NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part R6

Picture of a wheel

How to Spoke Shift 11x11 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 (56,57,58,59,60,61,62,63,64,65,66), i.e., 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 (50,51,52,53,54,61,68,69,70,71,72)(this page) or (49,50,51,52,53,61,69,70,71,72,73) (previous page).

Each will be treated separately since 4n + 3, where in this case n = 2 may be filled with all but one complementary set below:


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 
121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103 102
 
21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
 
101 100 99 98 97 96 95 94 93 92 91 90 89 48 87 86 85 84 83 82
 
41 42 43 44 45 46 47 48 49 50 5152 53 54 55 56 57 58 59 60
61
81 80 79 78 77 76 75 74 73 72 71 60 69 68 67 66 65 64 63 62

or with all the complementary numbers as was shown previously. In the 4n + 1 numbers, however, both squares use all the numbers in their complementary set.

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 (55,56,57,58,59) used to generate the left diagonal, 61 − 55 = 6.

3x3 template
c+Δ a b+Δ
a+2Δ b c
b-Δ c+2Δ a+Δ

A 11x11 Transposed Magic Square Using the Diagonals {44,45,46,47,48,61,74,75,76,77,78} and {59,58,57,56,55,61,67,66,65,64,63}

  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 = 11. For a 11x11 square the numbers in the center column correspond to 55 → 61 → 67 starting from the 5th row (Square A1).
  2. With 55, 67 and their complements generate a 3x3 square using Δ=6, b=61 and a=68 so that the sum of each column, row and diagonal of the 3x3 square sums up to 183, the sum of the internal 3x3 square within a 11x11 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. Then begin filling up the square by adding up the entries on the first row and subtracting from 671 (the magic sum for a 11x11 square). This affords the value 492 for example equals 123 x 4. See Figure A2.
  5. Repeat for row 2 except subtract the value from 549 (the magic sum for a 9x9 internal square). This gives a value of 369.
  6. Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 246.
  7. Repeat for row 4 except subtract the value from 305 (the magic sum for a 5x5 internal square). This gives a value of 123.
  8. Do the same for rows 11, 10, 9 and 8 obtaining, respectively, 484, 363, 242 and 121.
  9. Then repeat for columns 1, 2, 3 and 4 obtaining, respectively, 484, 363, 242 and 121.
  10. Finally repeat for columns 11, 10, 9 and 8 obtaining, respectively, 492, 369, 246 and 123.
  11. Fill the 4th & 8th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (36,35) &(33,34) and enter into Square A3.
  12. Fill the 3rd & 9th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (43,60),(21,2) & (37,49),(32,22) and enter into Square A4.
  13. Fill the 2nd & 10th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (30,29),(28,27),(26,25) & (23,24),(19,20),(17,18) and enter into Square A5.
  14. And finally fill the 1st & 11th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (16,15),(14,13),(12,11),(10,9) & (7,8),(5,6),(3,4),(0,1) and enter into Square A6.
  15. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers. The same for their complements.
  16. The complementary set (31,91) is not used in this square but is replaced by (0,122).
  17. Picture of squares
    Figure A
  18. Square A6 shows the 5 border squares in "border format".
  19. The complement table below also shows how the color pairs are layed out (for comparison with Square A6).
A1
 
 
 
 
  48 68 67
80 61 42
55 54 75
 
 
 
 
A2
44 72 63 492
  45 71 64 369
  46 70 65 246
  47 69 66 123
  48 68 67
84 8382 81 80 61 42 414039 38
55 54 74
  56 53 75 121
  57 52 76 242
  58 51 77 363
59 50 78484
484363242 121 123 246369492
A3
44 72 63
  45 71 64
  46 70 65
4736 69 87 66
33 48 68 67 89
84 8382 81 80 61 42 414039 38
8855 54 74 34
56 86 53 35 75
  57 52 76
  58 51 77
59 50 78
A4
44 72 63
  45 71 64
46 43 2170120 62 65
37 4736 69 87 66 85
3233 48 68 67 89 90
84 8382 81 80 61 42 414039 38
1008855 54 74 34 22
7356 86 53 35 75 49
57 79 101 52 260 76
  58 51 77
59 50 78
A5
44 72 63
45 30 2826 71 97 95 93 64
23 46 4321 70120 62 65 99
1937 4736 69 87 66 85 103
1732 33 48 68 67 89 90105
84 8382 81 80 61 42 414039 38
10410088 55 54 74 34 2218
10273 56 86 53 35 75 49 20
98 57 79 101 52 260 76 24
58 92 9496 51 2527 29 77
59 50 78
A6
4416 1412 1072 113111 10910763
745 30 2826 71 97 9593 64115
523 46 4321 70120 62 65 99117
319 37 4736 69 87 66 85 103119
01732 33 48 68 67 89 90105 122
84 8382 81 80 61 42 414039 38
121104100 88 55 54 74 34 22181
11810273 56 86 53 35 75 49 204
11698 57 79 101 52 260 76 246
11458 92 9496 51 25 2729 77 8
59106 108110 11250 911 1315 78
A7
4416 14 1210 72 113 111 109 107 63
745 30 2826 71 97 95 93 64 115
523 46 4321 70120 62 65 99 117
319 37 4736 69 87 66 85 103 119
017 3233 48 68 67 89 90105 122
8483 82 81 80 61 42 414039 38
121104 10088 55 54 74 342218 1
118 102 7356 86 53 35 75 49 20 4
11698 57 79 101 52 260 76 24 6
11458 92 9496 51 25 27 29 778
59 106 108 110 11250 9 1113 15 78
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
 
122 121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104
19 20 21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40 4142
 
103 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 8180
43 44 4546 4748 49 5051 5253 54 5556 57 5859 60
61
79 7877 76 7574 73 7271 7069 68 6766 65 6463 62

This completes the Part R6 of a 11x11 Magic Square Wheel Spoke Shift method. To go to Part R7 of an 13x13 square.
Go back to homepage.


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