NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part P3

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). The only numbers retained are (39,41,43) with the remaining three complements (40,42), (38,44) and (37,45) being replaced by another complementary pair from the list:


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 {30,31,32,33,41,49,50,51,52} and {38,37,36,35,41,47,46,45,44}

  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 and to the central column and row replace 37,38 and 40 with 28, 29 and 34, 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 A where 37 and 38 are now on the left diagonal.
  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 give the sum of pairs needed to fill up that line, as 83 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 158 (65 + 93).
  6. Repeat for row 3 except subtract the value from 205 (the magic sum for a 5x5 internal square). This gives a value of 79.
  7. Do the same for rows 7, 8 and 9 obtaining, respectively, 85, 170 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, (19,22) & (20,21) 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, (42,59),(12,1) & (10,11),(8,9) 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, (15,17),(16,17),(13,14) & (6,7),(4,5),(2,3) 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
 
 
 
  33 43 47
55 41 27
35 39 49
 
 
 
A2
30 54 44241
  31 53 45 158
  32 48 46 79
  33 43 47
5857 56 55 41 27 262524
35 39 49
  36 34 50 85
 37 29 51 170
38 28 52251
24316281 83166249
A3
30 54 44
  31 53 45
3219 48 60 46
20 33 43 47 62
5857 56 55 41 27 262524
6135 39 49 21
36 63 34 22 50
 37 29 51
38 28 52
A4
30 54 44
31 42 12 5381 23 45
103219 48 60 46 72
820 33 43 47 62 74
5857 56 55 41 27 262524
736135 39 49 219
7136 63 34 22 50 11
37 40 70 29 159 51
38 28 52
A5
30 15 1613 54 6865 64 44
6 31 42 12 5381 23 45 76
410 3219 48 60 46 72 78
28 20 33 43 47 62 7480
5857 56 55 41 27 262524
797361 35 39 49 21 93
7771 36 63 34 22 50 115
7537 40 70 29 159 51 7
38 67 6669 28 1417 18 52
A6
30 15 1613 54 6865 64 44
6 31 4212 5381 23 45 76
410 3219 48 60 46 72 78
2820 33 43 47 62 7480
5857 56 55 41 27 262524
797361 35 39 49 2193
777136 63 34 22 50 115
7537 40 70 29 159 51 7
38 67 6669 28 14 17 18 52
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 the Part P3 of a 9x9 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part P4.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com