NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part R3

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 (32,33,34,35,41,47,48,49,50)(this page) or (31,32,33,34,41,48,49,50,51) (next 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 {27,28,29,30,41,52,53,54,55} 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 35 → 41 → 47 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=47 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). 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 of (83x2).
  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, (30,34) & (15,16) 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, (14,13),(12,11) & (9,10),(7,8) 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, (40,31),(19,26),(21,20) & (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
 
 
 
  30 47 46
57 41 25
36 35 52
 
 
 
A2
27 50 43249
  28 49 44 166
  29 48 45 83
  30 47 46
6059 58 57 41 25 242322
36 35 52
  37 34 53 81
 38 33 54 162
39 32 55243
24316281 83166249
A3
27 50 43
  28 49 44
2918 48 65 45
15 30 47 46 67
6059 58 57 41 25 242322
6636 35 52 16
37 64 34 17 53
 38 33 54
39 32 55
A4
27 50 43
28 14 12 4971 69 44
92918 48 65 45 73
715 30 47 46 67 75
6059 58 57 41 25 242322
746636 35 52 168
7237 64 34 17 53 10
38 68 70 33 1113 54
39 32 55
A5
27 40 1921 50 6256 51 43
1 28 14 12 4971 69 44 81
39 2918 48 65 45 73 79
57 15 30 47 46 67 7577
6059 58 57 41 25 242322
767466 36 35 52 16 86
7872 37 64 34 17 53 104
8038 68 70 33 1113 54 2
39 42 6361 32 2026 31 55
A6
27 40 1921 50 6256 51 43
1 28 1412 4971 69 44 81
39 2918 48 65 45 73 79
5715 30 47 46 67 7577
6059 58 57 41 25 242322
767466 36 35 52 1686
787237 64 34 17 53 104
8038 68 70 33 1113 54 2
39 42 6361 32 20 26 31 55
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 Part R3 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part R4 of an 9x9 square.
Go back to homepage.


Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com