New Generalized Procedure of Loubère Method (Part I)

A Discussion of the New Method

An important general principle for generating odd magic squares by the De La Loubère 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 Loubère 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 7x7 regular Loubère squares is shown below as an example:

********************************************************************************************************************************************************

Normally the Loubère method involves a stepwise approach of consecutive numbers, i.e., 1,2,3... In previous methods I showed that these numbers do not have to be consecutive but may be placed semi-consecutively both frontwards or backwards as in Pendulum method I and Pendulum method II.

If the numbers are added non-consecutively,using a 7x7 example i.e. partially or fully random heptad 1,7,4,3,2,5,6 then it gets patricularly tricky keeping tracks of how the next subsequent numbers are added to complete the rest of the broken diagonals. I will show that there are two ways to accomplish this, both of which give different squares using the same heptad of opening numbers. Using the heptad above, the next number 8 can come after the 7 (the last number of the heptad) or after the 6 (the last number of the set). This method in fact may be used for any group of numbers (random or consecutive) to produce magic or semi-magic squares and may, therefore, be classified as general.

In all our examples we will be using 7x7 tables, so therefore, the heptad table is first produced by inserting in the first 7 numbers in the first row. To acquire the next row of heptad n (in this case 7) is added to all the previous numbers. This is continued until the heptad table is filled. See the first example below. Both translational and knight breaks are shown. ********************************************************************************************************************************************************

Construction of Generalized Procedure of (Random or Symmetrical) Loubère Square

7x7 Squares

1. Generate the heptad table.
2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
3. Move one cell right from the number 7 (the last number of the heptad).
4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.

1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
2. Move one cell down from the number 7 (the last number of the heptad).
3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.

1. This example is included to show that Method II: Group IA turns out to be non-magic.
2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
3. Move one cell right from the number 3 (the last number of the set of the heptad).
4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.

1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
2. Move one cell down from the number 3 (the last number of the set of the heptad).
3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.

Construction of Generalized Procedure of (Random or Symmetrical) Knight-Break Loubère Square

Below is again listed the heptad table I and is shown here for easier viewing. These are the only two groups that are either magic or semi-magic.

1. This group of squares is semi-magic.
2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
3. Using a knight break, move one cell up and two cells right from the number 3 (the last number of the set of the heptad).
4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.

1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
2. Using a knight break, move two cells down and one cell left from the number 3 (the last number of the set of the heptad).
3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.

Results of a typical Generalized Procedure of Loubère Odd n Square (n divible by 3)

Only the Method I:Group B and Method II:Group A squares are semi magic, the rest are non-magic.

This completes this section on De La Loubère all inclusive squares (Part I). The next section deals with new Méziriac general squares (Part II). To return to homepage.