New Generalized Procedure for Magic Squares (Part III)

Loubère and Méziriac Type Squares

A Discussion of the New Method

An important general principle for generating odd magic squares by the Méziriac method is that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n² + 1). The properties of these regular or associated Méziriac squares are:

1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n² + 1, i.e., or twice the number in the center cell and are complementary to each other.

The 5x5 regular Méziriac square is shown below as an example:

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Normally the Méziriac 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 on this page or down as in the next page (continuation) (Part IV). 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 Méziriac 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 to the Right

1. Place a 1 to the right of the center cell of the middle row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move one cell right.
4. Repeat the process until the square is filled as shown below in squares 1-5.
5. Move all columns 2 cellsto the right;this square is not magic.

1. Place a 1 to the right of the center cell of the middle row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move two cells right.
4. Repeat the process until the square is filled as shown below in squares 1-6.
5. This square is the original Méziriac square as shown in the introduction. This square is semi-magic in that the left main diagonal is magic but the broken diagonals differ by some multiple of n.

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

1. Place a 1 to the right of the center cell of the middle row.
2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
3. Move four cells right; same as one cell left .
4. Repeat the process until the square is filled as shown below in squares 1-6.
5. As shown below all columns need to be moved 1 space to the right for the square to be magic.

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

Summary of Generalized Procedure for Right Break 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: R(right). 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 three 7x7 examples are shown in white in the following table:

This completes this section on the new generalized procedure (Part I). The next section deals with Continuation of a new generalized procedure (Part IV). To return to homepage.