or middle square. In a 5x5, 13 is the median between 1 and 25. In a 7x7, 25 is the median between 1 and 49. We also came up with an algebraic formula to find the median number. If we have an n x n square, we started by squaring n to get n times n or n^2 (n squared). Then we added one to that square and divided that result by 2; so we get 

(n^2 + 1)/2 To make sure this works, let’s try it on a 5x5 square. (5^2 + 1)/2 = (25 + 1)/2 = 26/2 = 13

2. Next, the children noted that the last number will always be placed at the middle column bottom row.

3. Other observations were made such as the middle column always starts with 1 then adds (n +1) in each square below so in a 5x5, 1, 7, 13, 19, and finally, 25.

4. Another keen observation was that the corner squares always add up to four times the median number.

There are countless other observations. What else do you see?

I asked the children to find the median number first, and place it in the middle square so when they reach it, they will know if they are on track. After completing their magic square, they should find the magic number by adding any row, column or diagonal. But the one I like best is the middle column. It is also fun to add up all rows, columns and diagonal to prove that it is, in fact, a magic square.

For the older children, we looked at an algebraic formula for finding the magic number without first completing the magic square. This formula takes the dimension of the square, n x n. First we cube the n value (this means multiplying n x n x n) and then we add n. Finally, we divide that value by 2. The formula looks like this:

(n^3 + n)/2

The last two pages of the pdf challenge the children to find the magic number using this formula only. This pdf contains grids from 3x3 to 19x19 and they can go beyond if they dare.