NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part N4

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 (34,35,36,37) used to generate the left diagonal, 41 − 34 = 7.

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 {30,31,32,33,41,49,50,51,52} 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 40 → 41 → 42 starting from the 5th row (Square A1).
  2. With 33, 34 and their complements generate a 3x3 square using Δ=7, 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 29. The reason for this connectivity, 37 is 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 241 which may be give the sum of pairs needed to fill up that line, as 83 x 2 + 75 or 80x2 + 81. 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 (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 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, (28,27) & (21,22) 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, (20,19),(18,17)) & (15,16),(13,14) 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, (10,12),(9,11),(7,8) & (1,2),(3,4),(5,6) 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 42 48
56 41 26
34 40 49
 
 
 
A2
30 53 45241
  31 44 46 166
  32 43 47 83
  33 42 48
5958 57 56 41 26 252423
34 40 49
  35 39 50 81
 36 38 51 162
37 29 52251
24316281 83166249
A3
30 53 45
  31 44 46
3228 43 55 47
21 33 42 48 61
5958 57 56 41 26 252423
6034 40 49 22
35 54 39 27 50
 36 38 51
37 29 52
A4
30 53 45
31 20 18 4465 63 46
153228 43 55 47 67
1321 33 42 48 61 69
5958 57 56 41 26 252423
686034 40 49 2214
6635 54 39 27 50 16
36 62 64 38 1719 51
37 29 52
A5
30 10 97 53 7471 70 45
1 31 20 18 4465 63 46 81
315 3228 43 55 47 67 79
513 21 33 42 48 61 6977
5958 57 56 41 26 252423
766860 34 40 49 22 146
7866 35 54 39 27 50 164
8036 62 64 38 1719 51 2
37 72 7375 29 811 12 52
A6
30 10 97 53 7471 70 45
1 31 2018 4465 63 46 81
315 3228 43 55 47 67 79
51321 33 42 48 61 6977
5958 57 56 41 26 252423
766860 34 40 49 22146
786635 54 39 27 50 164
8036 62 64 38 1719 51 2
37 72 7375 29 8 11 12 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
 
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 the Part N4 of a 9x9 Magic Square Wheel Spoke Shift method. To go forward to 9x9 Part N5.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com