New Generalized Procedure for Magic Squares Continuation (Part II)

Loubère and Méziriac Type Squares

A Discussion of the New Method

Normally the Loubère method involves a stepwise approach of consecutive numbers, i.e., 1,2,3... In this report the initial number 1 does not have to be placed at the middle of the first row cell but may be placed in any available position on the first row or the center column to give modified Loubère squares. In addition, the translational moves following a break may be either to the right as in A new generalized procedure (Part I) or down as on this page. For simplicity 5x5 squares will be used for demonstration. Fully formed 7x7 examples will be shown at the end.

Using first principles we use one configuration of squares to generate all the other configurations in the set. For example, the first step involves always placing the initial number 1 onto the middle cell of the first row and doing the stepwise Loubère addition of numbers onto the square. If the square (in this case a 5x5) has the right number set 11,12,13,14,15 on the right main diagonal nothing has to be done. If these numbers are not on the main right diagonal then column or rows must be moved an m number of moves, where m is 1..n until the square is in the right configuration. This new configuration will be shown to produce both Loubère and Méziriac type Squares.

Construction of Generalized Procedure for Loubère and Méziriac Squares

A Break Vertically Down

1. Place a 1 in the center cell of the first row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move one cell down.
4. Repeat the process until the square is filled as shown below in squares 1-5.
5. As shown below no column or row need be moved and the square is the known regular magic shown at the beginning.

1. Place a 1 in the center cell of the first row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move two cells down.
4. Repeat the process until the square is filled as shown below in squares 1-6.
5. As shown below all rows need to be moved 2 spaces up for the square to be magic.

1. Place a 1 in the center cell of the first row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move three cells down.
4. Repeat the process until the square is filled as shown below in squares 1-6.
5. As shown below all rows need to be moved 1 space down for the square to be magic.

1. Place a 1 in the center cell of the first row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move four cells down.
4. Repeat the process until the square is filled as shown below in squares 1-6.
5. As shown below all rows need to be moved 1 space up but the square is not magic.

1. Place a 1 in the center cell of the first row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move five cells down. The next number 6 would land directly over the 5, so that this square is not magic.

Summary of Generalized Procedure for Break Down Squares

This table summarizes square sizes 5x5 to 11x11 for the generalized procedure above. Each of the cells corresponds to where the initial number 1 would go along with the number of moves after the break: D(down). The semi-magic squares are in light green color, while the non-magic squares are in orange color. Squares formed from n div by 3 forms only one magic square in the semi-magic set; the other n - 1 are semi-magic squares. In addition, the four 7x7 examples are shown in white in the following table:

This completes this section on the new generalized procedure (Part II). The next section deals with A new generalized procedure (Part III) starting with the Méziriac procedure. To return to homepage.