New Squares from the Modified De La Loubère Method Con't

Part IC

A Spinning wheel


A Discussion of New De La Loubère type squares: the cross L-leap algorithm

The previous page showed a modified De La Loubère squares of order 3. This page is a continuation in that the L-leap method is modified so that the insertion of certain numbers into the boundary cells must go through a variable arm length cross path. The cross is normal only for a 5x5 and is distorted or unsymmetrical for 7x7 or larger.

Modified Method IV using the cross L-leap approach

To expand a modified De La Loubère square using the cross 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 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 an L-leap, to the right lower corner insert a 5 and proceed up and across the square to the number 8, the first arm of the cross. Move to just below and to the left of the number 4 (the number 1, the center of the cross) and move up one cell and insert a 9, the second arm of the cross.
  4. Move back away from the 4 in an L-leap way (since moving towards the 4 inserts the wrong complement for 4) then take the complement of 10, i.e. 16, and this time L-leap down the square, filling in the complementar of 9, the number 17, the third arm of the cross, move up to 1 and to the last column to fill in the number 18, the fourth arm of the cross.
  5. L-leap down and across the square to 20, jump to 21 and then back to 22 as shown. This completes the square.

********************************************************************************************************************************************************
                   
    241 14    
   3 13 23    
   12 25 2    
                   
   ⇒   
            4 15
    241 14    
   3 13 23    
   12 25 2    
11                
   ⇒   
        9 4 15
8 241 14    
   3 13 23 7
612 25 2    
11             5
   ⇒   
    16 9 4 15
8 241 14    
   3 13 23 7
612 25 2    
11 10 17     5
   ⇒   
21 16 9 4 15
8 241 14 18
193 13 23 7
612 25 2 20
11 10 17 22 5


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 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 which places 5 into either of two corner cells. Note that in B the entry 5 goes to the empty left top cell not next to the 4:

       
21 8 19 6 15
16 241 14 10
93 13 23 17
412 25 2 22
11 18 7 20 5
                                 
5 20 7 18 15
10 241 14 16
173 13 23 9
2212 25 2 4
11 6 19 8 21
                                 
5 20 7 18 15
22 241 14 4
173 13 23 9
1012 25 2 16
11 6 19 8 21


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

Modified Method V using a modified Loubère square and a modified cross L-leap approach

A modified 3x3 De La Loubère square with a semi consecutive main diagonal along with the cross L-leap method is used to to fill in the boundary and generate an expanded magic 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 cross L-leap, first across the square then up/down.
  4. Using the required pair table below modify the cross L-leap and take the complement of 10, i.e. 16, but this time cross L-leap opposite to the number 8 and insert 16. cross L-leap across and fill in the first and last rows up to the number 19.
  5. cross L-leap opposite to 4 insert a 20 then cross L-leap up the square filling in all the empty cells.

********************************************************************************************************************************************************
                   
    241 14    
   3 13 23    
   12 25 2    
                   
   ⇒   
                11
    241 14    
   3 13 23    
   12 25 2    
15                
   ⇒   
    8     6 11
    241 14    
93 13 23    
412 25 2    
15     7     5
   ⇒   
    8     6 11
16 241 14 10
93 13 23 17
412 25 2    
15     9     7
   ⇒   
21 8 19 6 11
16 241 14 10
93 13 23 17
412 25 2 22
15 18 7 20 5


the last three examples in the series (D, E and F) are shown below after repeating the algorithm starting at the only three other positions on the square that produce magic squares and place the 5 into either of two corner cells. Note that in D, the number 5 goes to the bottom corner, while in E it goes to the top left corner:

       
21 8 19 6 11
4 241 14 22
93 13 23 17
1612 25 2 10
15 18 7 20 5
                                 
5 10 17 22 11
20 241 14 6
73 13 23 19
1812 25 2 8
15 16 9 4 21
                                 
5 22 17 10 11
20 241 14 6
73 13 23 19
1812 25 2 8
15 4 9 16 21


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

Expanded 5x5 Loubère Squares using the Modified Method IV and V Approaches

In these examles 5x5 Loubère squares are expanded into 7x7 using the cross L-leap methods:

