NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part R4

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, the spoke of the central column is no longer the adjacent numbers (37,38,39,40,41,42,43,44,45), i.e., ½(n2-n+2) to ½ n2+n, but may be chosen from any other consecutive group of numbers.
½(n2-n+2) to ½ n2+n, but may be chosen from any other consecutive group of numbers, which in our case may be (31,32,33,34,41,48,49,50,51) (this page) or (32,33,34,35,41,47,48,49,50) (previous page).

Furthermore, since the squares are of type 4n + 1 the rest of the numbers can be chosen from from the complimentary list below and do not require negative or zero numbers to complete the squares:


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 (34,35,36,37) used to generate the left diagonal, 41 − 36 = 5.

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 {26,27,28,29,41,53,54,55,56} and {39,38,37,36,41,46,45,44,43}

  1. To the center column of the internal 3x3 square fill numbers ½(n2-1) to ½(n2+3) in consecutive order 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 described above, as for example using n = 9. For a 9x9 square the numbers in the center column correspond to 34 → 41 → 48 starting from the 5th row (Square A1).
  2. With 36, 46 and their complements generate a 3x3 square using Δ=5, b=41 and a=48 so that the sum of each column, row and diagonal of the 3x3 square sums up to 123, the sum of the internal 3x3 square within a 9x9 square (Square A1).
  3. 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.
  4. To begin filling 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 may be will give the sum of pairs needed to fill up that line, as for example (83 x 3). However, it will be filled up with 87 and 81x2. See Figure A2.
  5. Repeat for row 2 except subtract the value from 287 (the magic sum for a 7x7 internal square). This gives a value of 166 as for example (83x2). However, it will be filled up with 87 + 79.
  6. Repeat for row 3 except subtract the value from 205 (the magic sum for a 5x5 internal square). This gives a value of 83.
  7. Do the same for rows 7, 8 and 9 obtaining, respectively, 81, 162 and 243.
  8. Then repeat for columns 1, 2 and 3 obtaining, respectively, 243, 162 and 81.
  9. Finally repeat for columns 7, 8 and 9 obtaining, respectively, 83, 166 and 249.
  10. Fill the 3rd & 7th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (20,19) & (17,18) and enter into Square A3.
  11. Fill the 2nd & 8th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (40,35),(13,16) & (14,15),(11,12) and enter into Square A4.
  12. Fill the 1st & 9th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (30,25),(9,10),(7,8) & (1,2),(3,4),(5,6) and enter into Square A5. Note that 3 sets of inputted numbers are not 83x3 but 91, 75 and 83 which still add up to 249.
  13. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers. The same for their complements.
  14. Picture of squares
    Figure A
  15. Square A6 shows the 4 border squares in "border format".
  16. The complement table below also shows how the color pairs are layed out (for comparison with Square A5).
A1
 
 
 
  29 48 46
58 41 24
36 34 53
 
 
 
A2
26 51 43249
  27 50 44 166
  28 49 45 83
  29 48 46
6160 59 58 41 24 232221
36 34 53
  37 33 54 81
 38 32 55 162
39 31 56243
24316281 83166249
A3
26 51 43
  27 50 44
2820 49 63 45
17 29 48 46 65
6160 59 58 41 24 232221
6436 34 53 18
37 62 33 19 54
 38 32 55
39 31 56
A4
26 51 43
27 40 13 5066 47 44
142820 49 63 45 68
1117 29 48 46 65 71
6160 59 58 41 24 232221
706436 34 53 1812
6737 62 33 19 54 15
38 42 69 32 1635 55
39 31 56
A5
26 30 97 51 7472 57 43
1 27 40 13 5066 47 44 81
314 2820 49 63 45 68 79
511 17 29 48 46 65 7177
6160 59 58 41 24 232221
767064 36 34 53 18 126
7867 37 62 33 19 54 154
8038 42 69 32 1635 55 2
39 52 7375 31 810 25 56
A6
26 30 97 51 7472 57 43
1 27 4013 5066 47 44 81
314 2820 49 63 45 68 79
51117 29 48 46 65 7177
6160 59 58 41 24 232221
767064 36 34 53 18126
786737 62 33 19 54 154
8038 42 69 32 1635 55 2
39 52 7375 31 8 10 25 56
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
 
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
 
26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
56 55 54 53 52 51 50 49 48 47 46 45 44 43 42

This completes Part R4 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part R5 of an 11x11 square.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com