METHOD A-1:VARIANTS 2, 3 and 4

Picture of a wheel

A Discussion of Variant 2, 3 and 4

For the 5x5 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 12 → 11 → 13 → 15 → 14. Variant 3 is set up in forward zigzag fashion and variant 4 in reverse zigzag fashion, the last two corresponding to 11 → 14 → 13 → 12 → 15 and 12 → 15 → 13 → 11 → 14, respectively. Alternatively, these may be shown as templates in the partial complementary tables where ⤩ points independently into two directions:

Variant 2
11 12
13
15 14
Variant 3
11 12
13
15 14
Variant 4
11 12
13
15 14

It is seen that variants 1,2 or 3,4 are opposite to each other through template inversion where its step by step construction is shown in Method A-2 . The other diagonal, column and row of the wheel are then added using the templates obtained for the reverse (variant 2), forward zigzag (variant 3) or reverse zigzag (variant 4), followed by filling in of the "non-spoke" numbers. The green colors at the end of variants 2, 3 and 4 show the total sums to complete the columns and rows. Filling in with the requisite pairs produces the magic squares of which the first and third are also border squares.

Variant 2
12 6 22 25
11 5 23
24 25 13 1 2
3 21 15
4 20 14 27
25 27
 
12 18 6 7 22
11 5 23
24 25 13 1 2
3 21 15
4 8 20 19 14
border square
12 18 6 7 22
1611 5 23 10
24 25 13 1 2
9 3 21 15 17
4 8 20 19 14
Variant 3
11 5 23 26
14 20 4
25 2 13 24 1
22 6 12
3 21 15 26
26 26
11 19 5 7 23
14 20 4
25 2 13 24 1
22 6 12
3 8 21 18 15
11 19 5 7 23
1714 20 4 10
25 2 13 24 1
9 22 6 12 16
3 8 21 18 15
Variant 4
12 6 22 25
15 21 3
24 1 13 25 2
23 5 11
4 21 14 27
25 27
 
12 18 6 7 22
15 21 3
24 1 13 25 2
23 5 11
4 8 21 19 14
border square
12 18 6 7 22
16 15 21 3 10
24 1 13 25 2
9 23 5 11 17
4 8 21 19 14
1 2 3 4 5 6 7 8 9 10 11 12
13
25 24 23 22 21 20 19 18 17 16 15 14

The next page uses variants 2, 3 and 4 to construct 7x7 squares.
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Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com