NEW MAGIC SQUARES WHEEL METHOD

Part IX

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 5x5 and 7x7 squares are magic but not the 3x3 and this square is classified as a partial border.

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 {37,38,39,40,41,42,43,44,45} and {2,19,28,35,41,47,54,63,80}

  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 2,5,8,11 and conjugates 80,77,74,71 in the right diagonal; numbers 1,4,7,10 and conjugates 81,78,75,72 in top to bottom center; and 3,6,9,12 and conjugates 70,73,76,79 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 4 7 1025 7113 6912 ... 373839 40
    41
    8178 75 728077 74 71 79 76 73 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
  4. A6 shows the square in border form.
A1
80 1 45
  63 18 44
  54 27 43
  47 36 42
320 29 34 41 48 536279
40 46 35
  39 55 28
 38 64 19
37 81 2
⇒
A2
80 1 45243 81x3
  63 18 44 24482+81x2
  54 27 43 245 82x2+81
  47 36 42 24482x2+80
320 29 34 41 48 536279
40 46 35 24882x2+84
  39 55 28 24782x2+83
 38 64 19 24883x2+82
37 81 2 24983x3
249248247 248244 245244243
⇒
A3
80 1 45
  63 18 44
5432 27 49 43
50 47 36 42 30
320 29 34 41 48 536279
3340 46 35 51
39 52 55 31 28
 38 64 19
37 81 2
⇒
A4
80 1 45
63 10 1218 6971 44
595432 27 49 43 23
5750 47 36 42 30 25
320 29 34 41 48 536279
263340 46 35 5156
2439 52 55 31 28 58
38 72 70 64 1311 19
37 81 2
⇒
A5
80 4 68 1 7375 77 45
68 63 10 121869 7144 14
6659 5432 27 49 43 23 16
6157 50 47 36 42 30 2521
320 29 34 41 48 536279
222633 4046 35 51 5660
1724 39 52 55 31 28 5865
1538 72 70 64 1311 19 67
37 78 7674 81 97 5 2
⇒
A6 Border
80 4 68 1 7375 77 45
68 63 10 121869 71 44 14
6659 5432 27 49 43 23 16
6157 50 47 36 42 30 2521
320 29 34 41 48 536279
222633 40 46 35 51 5660
1724 39 52 55 31 28 5865
1538 72 70 64 1311 19 67
37 78 7674 81 97 5 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 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

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