NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part N6

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). At least one pair of complements must be retained in the central column which may be (38,39,40,41,42,43,44) for the replacement of one, (39,40,41,42,43) for the replacement of two and (40,41,42) for the replacement of three sets of complementary pairs 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 (36,37,38,39) used to generate the left diagonal, 41 − 36 = 5.

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.

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 {32,33,34,35,41,47,48,49,50} 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 40 → 41 → 42 starting from the 5th row (Square A1).
  2. With 35, 36 and their complements generate a 3x3 square using Δ=5, b=41 and a=42 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. Instead of 37 replace by 25 and 38 replaced by 26. The reason for this connectivity, 37 and 38 are already part of the left diagonal. See Figure A.
  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 237 which may be give the sum of pairs needed to fill up that line, for example, 79x3. 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 154 (79x2).
  6. Repeat for row 3 except subtract the value from 205 (the magic sum for a 5x5 internal square). This gives a value of 75.
  7. Do the same for rows 7, 8 and 9 obtaining, respectively, 89, 174 and 255.
  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, (17,24) & (22,23) 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, (11,16),(5,10)) & (20,21),(18,19) 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, (12,15),(6,9),(1,4) & (13,14),(7,8),(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
 
 
 
  35 42 46
52 41 30
36 40 47
 
 
 
A2
32 57 43237
  33 56 44 154
  34 51 45 75
  35 42 46
5554 53 52 41 30 292827
36 40 47
  37 31 48 89
 38 26 49 174
39 25 50255
24316281 83166249
A3
32 57 43
  33 56 44
3417 51 58 45
22 35 42 46 60
5554 53 52 41 30 292827
5936 40 47 23
37 65 31 24 48
 38 26 49
39 25 50
A4
32 57 43
33 11 5 5672 66 44
203417 51 58 45 62
1822 35 42 46 60 64
5554 53 52 41 30 292827
635936 40 47 2319
6137 65 31 24 48 21
38 71 77 26 1016 49
39 25 50
A5
32 12 61 57 7873 67 43
13 33 11 5 5672 66 44 69
720 3417 51 58 45 62 75
218 22 35 42 46 60 6480
5554 53 52 41 30 292827
796359 36 40 47 23 193
7461 37 65 31 24 48 218
6838 71 77 26 1016 49 14
39 70 7681 25 49 15 50
A6
32 12 61 57 7873 67 43
13 33 115 5672 66 44 69
720 3417 51 58 45 62 75
21822 35 42 46 60 6480
5554 53 52 41 30 292827
796359 36 40 47 23193
746137 65 31 24 48 21
6838 71 77 26 1016 49 14
39 70 7681 25 4 9 15 50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
 
81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58
 
25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42

This completes the Part N6 of a 9x9 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part N7.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com