New Modified Wheel and De La Loubère Methods
A Discussion of These New Methods
The modified nxn wheel methods A1 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..
n^{2} complementary table. 3x3 squares of the same type produced by wheel method A1 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(n^{2}n+2) to ½(n^{2}+
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:
1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
 
1  2 
3  4 
5  6 
7  8 
9  10 
11  12 

 13  
 13 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 
 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 

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 A1: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 :

1  2 
3  4 
5  6 
7  8 
9  10 
11  12  
 13 

25  24 
23  22 
21  20 
19  18 
17  16 
15  14  
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}(n^{2} + 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(n^{2} + 1)  ½((n_{s})^{2} 1) and
that for the wheel method is
½(n^{2} + 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):
Number of Squares
 Loubère  Wheel  Loubère  Wheel  Loubère 
Wheel  Loubère  Wheel 
n  5  5 
7  7  9  9 
11  11 
n_{s}  
3  9  9 
21  21  37  37 
57  57 
    
13  17  29  33 
49  53 
7     
    17  29 
37  49 
9     
       
21  45 
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:
Square 1
41  48 
1  8 
27 
47  5 
7  26 
40 
4  6 
25  44 
46 
10  24 
43  45 
3 
23  42 
49  2 
9 


Square 2
40  47 
2  9 
27 
46  6 
8  26 
39 
5  7 
25  43 
45 
11  24 
42  44 
4 
23  41 
48  3 
10 


Square 3
39  46 
3  10 
27 
45  7 
9  26 
38 
6  8 
25  42 
44 
12  24 
41  43 
5 
23  40 
47  4 
11 

... 
Square 13
29  36 
13  20 
27 
35  17 
19  26 
28 
16  18 
25  32 
34 
22  24 
31  33 
15 
23  30 
37  14 
21 

1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
13  14 
15  16 
17  18 
19  20 
21  22 
23  24  
 25 
49  48 
47  46 
45  44 
43  42 
41  40 
39  38 
37  36 
35  34 
33  32 
31  30 
29  28 
27  26  
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 {1324} and {2637} and {25} then
we can construct the 13th, 15th and 17th squares. subtracting 12 from each entry gives the same subset produced in method A1 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(n^{2}4n + 7).
Square 1
23  7 
5  43 
47 
9  24 
6  46 
40 
49  48 
25  2 
1 
41  4 
44  26 
10 
3  42 
45  8 
27 

... 
Square 13
23  19 
17  31 
35 
21  24 
18  34 
28 
37  36 
25  14 
13 
29  16 
32  26 
22 
15  30 
33  20 
27 


Square 15
23  13 
19  37 
33 
21  24 
20  32 
27 
35  34 
25  16 
15 
29  18 
30  26 
22 
17  36 
31  14 
27 


Square 17
23  13 
21  37 
31 
15  24 
22  30 
34 
33  32 
25  18 
17 
35  20 
28  26 
16 
19  36 
29  14 
27 

 
11  7 
5  11 
23 
9  12 
6  22 
16 
25  24 
13  2 
1 
17  4 
20  14 
10 
3  18 
21  8 
15 


11  1 
7  25 
21 
9  12 
8  20 
15 
23  22 
13  4 
3 
17  6 
18  14 
10 
5  24 
19  2 
15 


11  1 
9  25 
19 
3  12 
10  18 
22 
21  20 
13  6 
5 
23  8 
16  14 
4 
7  24 
17  2 
15 

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.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com