How To Solve Magic Squares

(Magic Square: A square in which the sum of each row, column, and diagonal equals the same number.)

I've always wondered whether there was any trick to solving these. Today, my math teacher revealed the secret to me. I'm so thrilled that I finally learned how to do these that I have to share the secret.

I will use a 3×3 square as an example:

  1. Start with "1" in the top row, middle column.
    Example:
         1     
               
               

  2. Go up one and to the right one space. In this case, up one rolls down to the bottom row.
    If you're on the top row and have to go up, then move to the bottom row; if you're on the right-most column and have to go right, go to the left-most column. The only exception is in the upper-right corner.
    Example:
         1     
               
             2 

  3. Up one, right one. Since the "2" is in the right-most column, move up one and switch to the left-most column.
    Example:
         1     
     3         
             2 

  4. Up one, right one. Since the space up one, right one to the "3" is already occupied, automatically drop the "4" right below the "3."
    Example:
         1     
     3         
     4       2 

  5. Up one, right one.
    Example:
         1     
     3   5     
     4       2 

  6. Up one, right one.
    Example:
         1   6 
     3   5     
     4       2 

  7. Since the "6" is in the upper-right corner, drop the next number right below it.
    Example:
         1   6 
     3   5   7 
     4       2 

  8. Up one, right one. Roll over to the left-most column.
    Example:
     8   1   6 
     3   5   7 
     4       2 

  9. Up one, right one. Roll down to the bottom row. The last number is always in the bottom row, center column.
    Example:
     8   1   6 
     3   5   7 
     4   9   2 

Every row, column, and diagonal adds up to 15 in this 3×3 square.

My explanations suck, but it's a little easier as the squares get bigger. Here's what an 11×11 square looks like. The rows, columns, and diagonals add up to 671.

 68   81   94   107   120   1   14   27   40   53   66 
 80   93   106   119   11   13   26   39   52   65   67 
 92   105   118   10   12   25   38   51   64   77   79 
 104   117   9   22   24   37   50   63   76   78   91 
 116   8   21   23   36   49   62   75   88   90   103 
 7   20   33   35   48   61   74   87   89   102   115 
 19   32   34   47   60   73   86   99   101   114   6 
 31   44   46   59   72   85   98   100   113   5   18 
 43   45   58   71   84   97   110   112   4   17   30 
 55   57   70   83   96   109   111   3   16   29   42 
 56   69   82   95   108   121   2   15   28   41   54