New Modified Wheel and De La Loubère Methods

A Discussion of These New Methods

The modified nxn wheel methods A-1 and the modified De La Loubère are constructed using complementary tables of (n+2)x(n+2) or greater. This generates a series of De La Loubère or wheel magic squares that are related to one another via the main diagonal within the same complementary set. These n_s x n_s consists of a smaller subset of complementary numbers chosen from the larger 1.. n² complementary table. 3x3 squares of the same type produced by wheel method A-1 are identical to a modified De La Loubère but differ only in the way we view them. Not so the larger squares. De La Loubère's method is well known and is described using applets in De La Loubère.

To construct a modified 3x3 De La Loubère square we take a 5x5 or higher odd nxn complementary table and choose three consecutive numbers and their complements such as {1,2,3} and {23,24,25} and begin filling in the Loubère square. The left diagonal, however, is filled with the group of numbers ½(n²-n+2) to ½(n²+ n) of the 5x5 complement table, i.e. {12,13,14} followed by the rest of the complementary numbers as shown below on the left beginning at the blue cell 23 or at the right at blue cell 22:

The second set of squares is obtained by moving one number over on the complementary table, which previously had to be moved over by a pair of numbers Method A-1:Variant 1. Because there are more than enough numbers in the complementary table to form the smaller squares, this restriction no longer holds. One keeps shifting over until all the squares within this group are obtained as shown below for the final square which after subtracting 8 from each entry gives the original De La Loubère 3x3 square on the right. This is a property of these squares in that the last one in the series, being the only one with consecutive integers, is convertible to the original normal Loubère square :

To generate 3x3 modified Wheel squares using a 5x5 complementary table we use the three pair of spoke numbers along with {12,13,14} to produce the series of squares below with (identical to the modified De La Loubère method discussed above except for rotation):

Again the last square, by subtracting 8 from each entry, produces the regular 3x3 square. In addition, the magic sum S is equal to 39 for these squares and the squares are found to be identical to the internal squares produced in method B. Those produced by the de la Loubère method, however, are not part of any larger de la Loubère square, i.e. are not internal squares to that method. As mentioned in the introduction the smaller n, n_s, is included in the larger n making the equation for the magic sum S is ½n_s(n² + 1). Plugging the requisite numbers into this equation gives S of 39 and 125, respectively for the the 3x3 magic squares above and 125 for the 5x5 squares in the section depicted below. In addition, these calculations are found to be in agreement with the actual summations.

The Number of Squares Generated per Method

To determine the number of magic squares in each group requires the use of two equations. The equation for the number of possible De La Loubère squares is equal to ½(n² + 1) - ½((n_s)²- 1) and that for the wheel method is ½(n² + 1) - 2(n_s - 1) where n is used for generating the complementary table and n_s is used for generating the magic square. Using these equations the following table is constructed. It displays the number of Loubère magic squares and the number of wheel conformations (since because of interchangeability of pairs the number of squares is actually higher):

5x5 squares from the 7x7 complementary table

The 5x5 squares are where we deviate, since the Loubère squares are no longer equivalent to those constructed from the wheel method. Below is shown the series of 5x5 squares obtained by shifting one entry at a time on the 7x7 complement table:

The 13th and last square in this group is identical to well known normal de La Loubère this time by subtracting 12 from each entry. The modified 5x5 wheel squares are shown below. If we take the subsets {13-24} and {26-37} and {25} then we can construct the 13th, 15th and 17th squares. subtracting 12 from each entry gives the same subset produced in method A-1 as shown in the last three squares (note that the rightmost third cell entry shows the square number). Squares 14th and 16th, however, are not part of that group, since only these three conformations are allowed by the previous equation ¼(n²-4n + 7).

This completes this section on new wheel and De La Loubère methods. The next sections deal with the implementation of these methods to create new types of magic squares which can be expanded into larger squares (Part IA) by expanding the boundary. To return to homepage.