New Squares from the Modified De La Loubère Method

Part IB

A Square


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. This page switches the starting and final numbers around on the main diagonal but leaves the diagonal in the smaller square in either forward or reverse consecutive order.

Modified Method I using the L-leap approach

To expand a modified De La Loubère square using the L-leap approach :

  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 in an opposite manner to complete the main 5x5 diagonal and fill in the first cell in the second row with the number 4, since 3 was the last number on the complementary list.
  3. Perform a L-leap, first across the square then up/down.
  4. Take the complement of 10, i.e. 16, and this time L-leap up/down the square, filling in all the complementary cells up to the number 18 in the last row.
  5. L-leap to the upper left corner cell fill in with the number 19 thenL-leap across the square back to the last column and continue until the square is completed.

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1
                   
    241 14    
   3 13 23    
   12 25 2    
                   
   ⇒   
2
                11
4 241 14    
   3 13 23    
   12 25 2    
15                
   ⇒   
3
    10     8 11
4 241 14    
   3 13 23 5
612 25 2    
15     9     7
   ⇒   
4
    10 17 8 11
4 241 14    
   3 13 23 5
612 25 2    
15 16 9 18 7
   ⇒   
5
19 10 17 8 11
4 241 14 22
213 13 23 5
612 25 2 20
15 16 9 18 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


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The last three examples in the series (A, B and C) are shown below after repeating the algorithm starting at the only three other positions on the square that produce magic squares:

       
A
19 8 17 10 11
6 241 14 20
213 13 23 5
412 25 2 22
15 18 9 16 7
                                 
B
7 22 5 20 11
16 241 14 10
93 13 23 17
1812 25 2 8
15 4 21 6 19
                                 
C
7 20 5 22 11
18 241 14 8
93 13 23 17
1612 25 2 10
15 6 21 4 19


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Modified Method II using a modified anti-Loubère square and a modified L-leap approach

To expand a modified 3x3 De La Loubère square with a reverse consecutive main diagonal (anti-Loubère) and using the L-leap approach, we use a modified L-leap method to fill in the boundary to produce an expanded magic square from a non-magical anti-Loubère square:

  1. Construct a 3x3 Loubère non-magic square with the main diagonal reversed and incorporate into a larger 5x5 square.
  2. Fill in the two corner cells to complete the main 5x5 diagonal.
  3. Fill in the first cell in the fourth row with the number 4, since 3 was the last number on the complementary list and perform a L-leap, first across the square then up/down.
  4. Using the required pair table below modify the L-leap and take the complement of 10, i.e. 16, but this time L-leap opposite to the number 8 and insert 16. L-leap across and fill in the first and last rows up to the number 19.
  5. L-leap opposite to 4 insert a 20 then L-leap up the square filling in all the empty cells.

********************************************************************************************************************************************************
1
                   
    241 12    
   3 13 23    
   14 25 2    
                   
   ⇒   
2
                11
    241 12    
   3 13 23    
   14 25 2    
15                
   ⇒   
3
    8     10 11
6 241 12    
   3 13 23 5
414 25 2    
15     9     7
   ⇒   
4
    8 17 10 11
6 241 12    
   3 13 23 5
414 25 2    
15 16 9 18 7
   ⇒   
5
19 8 17 10 11
6 241 12 22
213 13 23 5
414 25 2 20
15 16 9 18 7


ROWSum of Required PairCOLUMN
2284
3263
4242


The next example in the series, D, is shown below after repeating the algorithm starting at the only other position on the square that produces magic squares:

D
7 20 5 22 11
18 241 12 10
93 13 23 17
1614 25 2 8
15 4 21 6 19


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

Modified Method III using a modified anti-Loubère square and a modified L-leap approach

To expand a modified 3x3 De La Loubère square with a partially reversed consecutive main diagonal where only that part of the modified Loubère square is reversed, we use the L-leap approach. This approach uses a modified L-leap method to fill in the boundary and produce an expanded magic square from a non-magical anti-Loubère square:

  1. Construct a 3x3 Loubère non-magic square with only the Loubère main diagonal reversed and incorporate into a larger 5x5 square.
  2. Fill in the two corner cells in normal fashion (non-reversed) to complete the main 5x5 diagonal.
  3. Fill in the second cell in the first row with the number 4, since 3 was the last number on the complementary list and perform a L-leap, first up/down the square then across.
  4. Using the required pair table above modify the L-leap and take the complement of 10, i.e. 16, but this time L-leap opposite to the number 8 and insert 16. L-leap across and fill in the first and last rows up to the number 19.
  5. L-leap opposite to 4 insert a 20 then L-leap up/down the square filling in all the empty cells.

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

1
                   
    241 12    
   3 13 23    
   14 25 2    
                   
   ⇒   
2
                15
    241 12    
   3 13 23    
   14 25 2    
11                
   ⇒   
3
    4     6 15
10 241 12    
   3 13 23 9
814 25 2    
11     5     7
   ⇒   
4
    4     6 15
10 241 12 18
173 13 23 9
814 25 2 16
11     5     7
   ⇒   
5
19 4 21 6 15
10 241 12 18
173 13 23 9
814 25 2 16
11 20 5 22 7


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

The next example in the series, E, is shown below after repeating the algorithm starting at the only other position on the square that produces magic squares:

E
7 16 9 18 15
22 241 12 6
53 13 23 21
2014 25 2 4
11 8 17 10 19


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This completes this section on new squares from modified wheel and De La Loubère methods. To continue using a modified version of this method 3x3 and 7x7 squares (Part IC). To go back to Part IA of new Loubère squares. To return to homepage.


Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com