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
  5. 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.
  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
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