New Squares from the Modified De La Loubère Method

Part IIA

A Square


Further Discussion of New De La Loubère type squares

The previous page showed how to use Modified De La Loubère squares of order 3 to prepare larger L-leap and wheel add ons. This page will show how to generate larger Loubère squares where the internal square is 5x5 expanded to 7x7 or a 7x7 expanded into a 9x9. The first set of squares will show one of the exceptions to the rule

  1. Incorporate the 5x5 grey square (starting with the number 7) using the modified Loubère method into a larger 7x7 square as shown below.
  2. Fill in the two corner cells to complete the main diagonal.
  3. Begin a L-leap on the boundary from the unused numbers in the 7x7 complementary table.
  4. Stop at the first cell in the second column inserting a 5, leap across to the right lower corner and insert a 6, then leap to the second cell in the first column and insert a 17, since all numbers from 7 to 16 have been used to generate the 5x5 square.
  5. Continue L-leaping right and left across until the first cell in the sixth row is reached.
  6. Take the complement of 21, i.e. 29, and this time L-leap backwards across the square, filling in all the complementary cells up to the left corner cell in the first column.
  7. L-leap down and up filling in all the cells with the right complementary numbers until the square is completed.
  8. At this point columns 1 and 7 do not total to the magic sum 175. Interchange these two numbers and the square is now magic.

       
1,2
                        28
    3542 7 14 27    
   41 11 13 26 34    
   10 12 25 3840    
   1624 37399    
   2336 43815    
22                        
   ⇒   
3,4
   5     3    1 28
17 3542 7 14 27    
   41 11 13 26 34    
   10 12 25 3840    
   1624 37399    
   2336 43815    
22     4     2     6
   ⇒   
5,6,7
445 46 3481 28
17 3542 714 27 33
3241 11 13 26 34 18
1910 12 25 3840 31
301624 37399 20
212336 43815 29
2245 4 47 249 6
   ⇒   
8
445 46 3481 28
17 3542 7 14 27 33
3241 11 13 26 34 18
1910 12 25 3840 31
201624 37399 30
212336 43815 29
2245 4 47 249 6


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Square group 9, i.e., beginning with the number 9 also behaves similarly. With the 30 and 20 in this position it is possible to move pairs of adjacent numbers around the boundary and still retain the magic. Remember that there are more than one magic squares in each group since any adjacent pair may be moved about on the boundary.

The next series of 7x7 squares expanded into 9x9s show two adjacent groups 10 and 11 where the former turns out to be magic and the latter not since the first and last columns add up to 390 and 348, respectively instead of 369. The only difference between the boundary numbers is that 72 and 10 replace 51 and 31, respectively. In addition, the way the boundary is set up an odd number follows an even number on the same line or column (except on the corners in some cases) for the square to be magic. As can be seen this rule is broken since 32 follows 72 and 50 follows 10.

       
Start at 10
                              45
    5362 71 1019 28 44    
    6170 16 1827 43 52    
   6915 17 26 42 5860    
   14 23 25 4157 59 68    
   2224 405665 6713    
   3039 556466 1221    
   3854 637211 2029    
37                           
      ⇒      
Magic
74 7 76 5 78 3 80145
9 5362 71 1019 28 4473
51 6170 16 1827 43 52 31
326915 17 26 42 5860 50
4914 23 25 4157 59 68 33
342224 405665 6713 48
473039 556466 122135
363854 637211 202982
37756 774792 81 8


********************************************************************************************************************************************************
       
Start at 11
                              45
    5261 70 1120 29 44    
    6069 17 1928 43 51    
   6816 18 27 42 5759    
   15 24 26 4156 58 67    
   2325 405564 6614    
   3139 546365 1322    
   3853 627112 2130    
37                           
      ⇒      
Non-Magic
74 7 76 5 78 3 80145
9 5261 701120 29 44 73
72 6069 17 1928 43 51 10
326816 18 27 42 5759 50
4915 24 26 4156 58 67 33
342325 405564 6614 48
473139 546365 1322 35
363853 627112 2130 46
37756 774792 81 8


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

A 5x5 Loubère Square Expanded into a 7x7 using the >Wheel Boundary Method

  1. Incorporate a # 2 group 5x5 Loubère modified square into a 7x7 square, where the blue colors correspond to a shift from low to high complementary numbers.
  2. Fill in the two corner cells to complete the main diagonal.
  3. Fill in the six boundary cells with a subset of three pairs of spoke numbers.
  4. Fill in the top and bottom non-spoke cells with the requisite pair of numbers in reverse order so that the columns and rows total to 175.
  5. Fill in the right and left non-spoke cells with the requisite pair of numbers in reverse order so that the columns and rows total to 175.
       
1,2
                        28
    4148 1 8 27    
   475 7 2640    
   4 6 25 4446    
   1024 43453    
   2342 4929    
22                        
   ⇒   
3
38        11        28
    4148 1 8 27    
   475 7 2640    
134 6 25 4446 37
   1024 43453    
   2342 4929    
22         39         12
   ⇒   
4
3814 16 113335 28
    4148 1 8 27    
   475 7 2640    
134 6 25 4446 37
   1024 43453    
   2342 4929    
2236 34 39 17 15 12
   ⇒   
5
3814 16 113335 28
32 4148 1 8 27 18
30475 7 2640 20
134 6 25 4446 37
211024 43453 29
192342 4929 31
2236 34 39 17 15 12


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


Attempt to Expand a 5x5 Loubère Square into a 7x7 using the Wheel Boundary Method

  1. Incorporate a 5x5 Loubère modified square into a 7x7 square, where the blue colors correspond to a shift from low to high complementary numbers.
  2. Fill in the two corner cells to complete the main diagonal.
  3. Cannot perform a spoke wheel boundary addition due to the lone pair (1,19) which must be paired off e.g. (1,49) and (2,48) adjacent pairs (see discussion of wheel method rules . This group of squares along with group square 11 (starting position 11) produce no expanded magic squares.
       
1,2
                        28
    4047 2 9 27    
   466 8 2639    
   57 25 4345    
   1124 42444    
   2341 48310    
22                        
      ⇒       No magic square group possible


This completes this section on new squares from the expanded De La Loubère methods. To continue with Part IB using 7x7 squares (Part IIB). To see previous method Part IC: Loubère Square using 3x3 and 5x5 squares or to return to homepage.


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