NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part W3

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 = 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 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 {68,72,76,80,41,2,6,10,14} and {16,12,8,40,41,42,74,70,66}

  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 = 4 thrice from the center cell of the 3x3 square to the top left cell of the 9x9 square. This gives, respectively, the values 76, 72 and 68 plus their complements. Do the same for the right diagonal giving 16, 12 and 8, 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, 4 is added to 46 using numeric superscripts and in column 7, 21 to 29, followed by their complements, using alphabetic superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square A4). For example, in row 2, 18 is added to 50 and 19 to 49. While in column 8, 23 is added to 45 and 24 to 44 followed by their complements, using numeric superscripts.
  6. Finally fill in the external 9x9 square (color cells) (Square A5). For example, in row 1, 17 is added to 57, 20 to 54 and 22 to 52. While in column 9, 26 is added to 48, 27 to 47 and 31 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. 4 ... 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
    78... 65 64 63 62 61 60 59 58 57 5655 54 53 52 51 50 49 48 47 46 45 44 43 42
    331 ... 91 151 152 92 9a 93153 154 91 94 95 92 9a 93 96151 152 94 95 331 153 154 96
  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
68 13 66
  72 9 70
  76 5 74
  80 1 42
1511 7 3 41 79 757167
40 81 2
  8 77 6
 12 73 10
16 69 14
A2 (Δ=1,δ=4)
68 13 6622274x3
  72 9 70 13668x2
  76 5 74 50 50
  80 1 42
1511 7 3 41 79 757167
40 81 2
  8 77 6 114114
 12 73 10 19296x2
16 69 1427090x3
270192114 50136222
A3
68 13 66
  72 9 70
764 5 46 74
61 80 1 42 21
1511 7 3 41 79 757167
5340 81 2 29
8 78 77 36 6
 12 73 10
16 69 14
A4
68 13 66
72 18 199 4950 70
59764 5 46 74 23
5861 80 1 42 21 24
1511 7 3 41 79 757167
385340 81 2 2944
378 78 77 36 6 45
12 64 63 73 3332 10
16 69 14
A5
68 17 2022 13 5254 57 66
56 72 18 19949 50 70 26
5559 764 5 46 74 23 27
5158 61 80 1 42 21 2431
1511 7 3 41 79 757167
393853 40 81 2 29 4443
3537 8 78 77 36 6 4547
3412 64 63 73 3332 10 48
16 65 6260 69 3028 25 14
A6
68 17 2022 13 5254 57 66
56 72 1819 949 50 70 26
5559 764 5 46 74 23 27
515861 80 1 42 21 2431
1511 7 3 41 79 757167
393853 40 81 2 294443
35378 78 77 36 6 4547
3412 64 63 73 3332 10 48
16 65 6260 69 30 28 25 14
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 W3 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part W4 of an 9x9 square.
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Copyright © 2015 by Eddie N Gutierrez