NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part S3

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).

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.


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

Furthermore, a new symbol δ specifies the difference between entries on the diagonals and center row and column where δ = 4 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 spaghetti type connections between paired non-spoke numbers, a coded system ( which I call "coded connectivity" as opposed to 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. 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 {80,76,72,68,41,14,19,6,2} and {4,8,12,40,41,42,70,74,78}

  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 to the leftmost bottom cell. Repeat (i.e. spiraling towards the center) for the next 15 numbers, 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 2).
  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 sums which add up to 70; four that add up to 66 and two which add up to 54.
  4. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square 3). For example 24 is added to 30 and 26 to 28, followed by their complements, using alphabetic superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square 4). For example 16 is added to 50 and 18 to 48 in the row. Add 20 to 46 and 22 to 44 in the columns followed by their complements, using numeric superscripts.
  6. Finally fill in the external 9x9 square (color cells) (Square 5). For example 17 is added to 53, 19 to 51 and 21 to 49 in the row. Add 3 to 47, 25 to 45 and 27 to 43 in the column followed by their complements, using numeric superscripts.
  7. Below is the coded connections to this square:
  8. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
    41
    66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42
    171 131 172 132 173 133 174 134 7a 135 3a 136 3a 131 7a 132 171 133 172 134 173 135 174 136
  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. Very Messy!
  10. Picture of squares
    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 78
  76 5 74
  72 9 70
  68 13
37 11 15 41 67 717579
69 14
  12 73 10
 8 77 6
4 81 2
A2 (Δ=1,δ=4)
80 1 78210 70x3
  76 5 74 13266x2
  72 9 70 54
  68 13 42
37 11 15 41 67 717579
40 69 14
  12 73 10 110110
 8 77 6 19698x2
4 81 228294x3
282196110 54132210
A3
80 1 78
  76 5 74
7224 9 30 70
56 68 13 42 26
37 11 15 41 67 717579
5440 69 14 28
12 58 73 52 10
 8 77 6
4 81 2
A4
80 1 78
76 16 18 548 50 74
627224 9 30 70 20
6056 68 13 42 26 22
37 11 15 41 67 717579
385440 69 14 2844
3612 58 73 52 10 46
8 66 64 77 3432 6
4 81 2
A5
80 17 1921 1 4951 53 78
59 76 16 18 548 50 74 23
5762 7224 9 30 70 20 25
5560 56 68 13 42 26 2227
37 11 15 41 67 717579
393854 40 69 14 28 4443
3736 12 58 73 52 10 4645
358 66 64 77 3432 6 47
4 65 6361 81 3331 29 2
A6
80 17 1921 1 4951 53 78
59 76 1618 548 50 74 23
5762 7224 9 30 70 20 25
556056 68 13 42 26 2227
37 11 15 41 67 717579
393854 40 69 14 284443
373612 58 73 52 10 4645
358 66 64 77 3432 6 47
4 65 6361 81 33 31 29 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 S3 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part S4 of an 11x11 square.
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Copyright © 2015 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com