NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

BORDER and SPOKE SHIFT Part A5

Picture of a wheel

How to Spoke Shift 9x9 Magic Squares

A magic square is an arrangement of numbers 1,2,3,... n2 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 n2 + 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(n2 + 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(n2-n+2) to ½(n2+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, instead of choosing the complementary numbers (37,38,39,40,41,42,43,44,45) from the list:

The new magic squares with n = 9 are constructed as follows using a complimentary table as a guide.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 
81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62
 
21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42

as the left diagonal another set (34,33,32,31,41,51,50,49,48) may be chosen as its replacement and the magic square is no longer dependent on ½(n2-n+2) to ½(n2+n) for the main diagonal. In addition, the square will be rotated 90° to the left to allow the lowest numbers to occupy the top of the central column. 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 (34,33,32,31), i.e., Δ = 41 − 31 in order to to generate the left diagonal,

3x3 template
c+Δ a b+Δ
a+2Δ b c
b-Δ c+2Δ a+Δ

A 9x9 Transposed Magic Square Using the Diagonals {68,69,70,71,41,11,12,13,14} and {34,33,32,31,41,51,50,49,48}

  1. Generate a 3x3 square using Δ=10, b=41 and a=1. (Square A1).
  2. 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.
  3. To begin fill up the square add up the entries on the first row and subtract from 369 (the magic sum for a 9x9 square). This affords the value 249 which divided by 3 gives the sum of pairs needed to fill up that line, which in this case is (83 x 3). See Figure A.
  4. Repeat for row 2 except subtract the value from 287 (the magic sum for a 7x7 internal square). This gives a value of 83x2.
  5. Repeat for row 3 except subtract the value from 205 (the magic sum for a 5x5 internal square). This gives a value of 83.
  6. Do the same for rows 7, 8 and 9 obtaining, respectively, 81, 162 and 243.
  7. Then repeat for columns 1, 2 and 3 obtaining, respectively, 243, 162 and 81.
  8. Finally repeat for columns 7, 8 and 9 obtaining, respectively, 83, 166 and 249.
  9. Fill the 3rd & 7th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (6,5),(7,8) and enter into Square A3.
  10. Fill the 2nd & 8th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (10,9),(16,15),(17,18),(19,20) and enter into Square A4.
  11. Fill the 1st & 9th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (26,25),(28,27),(30,29) & (35,36),(37,38),(39,40) and enter into Square A5.
  12. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers. The same for their complements.
  13. Picture of squares
    Figure A
  14. Square A6 shows the 4 border squares in "border format".
  15. The complement table below also shows how the color pairs are layed out (for comparison with Square A5).
A1
 
 
 
  71 1 51
21 41 61
31 81 11
 
 
 
A2
68 4 48249
  69 3 49 166
  70 2 50 83
  71 1 51
2423 22 21 41 61 605958
31 81 11
  32 80 12 81
 33 79 13 162
34 78 14243
24316281 83166249
A3
68 4 48
69 3 49
706 2 77 50
7 71 1 51 75
2423 22 21 41 61 605958
7431 81 11 8
32 76 80 5 12
33 79 13
34 78 14
A4
68 4 48
69 10 16 367 73 49
17706 2 77 50 65
197 71 1 51 75 63
2423 22 21 41 61 605958
627431 81 11 820
6432 76 80 5 12 18
33 72 66 79 159 13
34 78 14
A5
68 26 2830 4 5355 57 48
35 69 10 16 367 73 49 47
3717 706 2 77 50 65 45
3919 7 71 1 51 75 6343
2423 22 21 41 61 605958
426274 31 81 11 8 2040
4464 32 76 80 5 12 1838
4633 72 66 79 159 13 36
34 56 5452 78 2927 25 14
A6
68 26 2830 4 5355 57 48
35 69 1016 367 73 49 47
3717 706 2 77 50 65 45
39197 71 1 51 75 6343
2423 22 21 41 61 605958
426274 31 81 11 82040
446432 76 80 5 12 1838
4633 72 66 79 159 13 36
34 56 5452 78 29 27 25 14
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 26
 
81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56
 
27 28 29 30 3132 33 34 35 36 37 38 39 40
41
55 54 53 52 51 50 49 48 47 46 45 44 43 42

This completes the Part A of a 9x9 Magic Square Wheel Spoke Shift method. To go forward to 9x9 Part A6.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com