New Squares from the Modified De La Loubère Method

Part IA

A square

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 L-leap approach, the other a wheel approach.

The L-leap approach

To expand a modified De La Loubère square using the L-leap approach we take an nxn and increase the number of rows and columns by 2 or greater. After the smaller square is incorporated into a larger the method involves placing the next number in the omplementary table into either of four positions. These include either right of first top corner, left/right of top right corner or top of left bottom corner. All four examples are shown below using the 3x3 example:

  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 left cell adjacent to the 15 with the number 4, since 3 was the last number on the complementary list.
  4. Perform a L-leap, i.e. place one number in a cell leap across, in a z manner, up/down on the squareto 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 L-leaping 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 L-leap backwards across the square, filling in all the complementary cells up to the left corner cell in the first column.
  8. L-leap down and up filling in all the cells with the right complementary numbers until the square is completed.
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 L-leap 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 L-leap, i.e. place one number in a cell leaping 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 L-leaping 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 L-leap backwards(up/down) across the square, filling in all the complementary cells up to the bottom right corner cell in the fifth column.
  7. L-leap 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

The last two examples in the series (A and B) are shown below after repeating the algorithm with the difference being that the leaping across the square as in line 6 above is done towards the end of the construction:

A
19 4 21 6 15
10 241 14 16
173 13 23 9
812 25 2 18
11 22 5 20 7
  
B
7 18 9 16 15
20 241 14 6
53 13 23 21
2212 25 2 4
11 8 17 10 19

The wheel approach

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. L-leap 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

To continue this method using reverse main diagonal (Part IB) 3x3 squares. To go back to previous wheel and De la Loubère methods. To return to homepage.


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