Wheel Method A-1:Variant 5 for a 9x9 Square

Picture of a wheel

A Discussion of Variant 5

For this one 9x9 example (out of a total of 192 combinations), variant 5 is set up so that the group of numbers in the left diagonal ½ (n2-n+2) to ½(n2+n) are set up according to the following order 44 → 37 → 39 → 40 → 41 → 42 → 43 → 45 → 38, as shown in the partial complementary template starting at 1:

Variant 5
37 38 39 40
41
45 44 43 42
⇒
Order
2 9 3 4
5
8 1 7 6

Using this template (normal) the other diagonal, column and row of the wheel are filled in followed by filling in of the "non-spoke" numbers using 81, 82 or 83 as the only sum pairs as shown in the parity table. Note that using the second entry of this table would make E+E+E disappear from row/column 2 or 8. Using the second entry of 7 would leave row 4 with at least one unallowed 80.

PARITY table for variant 5
ROW OR COLUMNPAIR OF NUMBERSPARITY ALLOWED
183+82+82O+E+EYES
183+83+82O+O+ENO
282+82+82E+E+EYES
283+81+82O+O+ENO
381+81+82O+O+EYES
481+81+81O+O+OYES
683+83+83O+O+OYES
782+83+83E+O+OYES
783+83+81O+O+ONO
882+82+82E+E+EYES
883+81+82O+O+ENO
981+82+82O+E+EYES
  1. The wheel is first filled in (Squares I-IV).
  2. Fill in row/columns 4 and 6 (all O) (Square V).
  3. Fill in row/columns 2 and 8 (all E) (Square VI).
    4. Square I
    44
    37
    39
    40
    41
    42
    43
    45
    38
    		⇒
    Square II
    44 6
    37 77
    39 75
    40 74
    41
    8 42
    7 43
    5 45
    76 38
    		⇒
    Square III
    44 72 6
    37 9 77
    39 11 75
    40 1274
    41
    8 70 42
    7 71 43
    5 73 45
    76 10 38
    		⇒
    Square IV
    44 72 6
    37 9 77
    39 11 75
    40 12 74
    2 81 79 78 41 4 3 1 80
    8 70 42
    7 71 43
    5 73 45
    76 10 38
    		⇒
    Square V
    44 62 72 20 6
    37 60 9 22 77
    39 58 11 24 75
    686664 40 12 74 17 15 13
    2 81 79 78 41 4 3 1 80
    141618 8 70 42 65 67 69
    7 23 71 59 43
    5 2173 61 45
    76 191063 38
    		⇒
  4. Fill in the last row/columns 1, 3 and 7, 9 (O+E+E or E+O+O) noting that the rows and columns add up to the requisite pairs.
  5. The last square(VIII) shows the arrangement of the first numbers of each of the two pairs used. (in yellow). These numbers are also clustered symmetrically about the center cell.
  6. An alternative simpler route to this type of square can be used instead. This uses a a color coded method from start to generate a 9x9 square with different arrangements of non-spoke numbers.
Square VI
44 53 62 72 20 30 6
49 37 56 60 9 22 26 77 33
50 39 58 11 24 75 31
686664 40 12 74 17 15 13
2 81 79 78 41 4 3 1 80
141618 8 70 42 65 67 69
32 7 23 71 59 4351
34 5 25 2173 61 57 45 48
76 29 191063 52 38
⇒
Square VII
44 53 54 62 72 20 28 30 6
49 37 56 60 9 22 26 77 33
4650 39 58 11 24 75 31 35
686664 40 12 74 17 15 13
2 81 79 78 41 4 3 1 80
141618 8 70 42 65 67 69
3632 7 23 71 59 435147
34 5 25 2173 61 57 45 48
76 2927 191063 55 52 38
⇒
Square VIII
44 53 54 62 72 20 28 30 6
49 37 56 60 9 22 26 77 33
4650 39 58 11 24 75 31 35
686664 40 12 74 17 15 13
2 81 79 78 41 4 3 1 80
141618 8 70 42 65 67 69
3632 7 23 71 59 435147
34 5 25 2173 61 57 45 48
76 2927 191063 55 52 38
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 27 28 29 30 31 32 33 34 35 36 37 38 39 40
41
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 55 54 53 52 51 50 49 48 47 46 45 44 43 42

The next page contains a color coded method for forming these 9x9 magic squares.
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Copyright © 2008 (revised 2009) by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com