Magic Squares

  Magic squares are those wonderful mathematical constructions where the sums vertically, horizontally, and diagonally are all the same.

There is a lot of math behind creating a 3 x 3 magic square.  One way is to place the basic number in the center square but it should be greater than four because the numbers should be positive.

The grid to the right shows the mathematics involved in creating a 3 by 3 square.  The lowest number n equals is 5 so none of the squares are zero. 

In addition, if you want larger numbers you simply have to add a value to n for all squares so the square remains a magic square.

For a 4 x 4 magic square, its not that difficult.  Begin by creating a 4 x 4 square.  Starting at the upper left most square, write one, then count each square till you get to four which should be at the upper right corner.  Continue counting and placing numbers which fall along the diagonal.  This means you have used the numbers, 1, 4, 6, 7, 10, 11, 13, and 16.  Then start at the bottom and work upwards, filling in the unused numbers so 2 is next to 16 and 3 is next to 13.  5 is above the 16 and 9 is between the 5 and the 4 while the 8 is above the 13 and the 12 between the 8 and the 1.  That leaves the 15 in the first row next to the 1 and 14 in the top row next to the four.  So it all shows up as
1, 15, 14, 4
12, 6, 7, 9
8, 10, 11, 5
13, 3, 2, 16

The cool thing is that you can move the rows or columns around so the digits are in the same order in the rows and the columns in order to get the same squares.

Have fun letting your students attempt to find the method behind the 3 x 3 squares first before showing them the mathematics.  Then let them move on to the 4 x 4 square.

Let me know what you think.  I'd love to hear.  Have a great weekend.