NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part V1

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 numbers (and complements) are added consecutively, starting from 1, at the center top cell. Subsequent numbers are added to each of the diagonals and the center row. The left diagonal of the internal 3x4 square, however, deviates from this arrangement where the three numbers on this diagonal are ½(n2 − 1), ½(n2 + 1), ½(n2 + 3).


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

In addition, the symbol Δ which has been used to specify a number added to or subtracted from the constants a, b or c in the first internal 3x3 magic square will always equal 1.

Furthermore, a new symbol δ specifies the difference between entries on the diagonals and center row and column where δ = 12 in all our cases except for the left diagonal of the internal 3x3 square (as shown below):

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

To avoid what I call spaghetti type connections between paired non-spoke numbers, a coded system ( which I call "coded connectivity" as opposedto lined connectivity) employs a number and superscript where the number gives the difference between two paired numbers and the superscript shows which two numbers are paired together. For example, 111 says that this number is added to a second complementary number 111 separated by a distance of 11. From the complementary table above 1 + 71 is such an example. While, 7a means that this number is added to a non-complementary number 7a both which are 7 units apart. In addition, if we look at the complementary table above 21 corresponds to the sum of 1 + 80, while 2-1 to the sum of 2 + 81. When either of the two sums is required,the ( ) or the (-) shows which one is being used.

A 9x9 Transposed Magic Square Using the Diagonals {80,68,56,44,41,38,26,14,2} and {4,16,28,40,41,42,54,66,78}

  1. Add one to the first row center of a 9x9 square, 2 to the rightmost bottom cell, 3 to the center of the first column and 4 the leftmost bottom cell. Repeat (i.e. spiraling towards the center) up to the number 40, followed by their complementary numbers . This generates a 3x3 internal magic square (Square A1).
  2. Sum up the empty 1st row, the empty 9th; the 2nd, the 8th; the 3rd and the 7th rows. Do the same for the columns (green cells). The values are in the tenth column and are equal to the multiplied values in the eleventh column. That is including both colums and rows, there should be six, four and two sums all of which add up to 70. However, the sums of the 1st and 9th rows will have different sums due to the non availability of six 70's.
  3. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square A3). For example 5 is added to 65 and 6 to 64, followed by their complements, using numeric superscripts.
  4. Fill in similarly the internal 7x7 square (color cells) (Square A4). For example, in row 2 , 7 is added to 63 and 9 to 61. While in column 8, 20 is added to 50 and 22 to 48 followed by their complements, using numeric superscripts.
  5. Finally fill in the external 9x9 square (color cells) (Square A5). For example, in row 1, 23 is added to 47, 8 to 51 and 11 to 70. While in column 9, 24 is added to 46, 10 to 49 and 29 to 52 followed by their complements, using numeric superscripts.
  6. Below is the coded connections to this square where the colored "spoke" cells are not included in the coding:
  7. 5 6 7 8 9 10 11 12 ... 17 18 19 20 21 22 23 24 ... 29 30 31 32 33 34 35 36 ... 40
    41
    77 76 75 74 73 72 71 70 ... 65 64 63 62 61 60 59 58 ... 53 52 51 50 49 48 47 46 ... 42
    131 132 133 241 134 242 21 21... 131 132 133 135 134 136 137 138... 22 22 241 135 242 136 137 138
  8. Because of the messy connectivities using spaghetti lines its best to use the connectivity table and the table at the end of this page to assertain the connectivities.
  9. Square A6 shows the 4 border squares in "border format".
  10. The complement table below also shows how the color pairs are layed out (for comparison with Square A5).
A1
80 1 78
  68 13 66
  56 25 54
  44 37 42
315 27 39 41 43 556779
40 45 38
  28 57 26
 16 69 14
4 81 2
A2 (Δ=1,δ=12)
80 1 7821070+59+81
  68 13 66 14070x2
  56 25 54 70
  44 37 42
315 27 39 41 43 556779
40 45 38
  28 57 26 9494
 16 69 14 18894x2
4 81 228294+105+83
28218894 70140210
A3
80 1 78
  68 13 66
565 25 65 54
76 44 37 42 6
315 27 39 41 43 556779
1840 45 38 64
28 77 57 17 26
 16 69 14
4 81 2
A4
80 1 78
68 7 9 1361 63 66
62565 25 65 54 20
6076 44 37 42 6 22
315 27 39 41 43 556779
341840 45 38 6448
3228 77 57 17 26 50
16 75 73 69 2119 14
4 81 2
A5
80 23 811 1 7051 47 78
58 68 7 91361 63 66 24
7262 565 25 65 54 20 10
5360 76 44 37 42 6 2229
315 27 39 41 43 556779
303418 40 45 38 64 4852
3332 28 77 57 17 26 5049
3616 75 73 69 2119 14 46
4 59 7471 81 1231 35 2
A6
80 23 811 1 7051 47 78
58 68 79 1361 63 66 24
7262 565 25 65 54 20 10
536076 44 37 42 6 2229
315 27 39 41 43 556779
303418 40 45 38 644852
333228 77 57 17 26 5049
3616 75 73 69 2119 14 46
4 59 7471 81 12 31 35 2
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 Part V1 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part V2 of an 11x11 square.
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Copyright © 2015 by Eddie N Gutierrez