Method A-2:Template Inversion

Picture of a wheel

A Discussion of Method A-2

This method is similar to method A-1 except that the combination of left diagonal with the group of numbers ½ (n2-n+2) to ½(n2+n) is in one template configuration (normal or invert) while the rest of the "spoke" numbers are in the opposite template form (invert or normal). Both the normal and its invert have the same but opposite configuration. Using a 7x7 example we can set up the order table for one template and convert it by inversion through the (3,5) axis the into the order table of its opposite. The table of actual complement numbers, however, is only inverted at the central number, e.g., 25 in this case, so as to keep the order of the complement table in numerical order as was shown a few pages back. Other template mixes such as two normal and two inverts do not produce magic squares. Only the combinations above: one normal/ three inverts or one invert/three normals produce magic squares.

22 23 24
25
28 27 26
⇔  
6 3 1
4
2 5 7
⇓
22 23 24
25
28 27 26
⇔
1 3 6
4
7 5 2

Variants 1 and 2 are set up in a different "spoke" conformation as displayed in previous pages, this time beginning at number 5 in the complement table. Notice that a shift in the complement table of two pairs occur. Variant 1 is set up using the normal template as the left diagonal and the invert template in the right diagonal as well as the "spoke" numbers of the column and row. Variant 2 uses the invert template as the left diagonal and the rest of the "spoke" numbers using the normal template form. Filling in the parity table below one obtains the pairs (49,49), (50,50), (51,51), (49,49) and (51,51). Since it is not easy to determine where to put the pairs (49,50) or (50,51) I have found that it is best to insert the (49,49) and (51,51) pairs into the rows and columns where they go first, followed by the (50,50) pairs. This makes it easier to fill in the square.

 
ROWS/COLUMNS
1
2
3
5
6
7
Variant 1
PAIR OF NUMBERSPARITY
49+49O+O
50+50E+E
50+49E+O
51+50O+E
50+50E+E
51+51O+O
Variant 2
PAIR OF NUMBERSPARITY
50+50E+E
51+51O+O
50+49E+O
51+50O+E
49+49O+O
50+50E+E
Variant 1
24 11 42
28 37 10
23 12 41
45 7 44 25 6 43 5
9 38 27
40 13 23
8 39 26
⇒
24 35 33 11 16 14 42
31 28 37 10 19
29 23 12 41 21
45 7 44 25 6 43 5
20 9 38 27 30
1840 13 23 32
8 15 1739 34 36 26
⇒
24 35 33 11 16 14 42
31 28 1 37 49 10 19
2946 23 12 41 3 21
45 7 44 25 6 43 5
204 9 38 27 47 30
1840 48 13 2 23 32
8 15 1739 34 36 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 2
22 13 40
26 39 8
23 12 41
43 5 44 25 6 45 7
9 38 27
42 11 24
10 37 28
⇒
22 32 13 18 40
36 26 34 39 17 8 15
30 23 12 41 20
43 5 44 25 6 45 7
21 9 38 27 29
1442 16 11 33 24 35
1019 37 31 28
⇒
22 32 4813 2 18 40
36 26 34 39 17 8 15
4630 23 12 41 20 3
43 5 44 25 6 45 7
421 9 38 27 29 47
1442 16 11 33 24 35
1019 1 37 49 31 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

The next page contains method B .
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Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com