NEW MAGIC SQUARES WHEEL METHOD

Part V

Picture of a wheel

9x9 Magic Square Wheel

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).

A second modified facile method for the construction of wheel type magic squares is now available. The position of the spokes are rotated by 90° so that the left diagonal starts at the bottom left cell. The 5x5 square is first filled followed by the 7x7 and finally the 9x9. The 9x9 square as well as the 3x3, 5x5 and 7x7 squares are magic and thus classified as border. In addition the bordered square may be everted to give an opposite square which is no longer bordered.

The new magic squares with n = 9 are constructed as follows using a complimentary table as a guide.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 
81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62
 
21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42

A 9x9 Transposed Magic Square Using the Diagonals {40,39,38,37,41,45,44,43,42} and {8,7,6,5,41,77,76,75,74}

  1. The 9x9 square is to be filled with 33 numbers from the subset 1-12 and their complements 70-81 and the numbers 37-45. The spokes of the wheel are generated as follows: Numbers 37-45 in the left diagonal; numbers 8,7,6,5 and conjugates 77,76,75,74 in the right diagonal; numbers 4,3,2,1 and conjugates 81,80,79,78 in top to bottom center; and 12,11,10,9 and conjugates 73,72,71,70 in center horizontal (square A1). The addition of these pair of numbers and conjugates to the 9x9 square are shown below using directional pointed arrows:

    1 2 3 4 5 6 789 101112 ... 373839 40
    41
    8180 79 787776 75 74 73 72 71 70 ... 454443 42
    ...
  2. Sum up the rows and columns 1-4 and 6-9 and subtract from the magic sum 369. This gives the amounts required (shown in green Square A2). The last column shows the two amounts need to complete the row and column (shown in yellow).
  3. Using adjacent pair numbers from the complementary table above, fill in the non-spoke cells of the 5x5 square, then the 7x7 and finally the 9x9 using the inset below as a guide: (Square A3, A4 and A5).
  4. Picture of arrows
  5. A6 shows the square in border form.
A1
74 4 42
  75 3 43
  76 2 44
  77 1 45
1211 10 9 41 73 727170
37 81 5
  38 80 6
 39 79 7
40 78 8
A2
74 4 42249 83x3
  75 3 43 24882+80+83
  76 2 44 247 82x2+83
  77 1 45 24682x3
1211 10 9 41 73 727170
37 81 5 24682x3
  38 80 6 24582x2+81
 39 79 7 24481x2+82
40 78 8 24381x3
243244245 246246 247248249
A3
74 4 42
  75 3 43
7614 2 69 44
66 77 1 45 16
1211 10 9 41 73 727170
1537 81 5 67
38 68 80 13 6
 64 79 7
40 78 8
A4
74 4 42
75 18 20 363 65 43
607614 2 69 44 22
5866 77 1 45 16 24
1211 10 9 41 73 727170
231537 81 5 6759
2138 68 80 13 6 61
39 64 62 79 1917 7
40 78 8
A5
74 26 2830 4 5355 57 42
50 75 18 20363 65 43 32
4860 7614 2 69 44 22 34
4658 66 77 1 45 16 2436
1211 10 9 41 73 727170
352315 37 81 5 67 5947
3321 38 68 80 13 6 6149
3139 64 62 79 1917 7 51
40 56 5452 78 2927 25 8
A6 Border
74 26 2830 4 5355 57 42
50 75 18 20363 65 43 32
4860 7614 2 69 44 22 34
4658 66 77 1 45 16 2436
1211 10 9 41 73 727170
352315 37 81 5 67 5947
3321 38 68 80 13 6 6149
3139 64 62 79 1917 7 51
40 56 5452 78 2927 25 8
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 26
 
81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56
 
27 28 29 30 3132 33 34 35 36 37 38 39 40
41
55 54 53 52 51 50 49 48 47 46 45 44 43 42

Conversion of the 9x9 into its transposed opposite

Using the method of Part IV, the 9x9 transposed opposite generates a new square which is not a border square. Only the external square is magic.

This completes Part V of a 9x9 border Magic Square Wheel method.
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Copyright © 2015 by Eddie N Gutierrez