METHOD A-1:VARIANTS 2, 3 and 4 for 7x7 Squares

Picture of a wheel

A Discussion of Variant 2, 3 and 4

For the 7x7 examples, variant 2 is set up using the same method as variant 1 except that the left diagonal with the group of numbers ½ (n2-n+2) to ½(n2+n) in reverse order (top left corner to the right lower corner) from the numbers listed in the 7x7 complementary table. These numbers in the left diagonal correspond to 24 → 23 → 22 → 25 → 28 → 27 → 26. Variant 3 is set up in forward zigzag fashion and variant 4 in reverse zigzag fashion, the last two corresponding to 22 → 27 → 24 → 25 → 26 → 23 → 24 and 24 → 27 → 22 → 25 → 28 → 23 → 26, respectively. Alternatively, these may be shown as templates in the partial complementary tables where ⤩ and ⤧ point independently into two directions:

Variant 2
22 23 24
25
28 27 26
Variant 3
22 23 24
25
28 27 26
Variant 4
22 23 24
25
28 27 26

The other diagonal, column and row of the wheel are then added using the templates obtained for the invert (variant 2), forward zigzag (variant 3) or invert zigzag (variant 4), followed by filling in of the "non-spoke" numbers. Below are the the results of filling up 7x7 squares for variant 2, 3 and 4 which follow the method of variant1.

  1. The method uses parity to determine the pairs of squares to use as shown in the following three parity tables.
  2. Also a symmetrical color scheme may be used to determine where to put the initial numbers of each of the pairs on the square and simplifies the method.
  3. In these squares the legitimate colored cells to use are where light blue crosses pink or light blue crosses light blue as in variant 1.
  4. It must be noted that entries into the non-spoke cells (obtained from the complementary tables) are in the following order for variants 2 and 3 (using as an example) 10 → 39 → 11 → 40 as opposed to variants 1 and 3 where the order is 10 → 40 → 11 → 39.
  5.  
    ROWS
    1
    2
    3
    5
    6
    7
    Variant 2
    SUMΔ 175PAIRSPARITY
    779849+49O+O
    769950+49E+O
    7510050+50E+E
    7510050+50E+E
    7410150+51E+O
    7310251+51O+O
    Variant 3
    SUMΔ 175PAIRSPARITY
    7510050+50E+E
    7410150+51E+O
    779849+49O+O
    7310251+51O+O
    769949+50E+O
    7510050+50E+E
    Variant 4
    SUMΔ 175PAIRSPARITY
    779849+49O+O
    7410150+51E+O
    7510050+50E+E
    7510050+50E+E
    769949+50E+O
    7310251+51O+O
  6. As an example, variant 2 is shown with the entries from the parity table included in the last two rows and columns and shows the crossover cells.
Variant 2
175
24 7 46 7798
23 8 45 7699
22 9 44 75100
49 48 47 25 3 2 1 1750
6 41 28 75100
5 42 27 74101
4 43 26 73102
777675 175 7574 73175
9899 100 0100 101 102
24 39 37 7 12 10 46
35 23 8 45 15
33 22 9 44 17
49 48 47 25 3 2 1
16 6 41 28 34
14 5 42 27 36
4 11 13 43 38 40 26
24 39 37 7 12 10 46
35 23 31 8 18 45 15
3329 22 9 44 21 17
49 48 47 25 3 2 1
1620 6 41 28 30 34
14 5 19 42 32 27 36
4 11 13 43 38 40 26
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
Variant 3
22 7 46
27 42 5
24 9 44
49 2 47 25 3 48 1
6 41 26
45 8 23
4 43 28
22 35 7 15 46
27 33 42 17 5
3937 24 9 44 12 10
49 2 47 25 3 48 1
1113 6 41 26 38 40
45 16 8 34 23
4 14 43 36 28
22 30 35 7 15 20 46
32 27 33 42 17 5 19
3937 24 9 44 12 10
49 2 47 25 3 48 1
1113 6 41 26 38 40
18 45 16 8 34 23 31
4 21 14 43 36 29 28
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
Variant 4
24 9 44
27 42 5
22 7 46
47 2 49 25 1 48 3
4 43 28
45 8 23
6 41 26
24 379 12 44
27 32 42 19 5
3330 22 7 46 20 17
47 2 49 25 1 48 3
1621 4 43 28 29 34
45 18 8 31 23
6 13 41 38 26
24 39 379 12 10 44
35 27 32 42 19 5 15
3330 22 7 46 20 17
47 2 49 25 1 48 3
1621 4 43 28 29 34
14 45 18 8 31 23 36
6 11 13 41 38 40 26
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

The next page contains a 9x9 Variant .
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Copyright © 2008 (revised 2009) by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com