NEW MAGIC SQUARES WHEEL METHOD

BORDER and SPOKE SHIFT Part A3

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 left diagonal numbers don't have to be chosen from the consecutive group ½(n2-n+2) to ½(n2+n) but may be chosen from any other consecutive group of numbers. However, every spoke on the wheel consists of consecutive numbers and their complements. For example, instead of choosing the complementary numbers (22,23,24,25,26,27,28) from the list:


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

as the left diagonal another set (19,20,21,25,29,30,31) may be chosen as its replacement and the magic square is no longer dependent on ½(n2-n+2) to ½(n2+n) for the main diagonal. On this site a variant of the previous square is generated. In addition, the square will be rotated 90° to the left to allow the lowest numbers to occupy the top of the central column. The symbol Δ will be used to specify a number added to or subtracted from the constants a, b or c to generate the first internal 3x3 magic square. In this case Δ is, consequently, obtained from the first number of the set (19,20,21) used to generate the left diagonal, 25 − 19.

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

A 7x7 Transposed Magic Square Using the Diagonals {39,40,41,25,9,10,11} and {21,20,19,25,31,30,29}

  1. Generate a 3x3 square using Δ=6, b=25 and a=3. (Square A1).
  2. Generate Square A2 by adding consecutive numbers to the two diagonals and the central column and row (the "spoke") and include their complements from the complement list above.
  3. To begin filling up the square add up the entries on the first row and subtract from 175 (the magic sum for a 7x7 square). This affords the value 102 which divided by 2 gives the sum of pairs needed to fill up that line, which in this case may be (51 x 2) but because of the way the complement table is structured (in groups of three) is (46 + 56). See Figure A.
  4. Repeat for row 2 except subtract the value from 125 (the magic sum for a 5x5 square). This gives a value of 51.
  5. Repeat for rows 6 and 7 obtaining, respectively, 49 and 98.
  6. Then repeat for columns 1 and 2 obtaining, respectively, 98 and 49.
  7. Finally repeat for columns 6 and 7 obtaining, respectively, 51 and 102.
  8. Fill the 2nd & 6th and 1st & 7th rows with the pairs/complements from the complement list corresponding to the requisite sums, (2,1) & (24,18),(8,12) and enter into Square A3.
  9. Fill the 2nd & 6th and 1st & 7th columns with the pairs/complements from the complement list corresponding to the requisite sums, (13,14) & (6,7) and (22,23) and enter into Square A4.
  10. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers.
  11. Picture of squares
    Figure A
  12. Square A5 shows the 3 border squares in "border format".
  13. The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1 (Δ=6)
 
 
41 3 31
15 25 35
19 47 9
 
 
A2
39 5 29102
40 4 30 51
41 3 31
17 1615 25 35 3433
19 47 9
20 46 10 49
21 45 11 98
98 49 51 102
A3
39 24 8 5 3832 29
40 2 449 30
41 3 31
17 1615 25 35 3433
19 47 9
20 48 46 1 10
21 26 42 45 1218 11
A4
39 24 8 5 3832 29
6 40 2 449 30 44
22 13 41 331 37 28
17 1615 25 35 3433
2736 19 47 9 14 23
4320 48 46 1 10 7
21 26 42 45 1218 11
A5
39 248 5 3832 29
6 40 2 449 30 44
22 13 41 331 37 28
17 1615 25 35 3433
273619 47 9 14 23
4320 48 46 1 10 7
21 26 42 45 1218 11
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 A3 of a 7x7 Magic Square Wheel Spoke Shift method. To see the next 7x7 Part A4.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com