Modified Magic Square Wheel and De La Loubère Methods

New Squares from the Modified De La Loubère Method

Part IIB

A Square


A Discussion of New De La Loubère type squares (non-consecutive main diagonal)

The previous page showed how to prepare Modified De La Loubère squares of order 7 or 9 where the main diagonal is constructed using a complementary table of 7 or 9 and where the main diagonal was constructed from ½ (n2-n+2) to ½(n2+n) in consecutive order. this page switches the starting and final numbers around on the main diagonal and also uses different combinations for the 5x5 diagonal.

Modified Method I using the L-leap approach

To expand a modified De La Loubère square where the diagonal is not in consecutive order:

  1. Construct a 5x5 Loubère square (a semi-anti-Loubère square) using a non-consecutive order of numbers in the diagonal. Incorporate this square into a larger 7x7 square as shown below.
  2. Fill in the two corner cells in an opposite manner to complete the main 7x7 diagonal and fill in the first cell in the sixth row with the number 11, since 10 was the last number on the complementary list and perform a L-leap, first across the square then up/down
  3. .
  4. Using the table for sum of required pairs below, take the complement of 21, i.e. 29 and move 3 spaces to the left giving the required sum 47 for COLUMN 3 then perform an L-leap down the square, filling the second COLUMN.
  5. Move 2 spaces at a time to the right, adding the requisite numbers L-leap up, then move left to the corner cell.
  6. Repeat the process in a reverse manner first a move down then an L-leap across.
  7. Move 2 spaces at a time up, entering the requisite numbers then L-leap across left filling in the square.
********************************************************************************************************************************************************
1
                           
    4148 1 8 24    
   47 5 7 23 40    
   4 6 25 44 46    
   1027 43 45 3    
   26 42 49 2 9    
                           
   ⇒   
2
    17     19     21 22
15 4148 1 8 24    
   47 5 7 23 40 14
134 6 25 44 46    
   1027 43 45 3 12
1126 42 49 2 9    
28     18     20     16
   ⇒   
3
    17 29 19     21 22
15 4148 1 8 24    
   47 5 7 23 40 14
134 6 25 44 46    
   1027 43 45 3 12
1126 42 49 2 9    
28 30 18     20     16
   ⇒   
4
34 17 29 19 33 21 22
15 4148 1 8 24    
   47 5 7 23 40 14
134 6 25 44 46    
   1027 43 45 3 12
1126 42 49 2 9    
28 30 18 31 20 32 16


5
34 17 29 19 33 21 22
15 4148 1 8 24    
   47 5 7 23 40 14
134 6 25 44 46    
351027 43 45 3 12
1126 42 49 2 9 36
28 30 18 31 20 32 16
   ⇒   
6
34 17 29 19 33 21 22
15 4148 1 8 24 38
3947 5 7 23 40 14
134 6 25 44 46 37
351027 43 45 3 12
1126 42 49 2 9 36
28 30 18 31 20 32 16


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


********************************************************************************************************************************************************
ROWSum of Required PairCOLUMN
2536
3535
4504
5473
6472


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

The other starting position for the semi-anti-Loubère square is shown in A, while B and C show the two positions for 11 when the ends of the main diagonal are inverted, i.e., positions 22 and 28. the four squares use the same table above for required pairs:

A
16 36 12 37 14 38 22
32 4148 1 8 24 21
2047 5 7 23 40 33
314 6 25 44 46 19
181027 43 45 3 29
3026 42 49 2 9 17
28 11 35 13 39 15 34
            
B
16 30 18 31 20 32 28
38 4148 1 8 24 15
1447 5 7 23 40 39
374 6 25 44 46 13
121027 43 45 3 35
3626 42 49 2 9 11
22 17 29 19 33 21 34
            
C
34 11 35 13 39 15 28
21 4148 1 8 24 32
3347 5 7 23 40 20
194 6 25 44 46 31
291027 43 45 3 18
1726 42 49 2 9 30
22 36 12 37 14 38 16


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

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

To expand a modified 5x5 De La Loubère square with a partially reversed non-consecutive main diagonal where only that part of the modified Loubère square is reversed, i.e, 22 →27 → 24 → 25 → 26 →23 → 28, we use the L-leap approach. this type of square is a much difficult square to fill in due to algorithm employed:

  1. Construct a 5x5 Loubère non-magic square with only the Loubère main diagonal reversed and incorporate into a larger 7x7 square.
  2. Fill in the two corner cells of the 7x7 square to complete the main diagonal and fill in the final cell in the sixth row with the number 11, since 10 was the last number on the complementary list and perform a L-leap, first across the square then up/down.
  3. Using the required pair table below modify the L-leap and take the complement of 21, i.e. 29, but this time L-leap opposite to the number 17 and insert 29. L-leap across and filling in the cells as shown (yellow) up to the number 34.
  4. L-leap opposite to 11 insert a 35 then L-leap up/down the square as shown to fill in all the empty cells.

1
                           
    4148 1 8 23    
   47 5 7 26 40    
   4 6 25 44 46    
   1024 43 45 3    
   27 42 49 2 9    
                           
   ⇒   
2
16     18     20     28
    4148 1 8 23 15
1447 5 7 26 40    
   4 6 25 44 46 13
121024 43 45 3    
   27 42 49 2 9 11
22 17     19     21    
   ⇒   
3
16 29 18 31 20 33 28
    4148 1 8 23 15
1447 5 7 26 40    
   4 6 25 44 46 13
121024 43 45 3    
   27 42 49 2 9 11
22 17 32 19 30 21 34
   ⇒   
4
16 29 18 31 20 33 28
39 4148 1 8 23 15
1447 5 7 26 40 36
374 6 25 44 46 13
121024 43 45 3 38
3527 42 49 2 9 11
22 17 32 19 30 21 34


********************************************************************************************************************************************************
ROWSum of Required PairCOLUMN
2546
3505
4504
5503
6462


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

The other starting position for the semi-anti-Loubère square is shown in D, while E and F show the two positions for 11 when the ends of the main diagonal are inverted, i.e., positions 22 and 28. the four squares use the same table above for required pairs:

D
34 11 38 13 36 15 28
21 4148 1 8 23 33
3047 5 7 26 40 20
194 6 25 44 46 31
321024 43 45 3 18
1727 42 49 2 9 29
22 35 12 37 14 39 16
            
E
16 35 12 37 14 39 22
33 4148 1 8 23 21
2047 5 7 26 40 30
314 6 25 44 46 19
181024 43 45 3 32
2927 42 49 2 9 17
28 11 38 13 36 15 34
            
F
34 17 32 19 30 21 22
15 4148 1 8 23 39
3647 5 7 26 40 14
134 6 25 44 46 37
381024 43 45 3 12
1127 42 49 2 9 35
28 29 18 31 20 33 16


this completes this section on new squares from modified wheel and De La Loubère methods. To continue new modified wheel boundary methods (Part III). To go back to Part IIA of 7x7 new Loubère squares. To return to homepage.


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