There are many different ways to construct magic squares. Below you'll find one way for odd-order magic squares (3 by 3, 5 by 5, and so on) and another way for even-order magic squares (4 by 4, 6 by 6, and so on).
ODD ORDER MAGIC SQUARES
1. Start by placing 1 in the middle top square.
2. Continue by always placing the next number in the square
diagonally up and to the right.
If that square is outside the grid, wrap around the square and place the number.
If that square is already filled, put the next number in the square below.
Now isn't that easy?!
Here's how to do a 3 by 3 square:
Place 1 in the top middle square. Go diagonally up and to the right one square to place the two. Since that square is outside the grid, wrap around and place the two in the bottom right hand square. Think about the grid as if it were a tube with the top of the grid wrapped around so it touches the bottom of the grid.
Now go up and to the right to place the three. Since that square is off the grid, wrap around and place the three in the middle square of the first column. Once again you have to think about the grid as a tube except this time, the right side wraps around so it touches the left side.
Going up and to the right puts us in a square that is already occupied by a number. In that case, put the next number, four, below the three.
Up and to the right places the five and the six easily. The seven should go up and to the right of six. If you think about it really hard, you can figure out that by wrapping up you end up at 2's square and going over to the right you wrap to 4's square. Since that square is taken by the 4, drop down and place the 7 below the 6.
Wrap around to the left side for the eight and then around the bottom for the 9 and you've got a completed magic square. Now trace the number path to see the design that this solution makes. All 3 by 3 magic squares create the same design (including rotations and reflections).
There are at least 15 other methods of completing magic squares. Can you discover one on your own?
EVEN ORDER MAGIC SQUARES
1. Start by filling the magic square with the numbers in order, starting in the upper left corner with one, going across to the right, and beginning a new row when you reach the end.
2. Change the order of the numbers in the two diagonals.
Here's how to do a 4 by 4 magic square.
Fill out the square using the numbers from 1 to 16.
Switch the numbers on the diagonals--1 and 16, 11 and 6 on one diagonal; 7 and 10, 4 and 13 on the other. That's it!
There are other swapping methods for even order magic squares. Try swapping the numbers that aren't in the diagonals, columns, rows, or rotating the numbers in squares within the bigger square. After you've created a new solution, trace the numbers to see if it's a new design or a variation of one you've already done. Good Luck!