Making odd size magic squares

This is the basic technique that can be used to make magic squares whose side length is odd: 3, 5, 7, 9, …

  1. Draw the table.
  2. Put the number 1 below the center cell.
  3. Put the next number to the right on the row below, wrapping around when necessary.
  4. If the next cell is already full, put the next number two rows below, wrapping around when necessary.

So to make a 3 by 3 square, the first step is to draw the table and put the number 1 under the center cell.

Then put the number 2 to the right and below number 1. Number 2 ends up in the upper right cell, as it is necessary to wrap around when the move exceeds the last row of the table. Accordingly, number 3 ends in the middle left cell, after wrapping around when the move exceeds the right column of the table.

As the cell to the right and below number 3 is already filled, number 4 ends two rows below number 3. With the wrap around, this means the upper left cell of the table.

From then we can resume the below-right move, with one more transition by the 2-rows-below move when the next cell below-right 6 is already filled by 4.

When all the cells are filled, the result is a 3 by 3 magic square. The sums of the numbers in each row, each column and each diagonal are all equal to 15.

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One Response to Making odd size magic squares

  1. Pingback: Making canonical magic squares | Squared Magick

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