New Squares from the Modified De La Loubère Method

Part I

A Spinning wheel


A Discussion of New De La Loubère type squares

The previous page showed how to prepare Modified De La Loubère squares of order n where the main diagonal is constructed from a complementary table of (n+2)x(n+2) or greater. this page will show how we can take these modified magic squares and build them up into larger squares that take into account all of the (n+2)x(n+2) or greater numbers of the complementary table used. Two methods will be shown one using a sweep approach, the other a wheel approach.

To expand a modified De La Loubère square using the approach we take an nxn and increase the number of rows and columns by 2 or greater. Using the 3x3 example we:

  1. Incorporate the smaller square as constructed previously into a larger square as shown below using a 3x3 inserted into a 5x5.
  2. Fill in the two corner cells to complete the main diagonal.
  3. Fill in the first cell adjacent to the 15 in this example with the number 4, since 3 was the last number on the complementary list.
  4. Perform a sweep, i.e. place one number in a cell sweeping across up/down on the square to the next column or right/left to the next row and insert the next number.
  5. Stop at the first cell in the second column inserting a 6, leap across to the right lower corner and insert a 7, then leap to the second cell in the first column and insert an 8.
  6. Continue jumping right and left across until you reach the first cell in the fourth row.
  7. Take the complement of 10, i.e. 16, and this time jump backwards across the square, filling in all the complementary cells up to the left corner cell in the first column.
  8. Sweep down and up filling in all the cells with the right complementary numbers until the square is completed.

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1
                   
    241 14    
   3 13 23    
   12 25 2    
                   
   ⇒   
2
                15
    241 14    
   3 13 23    
   12 25 2    
11                
   ⇒   
3
            4 15
    241 14    
   3 13 23    
   12 25 2    
11                
   ⇒   
4
    6     4 15
    241 14    
   3 13 23    
   12 25 2    
11     5        
   ⇒   


5,6
    6     4 15
8 241 14    
   3 13 23 9
1012 25 2    
11     5     7
   ⇒   
7
19 6     4 15
8 241 14 18
173 13 23 9
1012 25 2 16
11     5     7
   ⇒   
8
19 6 21 4 15
8 241 14 18
173 13 23 9
1012 25 2 16
11 20 5 22 7


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


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

The above gives one way of filling up a square by the sweep approach. A second method fills it in an opposite manner gives a different variant:

  1. Use square 2 from above.
  2. Fill in the first down cell adjacent to the 15 in this example with the number 4.
  3. Perform a sweep, i.e. place one number in a cell sweeping across the square left/right to the next row and insert the next number.
  4. Stop at the fourth cell in the fifth column inserting a 6, leap across to the left upper corner and insert a 7, then leap to the fifth cell in the fourth column and insert an 8.
  5. Continue jumping up and down until you reach the fifth cell in the second column.
  6. Take the complement of 10, i.e. 16, and this time jump backwards(up/down) across the square, filling in all the complementary cells up to the bottom right corner cell in the fifth column.
  7. Sweep left and right filling in all the cells with the right complementary numbers until the square is completed.

       
1
                15
    241 14    
   3 13 23    
   12 25 2    
11                
   ⇒   
2
                15
    241 14 4
   3 13 23    
   12 25 2    
11                
   ⇒   
3,4
                15
    241 14 4
53 13 23    
   12 25 2 6
11                
   ⇒   
4,5
7     9     15
    241 14 4
53 13 23    
   12 25 2 6
11 10     8    
   ⇒   
6,7
7 16 9 18 15
22 241 14 4
53 13 23 21
2012 25 2 6
11 10 17 8 19


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

To expand a modified De La Loubère square using the wheel approach we take an nxn and increase the number of rows and columns by 2 or greater. Using the 3x3 example we:

  1. Incorporate the smaller square as constructed in new Loubère methods into a larger square as shown below using a 3x3 inserted into a 5x5.
  2. Fill in the two corner cells to complete the main diagonal.
  3. Fill in the spoke cells as shown in method A variant 1, however, reverse the addition due to rotation of the square by 90° degrees.
  4. Fill in the non-spoke cells using parity as shown previously, inserting the L numbers from the complementary table.
  5. Comparing this square to the 4 rotated version shows that the non-spoke numbers are opposite to the normal mode of addition as was shown in variant 1.
       
