NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part U2

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 = 8 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 {80,72,64,56,41,26,18,10,2} and {16,24,32,40,41,42,50,58,66}

  1. Add one to the first row center of a 9x9 square, 2 to the rightmost bottom cell and 3 to the center of the first column. Repeat (i.e. spiraling towards the center) up to the number 19, followed by their complementary numbers (Square A1).
  2. Add the numbers 40 and 42 to the empty two internal cells. This generates a 3x3 internal magic square (Square 1).
  3. 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 74. However, the sums of the 2nd and 3rd rows will have different sums due to the non availability of four 74's.
  4. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square 3). For example 4 is added to 70 and 5 to 69, followed by their complements, using numeric superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square 4). For example, 8 is added to 62 and 33 to 45 in the row. While in the column 22 is added to 48 and 35 to 43 followed by their complements, using numeric superscripts.
  6. Finally fill in the external 9x9 square (color cells) (Square 5). For example, 6 is added to 68, 7 to 67 and 21 to 53 in the row. While in the columns 23 is added to 51, 28 to 44 and 30 to 46 followed by their complements, using numeric superscripts.
  7. Below is the coded connections to this square where the colored "spoke" cells are not included in the coding:
  8. 4 5 6 7 8 ... 12 13 14 15 ... 20 21 22 23 ... 28 29 30 31 ... 33 34 35 36 37 38 39 40
    41
    78 77 76 75 74 ... 70 69 68 67 ... 62 61 60 59 ... 54 53 52 51 ... 49 48 47 46 45 44 43 42
    91 92 93 94 131 ... 91 92 93 94... 131 95 132 96... 111 95 71 96 ... 51 132 52 71 51 111 52
  9. Figure A shows how the connectivity would look using spaghetti lines for portion of the complementary table, not including the complements which are also part of the connections. These can get very messy!
  10. Picture of square connections
    Figure A
  11. Square A6 shows the 4 border squares in "border format".
  12. The complement table below also shows how the color pairs are layed out (for comparison with Square A5).
A1
80 1 66
  72 9 58
  64 17 50
  56 25 42
311 19 27 41 55 637179
40 57 26
  32 65 18
 24 73 10
16 81 2
A2 (Δ=1,δ=8)
80 1 66222 74x3
  72 9 58 14870+78
  64 17 50 74
  56 25 42
311 19 27 41 55 637179
40 57 26
  32 65 18 9090
 24 73 10 18094+81
16 81 227090x3
27018090 74148222
A3
80 1 66
  72 9 58
644 17 70 50
77 56 25 42 5
311 19 27 41 55 637179
1340 57 26 69
32 78 65 12 18
 24 73 10
16 81 2
A4
80 1 66
72 8 33 945 62 58
60644 17 70 50 22
4777 56 25 42 5 35
311 19 27 41 55 637179
391340 57 26 6943
3432 78 65 12 18 48
24 74 49 73 3720 10
16 81 2
A5
80 6 721 1 5367 68 66
59 72 8 33945 62 58 23
5460 644 17 70 50 22 28
5247 77 56 25 42 5 3530
311 19 27 41 55 637179
363913 40 57 26 69 4346
3834 32 78 65 12 18 4844
3124 74 49 73 3720 10 51
16 76 7561 81 2915 14 2
A6
80 6 721 1 5367 68 66
59 72 833 945 62 58 23
5460 644 17 70 50 22 28
524777 56 25 42 5 3530
311 19 27 41 55 637179
363913 40 57 26 694346
383432 78 65 12 18 4844
3124 74 49 73 3720 10 51
16 76 7561 81 29 15 14 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 U2 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part U3 of an 11x11 square.
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Copyright © 2015 by Eddie N Gutierrez