Magic Squares — Odd Ordered (Grades 1-6)

A magic square is an arrangement of the numbers from 1 to n^2 (n-squared) in an n x n 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.

For example, in a 3 x 3 magic square, each row, column and diagonal sum to 15. 15 is the magic number. I had the children start with a blank 3 x 3 grid and had them search for ways to place the nine digits in such a pattern. This is a difficult task. I showed them an algorithm that will always work for any grid of odd ordered n x n. This will work for a 5 x 5, 7 x 7, 9 x 9, and so on. Next week we will search for patterns in even ordered magic squares.

The procedure is simple but you have to follow the algorithm perfectly:

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.

The trick to get into a rhythm of moves and follow with your pencil through each move. After you have finished your magic square, the last number n x n must appear in the bottom row in the middle. The number in the middle of the square between the 1 and the n x n number must be the median (middle) number between 1 and the n x n number. The children say how to generate the median number by adding 1 and the n x n number and dividing the sum by two. Some students noticed that each multiple of the nth number appears right before a block (rule 5 above). These observations are good ways to check your progress.

After you complete your magic square, find the magic number by adding at least one row, one column and one diagonal. If your sums are the same, you have most likely found the magic number.

The attached pdf entitled Magic Square — Odd Ordered provides the rules for this method and a 25-step example for how this works on a 5 x 5 grid. Try this with the grids provided from 3 x 3 to 21 x 21. If you have a question of what to do, look at the five rules and look at the example. 

I also provided a pdf with answers to 3x3 to 13x13 magic squares with the magic number.