NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part M5

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, for n = 9, we can choose the complementary numbers (37,38,39,40,41,42,43,44,45,46) from the complimentary table below:

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

for the central column and not for the diagonal as was done for the regular wheel algorithm. 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 (33,34,35,36) used to generate the left diagonal, 41 − 33.

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(n2 − 1) are usable. Figure A shows the connectivity of a 9x9 set. However, when Δ is an an odd number for a (9x9 square) at least one number in the set must be less than or equal to 0. This is shown in for a 5x5 square Part M1 and a variant square 5x5 Part M2.

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 {29,30,31,32,41,50,51,52,53} and {36,35,34,33,41,49,48,47,46}

  1. Generate a 3x3 square using Δ=8, b=41 and a=42. (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 A2.
  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, (28,27) & (25,26) 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, (20,19),(18,17)) & (15,16),(13,14) 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, (12,11),(10,9),(8,7) & (1,2),(3,4),(5,6) 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
 
 
 
  32 42 49
58 41 24
33 40 50
 
 
 
A2
29 45 46249
  30 44 47 166
  31 43 48 83
  32 42 49
6160 59 58 41 24 232221
33 40 50
  34 39 51 81
 35 38 52 162
36 37 53243
24316281 83166249
A3
29 45 46
30 44 47
3128 43 55 48
25 32 42 49 57
6160 59 58 41 24 232221
5633 40 50 26
34 54 39 27 51
35 38 52
36 37 53
A4
29 45 46
30 20 18 4465 63 47
153128 43 55 48 67
1325 32 42 49 57 69
6160 59 58 41 24 232221
685633 40 50 2614
6634 54 39 27 51 16
35 62 64 38 1719 52
36 37 53
A5
29 12 108 45 7573 71 46
1 30 20 18 4465 63 47 81
315 3128 43 55 48 67 79
513 25 32 42 49 57 6977
6160 59 58 41 24 232221
766856 33 40 50 26 146
7866 34 54 39 27 51 164
8035 62 64 38 1719 52 2
36 70 7274 37 79 11 53
A6
29 12 108 45 7573 71 46
1 30 2018 4465 63 47 81
315 3128 43 55 48 67 79
51325 32 42 49 57 6977
6160 59 58 41 24 232221
766856 33 40 50 26146
786634 54 39 27 51 164
8035 62 64 38 1719 52 2
36 70 7274 37 7 9 11 53
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 11x11 Part M6.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com