NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part X2

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. The method consists of forming a 5x5 internal Wheel spoke magic square then filling in the external 1,2 and 8,9 rows and columns with the requisite non-spoke numbers as will be shown below.


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, the symbol δ (where δ = 4) specifies the difference between entries on the diagonals and center row and column not situated on the center 5x5 square (shown in square A1).

The non-wheel entries in rows and columns 1,2 and 8,9 are added according to a coded system ( which I call "coded connectivity" as opposed to lined connectivity) employing a number and superscript and 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 number is preceded by either a ( ) or by (-).

A 9x9 Transposed Magic Square Using the Diagonals {60,56,51,50,41,32,31,26,22} and {24,28,39,40,41,42,43,54,58}

  1. Fill in the 5x5 internal square with the 25 numbers from the subset 29 to 53 (square A1). The spokes of the wheel are generated as follows: Numbers 39-43 in the left diagonal; numbers 31,32 and conjugates 50,51 in the right diagonal; numbers 29,30 and conjugates 52,53 in top to bottom center; and 33,34 and conjugates 48,49 in center horizontal. The addition of these pair of numbers and conjugates to the 5x5 square are shown below using directional pointed arrows:

    29 30 31 32 33 34 3536 37 3839 40
    41
    5352 51 504948 4746 45 4443 42

  2. The 5x5 square is finished by filling in the non-spoke numbers from those numbers left over, i.e., 35 to 38 and their conjugates 44 to 47, giving a magic sum of 205.
  3. Subtract δ = 4 from 29 and add this number (25) to the center cell of row 2 and repeat again (25-4) = 21 and place this number in the center cell of row 1. Starting ith 21 place consecutive numbers as shown in Square A1 in a spiraling fashion up to the number 28, followed by their complementary numbers (Square A1). This completes the spokes for the square, along with the nonspoke numbers of the 5x5 square.
  4. Sum up the empty 1st row, the empty 9th; the 2nd, the 8th rows. Do the same for the columns (green cells). See Square A2.
  5. Fill in the internal 7x7 square with numbers generated using the new coding method (Square 3). For example in row 2 using numeric superscripts, 2 is added to 74 and 4 to 72 , followed by their complements. While in column 2 also using numeric superscripts, 11 is added to 77 and 18 to 70 also followed by their complements.
  6. Finally fill in the external 9x9 square (color cells) (Square A4). For example, in row 1, 3 is added to 75, 13 to 65 and 1 to 73. While in column 9, 67 is added to 19, 66 to 20 and 76 to 14 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. 1 2 34 5 678 9 10111213 14 15 161718 19 20... 40
    41
    8180 79 787776 75 74 73 7271706968 6766 65 64 63 62... 42
    9171 51 72 73 92 5171 91 72 73 74 52 92 53 5452 74 53 54 ...
  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 A5 shows the 3 border squares in "border format".
  11. The complementary table below also shows how the color pairs are layed out (for comparison with Square A4).
  12. This square differs from the magic square A5 generated in the previous page in that only the 5x5 internal squares differ. The entries in rows and columns 1,2,8,9 remain the same.
A1
60 21 58
  56 25 54
  5144 2938 43
  47 50 30 42 36
2327 33 34 41 48 495559
3540 52 32 46
  39 37 53 45 31
 28 57 26
24 61 22
A2 (δ=4)
60 21 58 23078x2+74
  56 25 54 15276x2
  5144 2938 43
  47 50 30 42 36
2327 33 34 41 48 495559
3540 52 32 46
  39 37 53 45 31
 28 57 26 17688x2
24 61 222622x86+90
262176 152230
A3
60 21 58
56 2 425 7274 54
775144 29 38 43 5
7047 50 30 42 36 12
2327 33 34 41 48 495559
183540 52 32 4664
1139 37 53 45 31 71
28 80 78 57 108 26
24 61 22
A4
60 3 131 21 7365 75 58
67 56 2 42572 74 54 15
6677 5144 29 38 43 5 16
7670 47 50 30 42 36 126
2327 33 34 41 48 495559
141835 40 52 32 46 6468
2011 39 37 53 45 31 7162
1928 80 78 57 108 26 63
24 79 6981 61 917 7 22
A5
60 3 131 21 7365 75 58
67 56 24 2572 74 54 15
6677 5144 29 38 43 5 16
767047 50 30 42 36 126
2327 33 34 41 48 495559
141835 40 52 32 466468
201139 37 53 45 31 7162
1928 80 78 57 108 26 63
24 79 6981 61 9 17 7 22
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 X2 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part X3 of an 11x11 square.
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Copyright © 2015 by Eddie N Gutierrez