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Method for Magic Squares

A magic square is a square grid of positive integers (whole numbers greater than zero) where each row, each column and each of the two main diagonals sums to the same number. Usually, the numbers in the cells are consecutive integers starting with 1, but this is not strictly necessary. The method for constructing a magic square depends on the number of rows and columns. There is one method for odd number of squares (3, 5, 7...), one for even numbers that are not multiples of 4 and one for even numbers that are multiples of 4. The latter two types are called singly even and doubly even.

Instructions

Magic Squares of Odd Order
- 1
  Write the number 1 in the top middle cell.
- 2
  Attempt to move to the right and upwards. If you are at the top row, move to the bottom; if you are at the right, move to the column farthest on the left. Write the number 2. (The two will be in the bottom row, one cell to the right of the middle). If the cell to the top right is already full, go one row down from that cell.
- 3
  Repeat Step 2 until the square is full.
Magic Squares of Doubly Even Order
- 4
  Write the number 1 in the upper left, then continue to number the squares from left to right and top to bottom, so that 2 is to the right of 1.
- 5
  Draw lines from the top left cell to the bottom right cell, the top right cell to the bottom left cell, the top middle cell to the middle cell on the left, the middle cell on the left to the middle cell on the bottom, the middle cell on the bottom to the middle cell on the right and the middle cell on the right to the middle cell on the top. The first two lines form an X and the last four form a diamond. In a square of order 4, only the X can be drawn.
- 6
  Reverse the order of the numbers along each line. For example, in a square of order 4, the diagonal from the top left to the bottom right goes through the numbers 1, 6, 11 and 16. Switch the 1 with the 16 and the 6 with the 11, then do the same for the other diagonal.
Magic Squares of Singly Even Order
- 7
  Divide the square into 4x4 squares. There will be an odd number of 4x4 squares.
- 8
  Number the 4x4 squares as if they were an odd order magic square. Put these numbers in pencil.
- 9
  Divide the number of rows in the magic square by 4 and round down. Call this m. For example, if there are 6 rows then 6/4 = 1.5 rounded down = 1.
- 10
  Label the 4x4 squares in the top m+1 rows with L, the next row with U and the bottom m+1 rows with X. Switch the middle U with the L above it.
- 11
  Starting with the 4x4 square labeled 1 in step 2, fill in consecutive integers in the pattern of the letter (L, U or X) that is associated with that 4x4 square. The 4x4 square labeled 1 will have 1, 2, 3 and 4; the 4x4 square labeled 2 will have 5, 6, 7 and 8. If the 4x4 square has an L, put the numbers starting in the upper right, then lower left, then lower right, then upper left. If the 4x4 square has a U, then start in the upper left, then the lower left, then the lower right, then the upper right. Finally, if the 4x4 square has an X, start in the upper left, then the lower right, then the lower left then the upper right. Do this for each 4x4 square in order.

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