NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part W4

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 = 3 and 4 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.

This page unlike a previous method employs an internal 3x3 square containing the consecutive numerals 1, 2 and 3. In addition, the numerals for the 5x5 and 7x7 squares are incrementally added starting with the number 5 and increasing to the number 12. See Square A1 below. The rest of the square is filled out as follows:

A 9x9 Transposed Magic Square Using the Diagonals {69,73,77,80,41,2,5,9,13} and {15,11,7,40,41,42,75,71,67}

  1. Add one to the third row center of an internal 3x3 square, 2 to its rightmost bottom cell and 3 to the center of its first column. Repeat (i.e. spiraling towards the center) up to the number 16, followed by their complementary numbers (Square A1).
  2. Add the numbers 40 and 42 to the diagonal of the 3x3 internal square generating a 3x3 internal magic square. Fill up the left diagonal subtracting δ = 3 and 4, e.g. first δ = 3 from 80 then δ = 4 from 77 then from 73 to the top left cell of the 9x9 square. This gives, respectively, the values 77, 73 and 69 plus their complements. Do the same for the right diagonal to affors 15, 11 and 7, respectively, and their complements. (Square A1).
  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). See Square A2.
  4. Fill in the internal 5x5 square (light green cells) with numbers generated using the new coding method (Square 3). For example in column 3, 23 is added to 26 using numeric superscripts and in column 7, 24 to 25, followed by their complements, using alphabetic superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square A4). For example, in row 2, 16 is added to 51 and 21 to 47. While in column 8, 17 is added to 50 and 22 to 46 followed by their complements, using numeric superscripts.
  6. Finally fill in the external 9x9 square (color cells) (Square A5). For example, in row 1, 18 is added to 49, 27 to 54 and 20 to 53. While in column 9, 19 is added to 48, 37 to 44 and 30 to 43 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. 1617 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 5655 54 53 52 51 50 49 48 47 46 45 44 43 42
    161162 163 164 101 151 15-14a 2a 2a 4a 21 22 101 102 161162 163 164 151 15-1 22 22102
  9. 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.
  10. Square A6 shows the 4 border squares in "border format".
  11. The complementary table below also shows how the color pairs are layed out (for comparison with Square A5).
A1
69 12 67
  73 8 71
  77 4 75
  80 1 42
1410 6 3 41 79 767268
40 81 2
  7 78 5
 11 74 9
15 70 13
A2 (Δ=1,δ= 3 and 4)
69 12 6722167+81+73
  73 8 71 13567+68
  77 4 75 49 49
  80 1 42
1410 6 3 41 79 767268
40 81 2
  7 78 5 115115
 11 74 9 19397+93
15 70 13271271
271193115 49135221
A3
69 12 67
  73 8 71
7723 4 26 75
58 80 1 42 24
1410 6 3 41 79 767268
5740 81 2 25
7 59 78 56 5
 11 74 9
15 70 13
A4
69 12 67
73 16 218 4751 71
657723 4 26 75 17
6058 80 1 42 24 22
1410 6 3 41 79 767268
365740 81 2 2546
327 59 78 56 5 50
11 66 61 74 3531 9
15 70 13
A5
69 18 2720 12 5354 49 67
63 73 16 21847 51 71 19
4565 7723 4 26 75 17 37
5260 58 80 1 42 24 2230
1410 6 3 41 79 767268
393657 40 81 2 25 4643
3832 7 59 78 56 5 5044
3411 66 61 74 3531 9 48
15 64 5562 70 2928 33 13
A6
69 18 2720 12 5354 49 67
63 73 1621 847 51 71 19
4565 7723 4 26 75 17 37
526058 80 1 42 24 2230
1410 6 3 41 79 767268
393657 40 81 2 254643
38327 59 78 56 5 5044
3411 66 61 74 3531 9 48
15 64 5562 70 29 28 33 13
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 W4 of a 11x11 Magic Square Wheel Spoke Shift method. To go to Part W5 of an 11x11 square.
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Copyright © 2015 by Eddie N Gutierrez