NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part Q3

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. However, the spoke on the consisting of the central column is no longer the adjacent numbers (37,38,39,40,41,42,43,44,45), where the number 39 correponding to ½(n2-3) is retained in (39,41,43). The next number on the list is ½(n2-5), i.e., 38, the central column is now (..38,39,41,43,44..) where (40,42) and (37,45) no longer belong on the "spoke" but are replaced by two other pairs of complementary numbers (the ..) (see Square A2 below). Furthermore, the rest of the numbers belonging in the square are generated from the list 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 (34,35,36,37) used to generate the left diagonal, 41 − 35 = 6.

In addition, both 4n + 1 and 4n + 3 squares may be filled with the entire complement set.

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 {37,36,35,34,41,48,47,46,45}

  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 39 → 41 → 43 starting from the 5th row (Square A1).
  2. With 33, 35 and their complements generate a 3x3 square using Δ=6, b=41 and a=43 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. To the central column replace 37 and 40 by 28 and 33, viz, (the "spoke") numbers, and include their complements from the complement list above. The reason for this connectivity will become apparent as we proceed. See Figure A2 where 37 is now on the left diagonal and 40 can be anywhere else on the square.
  4. 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 241 which may be will give the sum of pairs needed to fill up that line, as for example (81 x 2 + 79). 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 162 of (81x2) or (63 + 99).
  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, 166 and 251.
  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, (27,26) & (19,20) 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, (21,40),(18,1) & (16,17),(14,15) 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, (2,5),(3,4),(6,7) & (8,9),(10,11),(12,13) and enter into Square A5.
  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
 
 
 
  32 43 48
57 41 25
34 39 50
 
 
 
A2
29 54 45241
  30 49 46 162
  31 44 47 83
  32 43 48
6059 58 57 41 25 242322
34 39 50
  35 38 51 81
 36 33 52 166
37 28 53251
24316281 83166249
A3
29 54 45
  30 49 46
3127 44 56 47
19 32 43 48 63
6059 58 57 41 25 242322
6234 39 50 20
35 55 38 26 51
 36 33 52
37 28 53
A4
29 54 45
30 21 18 4981 42 46
163127 44 56 47 66
1419 32 43 48 63 68
6059 58 57 41 25 242322
676234 39 50 2015
6535 55 38 26 51 17
36 61 64 33 140 52
37 28 53
A5
29 2 36 54 7578 77 45
8 30 21 18 4981 42 46 74
1016 3127 44 56 47 66 72
1214 19 32 43 48 63 6870
6059 58 57 41 25 242322
696762 34 39 50 20 1513
7165 35 55 38 26 51 1711
7336 61 64 33 140 52 9
37 80 7976 28 74 5 53
A6
29 2 36 54 7578 77 45
8 30 2118 4981 42 46 74
1016 3127 44 56 47 66 72
121419 32 43 48 63 6870
6059 58 57 41 25 242322
696762 34 39 50 201513
716535 55 38 26 51 1711
7336 61 64 33 140 52 9
37 80 7976 28 7 4 5 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 27
 
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 55
 
28 29 30 3132 33 34 35 36 37 38 39 40
41
54 53 52 51 50 49 48 47 46 45 44 43 42

This completes Part Q3 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part Q4 of an 11x11 square.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com