  1. Construct a 5x5 Loubère square and incorporate into a larger 7x7 square. Fill in the two corner cells in normal fashion (non-reversed) to complete the main 7x78 diagonal. Fill in the sixth cell in the first row with the number 11, since 10 was the last number on the complementary list.
  2. Perform a L-leap to the bottom right cell.
  3. Carry out an L-leap up and across to 17, the first arm of the cross, then to 18 the second arm by way of 8 (the center of the cross) then back to 21, enter 29 its complement then perform an L-leap to 32 and the to 33 the third and fourth arms also by by way of 8.
  4. Continue L-leaping down the square until all the empty cells are filled and fill in the last corner cell and finally 39.

********************************************************************************************************************************************************
                   11 28
    4148 1 8 27    
   47 5 7 26 40   
   4 6 25 44 46    
    10 24 43 45 3   
    23 42 49 2 9   
22                       
   ⇒   
                1811 28
17 4148 1 8 27    
   47 5 7 26 40 16
154 6 25 44 46    
    10 24 43 45 314
13 23 42 49 2 9   
22                    12
   ⇒   
    29 20 31 1811 28
17 4148 1 8 27 33
   47 5 7 26 40 16
154 6 25 44 46    
    10 24 43 45 314
13 23 42 49 2 9   
22 21 30 1932    12
   ⇒   
38 29 20 31 1811 28
17 4148 1 8 27 33
3447 5 7 26 40 16
154 6 25 44 4635
36 10 24 43 45 314
13 23 42 49 2 937
22 21 30 1932 3912


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

three other examples in this series (G, H and I) are shown below after repeating the algorithm starting at the only three other positions on the square that produce magic squares which places 12 into either of two corner cells. Note that in G, the number 12 goes to the bottom corner, while in H and I it goes to the top left corner:

       
38 11 18 31 2029 28
17 4148 1 8 27 33
3447 5 7 26 40 16
154 6 25 44 4635
36 10 24 43 45 314
13 23 42 49 2 937
22 39 32 1930 2112
                  
12 37 14 35 1633 28
39 4148 1 8 27 11
3247 5 7 26 40 18
194 6 25 44 4631
30 10 24 43 45 320
21 23 42 49 2 929
22 13 36 1534 1738
                  
12 37 14 35 1633 28
21 4148 1 8 27 29
3047 5 7 26 40 20
194 6 25 44 4631
32 10 24 43 45 318
39 23 42 49 2 911
22 13 36 1534 1738


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

The last four examples in this series (J, K, L and M) shown below begin with the number 28 in the main diagonal and switch positions between 12 and 38:

       
38 17 34 15 3613 22
11 4148 1 8 27 39
1847 5 7 26 40 32
314 6 25 44 4619
20 10 24 43 45 330
29 23 42 49 2 921
28 33 16 3514 3712
       
38 17 34 15 3613 22
29 4148 1 8 27 21
2047 5 7 26 40 30
314 6 25 44 4619
18 10 24 43 45 332
11 23 42 49 2 939
28 33 16 3514 3712
       
12 39 32 19 3021 22
37 4148 1 8 27 13
1447 5 7 26 40 36
354 6 25 44 4615
16 10 24 43 45 334
33 23 42 49 2 917
28 11 18 3120 2938
       
12 21 30 19 3239 22
37 4148 1 8 27 13
1447 5 7 26 40 36
354 6 25 44 4615
16 10 24 43 45 334
33 23 42 49 2 917
28 29 20 3118 1138


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

this completes this section on new squares from modified wheel and De La Loubère methods. To continue the method of IA using 7x7 and 9x9 squares (Part IIA). To go back to Part IB of new Loubère squares. To return to homepage.


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