NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part W2

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 = 4 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.

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 7x7 Magic Square Using the Diagonals {41,45,48,25,2,5,9} and {11,7,24,25,26,43,39}

  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 12, 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 left diagonal subtracting δ = 4 twice from the center cell of the 3x3 square to the top left cell of the 7x7 square. This gives, respectively, the values 45 and 41 plus their complements. Do the same for the right diagonal giving 7 and 11, respectively, and their complements. (Square A1).
  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, for the row, 13 is added to 20 using alphabetic superscripts and for the column, 15 to 18, followed by their complements using alphabetic superscripts.
  5. Fill up the 7x7 square with 2 sets of row numbers (12 + 31) and (16 + 28) totaling 87 and equal to 44 + 43 by using the numeric superscripts. For the column the two sets of numbers totaling (14 + 29) and (17 + 27) totaling 87 and equal to 43 + 44, also using numeric superscripts. The coded table below shows this along with colored "spoke" cells, which are not included in the coding:

  6. 12 13 14 15 16 17 18 19 20 21 22 23 24
    25
    38 37 36 35 34 33 32 31 30 29 28 27 26
    81 8a 82 4a 71 72 4a 81 8a 82 71 72
  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. The a above the two cells correspond to the alphanumeric connections.
  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
41   8 39
45 4  43
48 1 26
10 63 25 47 4440
24 49 2
7 46 5
11 42 9
A2 (Δ=1,δ=4)
41   8 39 87
45 4  43 33
48 1 26
10 63 25 47 4440
24 49 2
7 46 5 67
11 42 9 113
113 67 33 87
A3
41 8 39
45 13 420 43
35 48 1 26 15
10 63 25 47 4440
32 24 49 2 18
7 37 46 30 5
11 42 9
A4
41 12 16 8 2831 39
36 45 13 420 43 14
33 35 48 126 15 17
10 63 25 47 4440
2332 24 49 2 18 27
217 37 46 30 5 29
11 38 34 42 2219 9
A5
41 1216 8 2831 39
36 45 13 420 43 14
33 35 48 126 15 17
10 63 25 47 4440
233224 49 2 18 27
217 37 46 30 5 29
1138 34 42 2219 9
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 W2 of a 7x7 Magic Square Wheel Spoke Shift method. To go to Part W3 of an odd 9x9 square.
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Copyright © 2015 by Eddie N Gutierrez