NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part R1
How to Spoke Shift 7x7 Magic Squares
A magic square is an arrangement of numbers 1,2,3,... n^{2} where every row,
column and diagonal add up to the same magic sum S and n is also the order
of the square. A magic square having all pairs of cells diametrically equidistant
from the center of the square and equal to the sum of the first and last terms of the series
n^{2} + 1 is also called associated or symmetric. In addition,
the center of this type of square must always
contain the middle number of the series, i.e., ½(n^{2} + 1).
This site introduces a new methods used for the construction of border wheel type squares except that the initial spoke parts are added in a somewhat
different manner than in the original wheel method. The method consists of forming an internal 3x3 magic square, then generating all subsequent border magic squares.
as was done in the original method. The difference between this type of square and the original is that the left diagonal
numbers don't have to be chosen from the consecutive group ½(n^{2}n+2) to ½(n^{2}+n)
but may be chosen from any other consecutive group of numbers. However, the spoke of the central column is no longer the adjacent numbers
(22,23,24,25,26,27,28), i.e.,
½(n^{2}n+2) to ½ n^{2}+n,
but may be chosen from any other consecutive group of numbers, which in our case may be (17,18,19,25,31,32,33)(this page) or (18,19,20,25,30,31,32)
(next page).
Each will be treated separately since 4n + 3, where in this case
n = 1 may be filled with every number in the complementary set below:
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 
or with one number being either 0 or 1 and its complement.
In the 4n + 1 numbers, however, both squares use all the numbers in their complementary set.
In addition, the symbol Δ will be used to specify a number added to or subtracted from the constants a, b or c to generate the first
internal 3x3 magic square.
In this case Δ is, consequently, obtained from the first number of the set (21,25,29) used to generate the
left diagonal, 25 − 21 = 4.
3x3 template
c+Δ  a 
b+Δ 
a+2Δ 
b  c 
bΔ 
c+2Δ  a+Δ 
A 7x7 Transposed Magic Square Using the Diagonals {13,14,15,25,35,36,37} and {23,22,21,25,29,28,27}

To the center column of the internal 3x3 square fill numbers 19,25,31 starting at the bottom middle cell of the 3x3 internal square
and proceeding to the top middle cell of the 3x3 internal square
using the numbers listed in the complementary table. Thus the numbers follow the format of Square A1.
 With 21, 29 generate a 3x3 square using Δ=4, b=25 and a=31 so that the sum of each column, row and diagonal of the 3x3 square
sums up to 75. (Square A1).
 Generate Square A2 by adding consecutive numbers to the two diagonals and the central column and row
(the "spoke") and include their complements from the complement list above.
 To begin filling up the square add up the entries on the first row and subtract from 175 (the magic sum for a 7x7 square). This affords the value 102.
The sum of pairs needed to fill up that line is obtained by investigating sums which may be possible. In this case (58 + 44) is a possible set.
See Figure A2.
 Repeat for row 2 except subtract the value from 125 (the magic sum for a 5x5 square). This gives a value of 51.
 Repeat for rows 6 and 7 obtaining, respectively, 49 and 101.
 Then repeat for columns 1 and 2 obtaining, respectively, 98 and 49.
 Finally repeat for columns 6 and 7 obtaining, respectively, 51 and 102.
 Fill the 1^{st} & 7^{th} and 2^{nd} & 6^{th} rows with the pairs/complements from the complement list corresponding to
the requisite pairs, (24,17), (1,7) and (4,3) and enter into Square A3.
 Fill the 1^{st} & 7^{th} and 2^{nd} & 6^{th} columns with the pairs/complements from the complement list corresponding to
the requisite pairs, (12,20), (8,2) and (5,6) and enter into Square A4.
 Figure A shows the connectivity between numbers in the complementary table where the red and blue bars are the
"spoke" numbers.
Figure A
 Square A5 shows the 3 border squares in "border format".
 The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1 (Δ=4)
  
  

  
  

  15  31 
29   
  39 
25  11 
 
  21 
19  35 
 
  
 
 
  
  


⇒ 
A2
13   
33   
27  102 
 14  
32   28 
 51 
  15  31 
29    
41  40  39 
25  11 
10  9  
  21 
19  35 
  
 22  
18  
36   49 
23   
17   
37  98 
98  49 
 
 51 
102  

⇒ 
A3
13  24  1 
33  43  34 
27 
 14  4 
32  47  28 

  15  31 
29   
41  40  39 
25  11 
10  9 
  21 
19  35 
 
 22  46 
18  3 
36  
23  26  49 
17  7  16 
37 

 ⇒ 
A4
13  24  1 
33  43  34 
27 
12  14  4 
32  47  28 
38 
8  5  15 
31  29 
45  42 
41  40  39 
25  11 
10  9 
48  44  21 
19  35 
6  2 
30  22  46 
18  3 
36  20 
23  26  49 
17  7  16 
37 

⇒ 
A5
13  24  1 
33  43  34 
27 
12  14  4 
32  47  28 
38 
8  5  15 
31  29 
45  42 
41  40  39 
25  11 
10  9 
48  44  21 
19  35 
6  2 
30  22  46 
18  3 
36  20 
23  26  49 
17  7  16 
37 

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 
This completes Part R1 of a 7x7 Magic Square Wheel Spoke Shift method. To go to Part R2 of an odd 7x7 square.
Go back to homepage.
Copyright © 2014 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com