NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part S2

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.

The 7x7 square described in this page can be compared to the previous 7x7 which retains the same "spoke" numbers.

A second 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 generates 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 16 is added to 18 and 15 to 19, 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 81 5a 3a 4a 3a 5a 4a 81 111 112
  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 (37 + 43), 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 already shows spagetti like connectivity creeping in especially if the complementary connections are also included (as seen in Figure A). The coded method, therefore, simplifies the connectivity.
  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 16 518 42
35 40 9 26 15
3 711 25 39 4347
31 24 41 10 19
8 34 45 32 6
4 49 16 2
A4
48 12 13 1 2728 46
33 44 16 518 42 17
36 35 40 926 15 14
3 711 25 39 4347
2131 24 41 10 19 29
308 34 45 32 6 20
4 38 37 49 2322 2
A5
48 1213 1 2728 46
33 44 16 518 42 17
36 35 40 926 15 14
3 711 25 39 4347
213124 41 10 19 29
308 34 45 32 6 20
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 S2 of a 7x7 Magic Square Wheel Spoke Shift method. To go to Part S3 of an odd 9x9 square.
Go back to homepage.


Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com