New Squares from the Modified Wheel Boundary or Border Method
Part IIIA
A Discussion of the new Wheel type squares
the previous page showed how to prepare Modified Wheel 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 Lleap approach, the other a
wheel approach.
the Lleap approach
To expand a modified wheel square using the Lleap
approach (leaping across the square in a z manner) we take an nxn and increase
the number of rows and columns by 2 or greater. Using the 3x3 example we:
 Incorporate the smaller square as constructed previously into a larger square as shown below using a 3x3 inserted into a 5x5.
 Fill in the two corner cells to complete the main diagonal.
 Fill in the first cell adjacent to the 15 in this example with the number 4, since 3 was the last number on the complementary list.
 Perform a Lleap, i.e. place one number in a cell Lleaping across up/down on the square
to the next column or right/left to the next row and insert the next number.
 Stop at the fifth cell in the second column inserting a 6, leap across to the right upper corner and insert a 7, then leap to the fourth cell in the first
column and insert an 8.
 Continue Lleaping right and left across until you reach the first cell in the second row.
 Take the complement of 10, i.e. 16, and this time Lleap right and left across the square, filling in all the complementary cells up to the left lower corner cell in the
first column.
 Lleap down and up filling in all the cells with the right complementary numbers until the square is completed.

⇒ 
2
11   
 
 12  3 
24  
 25  13 
1  
 2  23 
14
 
  
 15 

⇒ 
3
11   
 
 12  3 
24  
 25  13 
1  
 2  23 
14
 
  
4  15 

⇒ 
4
11   5 
 
 12  3 
24  
 25  13 
1  
 2  23 
14
 
 6  
4  15 

⇒ 
5,6
11   5 
 7 
10  12  3 
24  
 25  13 
1  9 
8  2  23 
14
 
 6  
4  15 

⇒ 
7
11   5 
 7 
10  12  3 
24  16 
17  25  13 
1  9 
8  2  23 
14
 18 
19  6  
4  15 

⇒ 
8
11  20  5 
22  7 
10  12  3 
24  16 
17  25  13 
1  9 
8  2  23 
14
 18 
19  6  21 
4  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 
The above gives one way of filling up a square by the Lleap approach. A second method fills it in an opposite manner gives a different
variant:
 Use square 2 from above.
 Fill in the first up cell adjacent to the 15 in this example with the number 4.
 Perform a Lleap, i.e. place one number in a cell Lleaping across the square left/right
to the next row and insert the next number.
 Stop at the second cell in the fifth column inserting a 6, leap across to the left lower corner and insert a 7, then leap to the first cell in the fourth column
and insert an 8.
 Continue Lleaping up and down until the first cell in the second column is reached.
 Take the complement of 10, i.e. 16, and this time Lleap backwards(up/down) across the square, filling in all the complementary cells up to the top right
corner cell in the fifth column.
 Lleap left and right filling in all the cells with the right complementary numbers until the square is completed.
1
11   
 
 12  3 
24  
 25  13 
1  
 2  23 
14 

  
 15 

⇒ 
2
11   
 
 12  3 
24  
 25  13 
1  
 2  23 
14 
4 
  
 15 

⇒ 
3,4
11   
 
 12  3 
24  6 
5  25  13 
1  
 2  23 
14  4 
7   
 15 

⇒ 
4,5
11  10  
8  
 12  3 
24  6 
5  25  13 
1  
 2  23 
14  4 
7   9 
 15 

⇒ 
6,7
11  10  17 
8  19 
20  12  3 
24  6 
5  25  13 
1  21 
22  2  23 
14  4 
7  16  9 
18  15 

The wheel border approach:Method C (Part IIIB)
Method C produces two types of magic squares, depending on the whether the single triplet {a,b,c} is added at the end to the boundary of the nxn
square, as for example in the 7x7 partial square 23 below (Example A) generated from the triad {(1,2),(3,4),5,6)} and complements or the triplet is added to square first,
followed by the same triad. The latter approach while not really filling in the external border, however, is part of method C, the only difference is that two of the
triplet pairs must be added in an opposite manner (see next page Example B).
For example, for a 5x5 square II (below), the two triplets {1,2,3} and {4,5,6} along with their complements {25,24,23} and {22,21,20} are taken from
the complementary table below and used to construct the "spokes".
To expand a modified wheel square using the wheel
approach we take an nxn square
and increase the number of rows and columns by 2 or greater. Using the 3x3 example we:
 Incorporate the smaller square as constructed in new Loubère methods into a larger square as shown below using
a 3x3 inserted into a 5x5.
 Fill in the spoke cells.
 Fill in the nonspoke cells as shown in method A variant 1.

⇒ 
2
11   6 
 21 
 12  3 
24  
22  25  13 
1  4 
 2  23 
14
 
5   20 
 15 

⇒ 
3
11  19  6 
8  21 
17  12  3 
24  9 
22  25  13 
1  4 
10  2  23 
14
 16 
5  7  20 
18  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 
Note that with this method using a 3x3 expanded into a 5x5 the same or similar results are obtained as in method B. However, when
nxn is greater than 3x3 the results are not the same, and thus the reason for method C.
The following example A shows the construction of a 5x5 internal partial square expanded into a 7x7 using the
wheel method.
Note that an internal 3x3 magic square is produced followed by a 5x5 magic square at the first border, then a 7x7 magic square at the second border.
 Fill in the main diagonal with consecutive numbers 2228.
 Fill in the spoke cells for the 5x5.
 Either complete the internal square or fill in the outer 7x7 spoke cells.
 Fill in the internal 5x5 nonspoke cells.
 Fill in the outer boundary nonspoke cells as shown in method A variant 1 first top/bottom rows
then side left/right columns.

⇒ 
2
22   
 
 
 23  
5  
47  
  24 
6  46  

 49  48 
25  2  1 

  4 
44  26  

 3  
45   27 

  
  
28 

⇒ 
3
22   
13  
 38 
 23  
5  
47  
  24 
6  46  

39  49  48 
25  2  1 
11 
  4 
44  26  

 3  
45   27 

12   
37   
28 

4
22   
13  
 38 
 23  7 
5  43 
47  
 9  24 
6  46  40 

39  49  48 
25  2  1 
11 
 41  4 
44  26  10 

 3  42 
45  8  27 

12   
37   
28 

⇒ 
5a
22  36  34 
13  17 
15  38 
 23  7 
5  43 
47  
 9  24 
6  46  40 

39  49  48 
25  2  1 
11 
 41  4 
44  26  10 

 3  42 
45  8  27 

12  14  16 
37  33  35 
28 

⇒ 
5b
22  36  34 
13  17 
15  38 
32  23  7 
5  43 
47  18 
30  9  24 
6  46  40 
20 
39  49  48 
25  2  1 
11 
21  41  4 
44  26  10 
29 
19  3  42 
45  8  27 
31 
12  14  16 
37  33  35 
28 

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 

The following table summarizes the pairs of numbers and their parities required to fill in the empty rows or columns
(spoke numbers)
for last square 5 above, where two numbers e.g., 36/15 or 32/18 correspond to a pair.
Parity/Pair table for the 7x7 Square 5 above
ROWS/COLUMNS  PAIR OF NUMBERS  PARITY 
1  51+51  O+O 
2  50+50  E+E 
3  50+49  E+O 
5  51+50  O+E 
6  50+50  E+E 
7  49+49  O+O 
The next page shows a continuation of method C con't.
To return to previous page (Part II) using 7x7 squares or to return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com