1
                   
    241 14    
   3 13 23    
   12 25 2    
                   
   ⇒   
2
                15
    241 14    
   3 13 23    
   12 25 2    
11                
   ⇒   
3
21     4     15
    241 14    
63 13 23 20
   12 25 2    
11     22     5
   ⇒   
4
21 7 4 18 15
10 241 14 16
63 13 23 20
1712 25 2 9
11 19 22 8 5
   ≡   
4 rotated
11 17 6 10 21
19 123 24 7
2225 13 1 4
82 23 14 18
5 9 20 16 15


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 expand a modified De La Loubère square using a second wheel approach we take an nxn and increase the number of rows and columns by 2 or greater. Using the 3x3 example we:

  1. Use square 2 above, but reverse the 5 and 21 corner positions.
  2. Fill in the rest of the spoke numbers in a reverse manner than was done above for example 3 square 3.
  3. Fill in the non-spoke cells using parity as shown previously, inserting the L numbers from the complementary table.
       
1
5             15
    241 14    
   3 13 23    
   12 25 2    
11             21
   ⇒   
2
5     20     15
    241 14    
223 13 23 4
   12 25 2    
11     6     21
   ⇒   
3
5 7 20 18 15
10 241 14 16
223 13 23 4
1712 25 2 9
11 19 6 8 21


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

A property of these squares is that another magic square may be generated from the original by shifting pairs of numbers in the external boundary. Only pairs are permitted since since if only one number and its complement are shifted the square is no longer magic. To perform this:

  1. Insert a 3x3 modified Loubère square into a 5x5 square (in this case the starting number is 3).
  2. Fill in the two corner cells to complete the main diagonal.
  3. Sweep in the boundary numbers.
  4. Remove the first two pairs (1,2) and and move all the numbers across to fill in the empty position.
  5. Fill in the last two positions with the (1,2) pairs.

       
1,2
                15
    223 14    
   5 13 21    
   12 23 4    
11                
   ⇒   
3
19 6 24 1 15
8 223 14 18
175 13 21 9
1012 23 4 16
11 20 2 25 7
   ⇒   
4,5
17 8 19 6 15
10 223 14 16
255 13 21 1
212 23 4 24
11 18 7 20 9


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

The Number of Squares Generated for both Loubère's Method

The number of magic squares in each group may be determined by the use of two equations. the equation for the number of total groups, magic and non-magic De La Loubère using the sweep approach uses the modified equation ½(n2 + 1) - ½(ns2- 1) - 1 which after simplification becomes ½(n2 - ns2). A group consists of the internal Loubère square along with the external sweep boundary, i.e., making up the entire larger square. Although the internal square remains the same, the boundary may differ taking on a variety of values, some of which give rise to magic squares or to non-magic squares. this will be easily seen on the next page using and internal 7x7 modified Loubère square inside a 9x9 square.

The total number of magic De La Loubère squares using the wheel approach to finish off the square (the third example), however, uses the equation ns(n - 1), to account for the larger number of possibilities of generating these squares. Again ns and n correspond to the smaller and the larger square, respectively. Since this last equation gives only the total number of groups of squares, the following tables show the distribution of magic squares for the entire group, i.e ns = 3, 5, 7, ... only that half of the table is shown (to conserve space and show how the numbers are related to the rest of the entries). To obtain the rest of the entries the rest of the table is expanded using symmetry. See example following the tables.

Number of Loubère Square Groups with the wheel outer structure
magic square 12 3456 78910
ns      n n0 = 3
3        5n0 n0-3 n0-1n0-2- -----
5        7n0+2 n0-3 n0+1n0-2 n0 n0-1----
7        9n0+4 n0-3 n0+3n0-2 n0+2 n0-1n0+1n0 --
9       11ns+6 n0-3 n0+5n0-2 n0+4 n0-1n0+3n0 n0+2n0+1


********************************************************************************************************************************************************
Number of Loubère Square Groups with the wheel outer structure (calculated from the above table)
magic square 12 3456 78910
ns      n   
3        5  3    0     2    1     -     -    -    -     -    -  
5        750 413 2----
7        970 615 243 --
9       1190 817 263 54


For example the entry for line 1 is 3 0 2 1 1 2 0 3. the sum of these numbers is 12 identical to the answer obtained from the equation ns(n - 1). this total is doubled since the two corner positions on the external boundary may be inverted as shown in example 4.

this completes this section on new squares from modified wheel and De La Loubère methods. To continue this method using 7x7 squares. To return to home.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com