NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part S1

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. 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 7x7Magic Square Using the Diagonals {48,44,40,25,10,6,2} and {4,8,25,25,26,42,46}

  1. Add one to the first row center of a 7x7 square, 2 to the rightmost bottom cell, 3 to the center of the first column and 4 to the leftmost bottom cell. Repeat (i.e. spiraling towards the center) for the next 11 numbers, followed by their complementary numbers (Square A1).
  2. Add the numbers 24 and 26 to the empty two internal cells. This generated a 3x3 internal magic square (Square 2). Then sum up the empty first and last rows and second and sixth rows. Do the same for the columns (green cells).
  3. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square 3). For example 14 is added to 20 and 16 to 18, followed by their complements, using the alphabetic superscripts.
  4. 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
    111 112 7a 71 3a 3b 3a 3b 7a 71 111 112
  5. Fill up the 7x7 square with 2 sets of row numbers totaling 80 (40 + 40) [using the numeric superscripts]. For the column numbers 80 (36 + 44), use a mix of numeric and alphabetic superscripts from the coded table.
  6. 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.
  7. Picture of squares
    Figure A
  8. Square A5 shows the 3 border squares in "border format".
  9. The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1
48   1 46
44 5  42
40 9
3 711 25 39 4347
41 10
8 45 6
4 49 2
A2 (Δ=1,δ=4)
48   1 46 80
44 5  42 34
40 9 26
3 711 25 39 4347
24 41 10
8 45 6 66
4 49 2 120
120 66 34 80
A3
48 1 46
44 14 520 42
34 40 9 26 16
3 711 25 39 4347
32 24 41 10 18
8 36 45 30 6
4 49 16 2
A4
48 12 13 1 2728 46
33 44 14 520 42 17
35 34 40 926 16 15
3 711 25 39 4347
2132 24 41 10 18 29
318 36 45 30 6 19
4 38 37 49 2322 2
A5
48 1213 1 2728 46
33 44 14 520 42 17
35 34 40 926 16 15
3 711 25 39 4347
213224 41 10 18 29
318 36 45 30 6 19
4 38 37 49 2322 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 S1 of a 7x7 Magic Square Wheel Spoke Shift method. To go to Part S2 of an odd 7x7 square.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com