As I mentioned earlier, this method was brought by De la Loubere to Europe from Siam in 1687 or 1688. The best way to describe this method is through an example. The following is an example of how to use the staircase method to find a normal magic square of order 5.
1. If you want to create a normal magic square of order n (where n is odd), the staircase method requires that you begin with an empty square of order n+1. In this case we want to create a normal magic square of order 5. So we begin with an empty square of order 6. The lower left seven rows and columns will make up our desired normal magic square of order 5. The first step is to darken in the upper right entry of the big square. Consider this space occupied in the future. Next enter a 1 in the middle column of the top row of the small (grey) square.
| 1 | |||||
2. The idea behind the staircase method is to fill in the entries of the large square with consecutive natural numbers by stepping upward from left to right. Doing so at this point we get the following.
| 2 | |||||
| 1 | |||||
3. We can not go upward further. The staircase method requires us in this case to shift the last recorded number downward to the last row of the column and continue stepping.
| 2 | |||||
| 1 | |||||
| 4 | |||||
| 3 | |||||
| 2 |
4. When you can no longer step to the right, you shift the last recorded number left to the leftmost column of that row and continue stepping.
| 2 | |||||
| 1 | |||||
| 5 | |||||
| 4 | 4 | ||||
| 3 | |||||
| 2 |
5. When you reach a spot where the stepping is blocked by an occupied entry, enter the next entry below the last recorded entry and continue stepping. In this example, once the 5 is recorded, the stepping is blocked by the 1 and 2. So we drop down one entry below the 5, enter a 6, and continue stepping.
| 2 | 9 | ||||
| 1 | 8 | ||||
| 5 | 7 | ||||
| 4 | 6 | 4 | |||
| 3 | |||||
| 2 |
6. Using the rules in 1-5 we can finish the stepping process.
| 18 | 25 | 2 | 9 | ||
| 17 | 24 | 1 | 8 | 15 | 17 |
| 23 | 5 | 7 | 14 | 16 | 23 |
| 4 | 6 | 13 | 20 | 22 | 4 |
| 10 | 12 | 19 | 21 | 3 | 10 |
| 11 | 18 | 25 | 2 | 9 |
7. We get the following normal magic square of order 5.
| 17 | 24 | 1 | 8 | 15 |
| 23 | 5 | 7 | 14 | 16 |
| 4 | 6 | 13 | 20 | 22 |
| 10 | 12 | 19 | 21 | 3 |
| 11 | 18 | 25 | 2 | 9 |
Here's another method for creating magic squares that is very similar to the staircase method:
1. Construct a square grid of order 2 more than that of the magic square you would like to create. I am going to create a magic square of order 3 so I constructed a square grid of order 5.
2. Choose any real number and put it in the middle cell of the left most column. I chose 1.
3. Choose another real number to add to this number and put the sum in the upper left cell of the grey cells. I chose 2.
4. Add the same number to this result and put the sum in the center cell of the top row.
5. Choose another real number to add to the number in the middle cell of the left column. Record the sum in the lower left grey cell. I chose 6.
6. Add the same number to this number and record the sum in the middle cell of the bottom row.
7. Complete the upward diagonals by adding the same number that you chose to add in step three.
8. Check the downward diagonals by making sure that the same number is added that you chose to add in step five.
9. Complete the magic square by shifting the numbers in the red squares into the grey cells on the opposite sides of the square.