NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part M2
How to Spoke Shift 5x5 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, every spoke on the wheel consists of consecutive numbers and their complements.
For example, the complementary numbers (11,12,13,14,15) from the list:
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 
are placed in the central column and not on the diagonal.
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 (9,10) used to generate the
left diagonal, 13 − 8.
In addition, 4n + 1 number behave differently from
4n + 3
in that in the former at least one number in the set must be less than or equal to 0, while in the latter all numbers from 1 to
½(n^{2} − 1) are usable in half of the cases but not in the other half which then require numbers ≤ 0.
3x3 template
c+Δ  a 
b+Δ 
a+2Δ 
b  c 
bΔ 
c+2Δ  a+Δ 
A 5x5 Magic Square Using the Pairs {6,7} and {9,8}
 The center column is filled with the group of numbers ½
(n^{2}n+2) to ½(n^{2}+n) in consecutive
order starting at the bottom cell and proceeding to the top cell from the numbers listed in the complementary table described above, for example using n = 5.
For a 5x5 square the numbers in the center column correspond to 11 → 12 → 13 → 14 → 15 starting from the bottom (Square A1).
 Generate a 3x3 square using Δ=5, b=13 and a=14. (Square A2).
 Generate Square 3 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 65 (the magic sum for a 5x5 square). This affords the value 27
See Figure A3.
 Repeat for row 5 giving a value of 25.
 Then repeat for columns 1 and 5 obtaining, respectively, 25 and 27.
 Fill the 1^{st} and 5^{th} rows with the pairs/complements from the complement list corresponding to
the requisite sums for (4,3) and enter into Square A4.
 Fill the 1^{st} and 5^{th} columns with the pairs/complements from the complement list corresponding to
the requisite sums for (1,0) and also enter into Square A4. Note that (5,10) and their complements cannot be used for completing the square since they are adjacently unmatched.
 Figure A shows the connectivity between numbers in the complementary table where the red bars are the
"spoke" numbers.
Figure A
 The result of these operations is a wheel with a shifted spoke where the numbers in the diagonal of the regular wheel 11 → 12 → 13 → 14 → 15.
have been transposed or shifted to a column.
 The square that is produced via this method (A4) is a border square, since the internal 3x3 square has a Sum of 39 while the external has a Sum of 65.

⇒ 
A2
 
15  

 7 
14  18 

 24 
13  2 

 8 
12  19 

 
11  


⇒ 
A3
6  
15  
17  27 
 7 
14  18 
 
25  24 
13  2 
1  
 8 
12  19 
 
9  
11  
20  25 
25   
 27  

 ⇒ 
A4
6  4 
15  23 
17 
1  7 
14  18 
27 
25  24 
13  2 
1 
26  8 
12  19 
0 
9  22 
11  3 
20 

1  0 
1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
 13 
27  26 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 
This completes Part M2 of 5x5 Magic Square Wheel Spoke Shift method. To go to Part M3 of an odd 7x7 square match.
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Copyright © 2014 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com