New Modified Wheel and De La Loubère Methods

A Spinning wheel

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 ns x ns 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:

114
3 13
12 2
24114
3 13 23
12 25 2
  
214
4 13
12 3
23214
4 13 22
1224 3
 ...
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 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 :

914
11 13
12 10
16914
11 13 15
12 17 10
816
357
49 2
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):

 
12324
25 13 1
223 14
 
12423
24 13 2
3 22 14
 
12522
23 13 3
4 21 14
...
121116
17 13 9
1015 14
438
95 1
27 6

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, ns, is included in the larger n making the equation for the magic sum S is ½ns(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((ns)2- 1) and that for the wheel method is ½(n2 + 1) - 2(ns - 1) where n is used for generating the complementary table and ns 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
 LoubèreWheelLoubèreWheelLoubère WheelLoubèreWheel
n 55 7799 1111
ns   
399 21213737 5757
-- 13172933 4953
7-- --1729 3749
9-- ---- 2145

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 18 27
47 5 7 26 40
4 6 25 44 46
10 24 43 45 3
2342 49 2 9
 
Square 2
40 47 29 27
46 6 8 26 39
5 7 25 43 45
11 24 42 44 4
2341 48 3 10
 
Square 3
39 46 310 27
45 7 9 26 38
6 8 25 42 44
12 24 41 43 5
2340 47 4 11
...
Square 13
29 36 1320 27
35 17 19 26 28
16 18 25 32 34
22 24 31 33 15
2330 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 {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).

Square 1
23 7 543 47
924 6 46 40
49 48 252 1
41 4 44 26 10
342 45 8 27
...
Square 13
23 19 1731 35
2124 18 34 28
3736 2514 13
29 16 32 26 22
1530 33 20 27
 
Square 15
23 13 1937 33
2124 20 32 27
35 34 2516 15
29 18 30 26 22
1736 3114 27
 
Square 17
23 13 2137 31
1524 22 30 34
33 32 2518 17
35 20 28 26 16
1936 2914 27
11 7 511 23
912 6 22 16
2524 132 1
17 4 20 14 10
318 21 8 15
 
11 1 725 21
912 8 20 15
2322 134 3
17 6 18 14 10
524 19 2 15
 
11 1 925 19
312 10 18 22
2120 136 5
23 8 16 14 4
724 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. E-Mail: Fiboguti89@Yahoo.com