NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part U1

Picture of a wheel

How to Spoke Shift 7x7 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). This generates a new type of wheel structure leaving the non-spoke numbers to be filled in with the complementary set below:


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
49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

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.

3x3 template
c+Δ a b+Δ
a+2Δ b c
b-Δ c+2Δ a+Δ

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.

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. From the complementary table above 1 + 39 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 + 48, while 2-1 to the sum of 2 + 49. When either of the two sums is required,the ( ) or the (-) shows which one is being used.

A 7x7 Magic Square Using the Diagonals {48,40,32,25,18,10,2} and {8,16,24,25,26,34,42}

  1. Add one to the first row center of a 7x7 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 24 and 26 to the diagonal of the 3x3 internal square generating a 3x3 internal magic square. Fill up the right diagonal subtracting δ = 8 twice from the bottom left cell of the 3x3 square down to the bottom left cell of the 7x7 square. This gives, respectively, the values 16 and 8 plus their complements (Square 1).
  3. Then sum up the empty first and last rows and second and sixth rows. Do the same for the columns (green cells).
  4. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square 3). For example, 6 is added to 36 for the row and 15 to 27 for the column, followed by their complements and using numeric superscripts.
  5. Fill up the 7x7 square with 2 sets of row numbers totaling 84 (42 + 42) [using the numeric superscripts]. For the column numbers the two sets of numbers totaling 84 are equal to 49 + 35, also using numeric superscripts. The coded table below shows this along with colored "spoke" cells, which are not included in the coding:

  6. 4 5 6 7 ... 12 13 14 15 ... 20 21 22 23 24
    25
    46 45 44 43 ... 38 37 36 35 ... 30 29 28 27 26
    91 92 93 161 91 92 93 94 21 21 161 94
  7. Figure A shows the connectivity between numbers in the complementary table ("spoke" numbers not shown). The numbers are shown as regular lined connectivity which is simplified but in reality is spaghetti like if the complementary connections are also included.
  8. Picture of squares
    Figure A
  9. Square A5 shows the 3 border squares in "border format".
  10. The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1
48   1 42
40 9  34
32 17 26
3 1119 25 31 3947
24 33 18
16 41 10
8 49 2
A2 (Δ=1,δ=8)
48   1 42 84
42 9  34 42
32 17 26
3 1119 25 31 3947
24 33 18
16 41 10 58
8 49 2 116
116 58 42 84
A3
48 1 42
40 6 936 34
35 32 17 26 15
3 1119 25 31 3947
23 24 33 18 27
16 44 41 14 10
8 49 2
A4
48 4 5 1 3738 42
30 40 6 936 34 20
43 35 32 1726 15 7
3 1119 25 31 3947
2223 24 33 18 27 28
2116 44 41 14 10 29
8 46 45 49 1312 2
A5
48 45 1 3738 42
30 40 6 936 34 20
43 35 32 1726 15 7
3 1119 25 31 3947
222324 33 18 27 28
2116 44 41 14 10 29
846 45 49 1312 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
49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

This completes Part U1 of a 7x7 Magic Square Wheel Spoke Shift method. To go to Part U2 of an odd 9x9 square.
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Copyright © 2015 by Eddie N Gutierrez