MAGIC SQUARES WHEEL TRANSPOSITION OF TWO WHEEL VARIANTS

Picture of numbers

A Discussion of the Magic Square Wheel Method

Variant I

Variant 1 of the 7x7 magic square (B8) generated in section on wheel type squares is numbered here as Square B1. This magic square was constructed using a complimentary table as a guide. The square has a magic sum S of 175 with the internal subsquares non-magic. It has been found that doing several column and row transpositions can convert B1 into a new square having the diagonals and the center column and row inverted. In a sense B1 has been imploded or everted into B3, i.e., B1 and B3 are opposites. The new square (B3) is also a border square where the internal subsquares are also magic.

Transposition of B1

  1. Take square B1 and transpose (column 1 with column 3) and (column 5 with column 7) to get Square B2.
  2. Take square B2 and transpose (row 1 and row 3) and (row 5 with row 7) to get Square B3.
  3. B3 is a border square where the internal subsquares are also magic. In this case where the magic sums are 75, 125 and 175, respectively, for the 3x3, 5x5 and 7x7.
B1
22 10 12 7 38 40 46
16 23 18 8 32 45 33
14 20 24 9 44 29 35
49 48 47 25 3 2 1
36 30 6 41 26 21 15
34 5 31 42 19 27 17
4 39 37 43 13 11 28
B2
12 10 22 7 46 40 38
18 23 16 8 33 45 32
24 20 14 9 35 29 44
47 48 49 25 1 2 3
6 30 36 41 15 21 26
31 5 34 42 17 27 19
37 39 4 43 28 11 13
B3
24 20 14 9 35 29 44
18 23 16 8 33 45 32
12 10 22 7 46 40 38
47 48 49 25 1 2 3
37 39 4 43 28 11 13
31 5 34 42 17 27 19
6 30 36 41 15 21 26
Complementary Table
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 II

Variant 4 of a 7x7 magic square generated in section on wheel type squares is shown as Square C1. This magic square was constructed using a complimentary table as a guide. The square C1 is a border square where the external and two internal squares have magic sums of 17, 125 and 75, respectively. Performing several column and row transpositions can convert C1 into a new square having the diagonals and the center column and row inverted. In a sense C1 has been imploded or everted into C3, i.e., C1 and C3 are opposites. The new square (C3), however, is not a border square.

Transposition of C1

  1. The complementary table for C1 is identical to above but with the light green cells in light blue.
  2. C1 is a border square where the internal subsquares are also magic. In this case where the magic sums are 75, 125 and 175, respectively, for the 3x3, 5x5 and 7x7. Take square C1 and transpose (column 1 with column 3) and (column 5 with column 7) to get Square C2.
  3. Take square C2 and transpose (row 1 and row 3) and (row 5 with row 7) to get Square C3.
  4. C3 is not a border square but only magic externally.
C1
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
C2
37 39 24 9 44 10 12
32 27 35 42 15 5 19
22 30 33 7 17 20 46
49 2 47 25 3 48 1
4 21 16 43 34 29 28
18 45 14 8 36 23 31
13 11 6 41 26 40 38
C3
22 30 33 7 17 20 46
32 27 35 42 15 5 19
37 39 24 9 44 10 12
49 2 47 25 3 48 1
13 11 6 41 26 40 38
18 45 14 8 36 23 31
4 21 16 43 34 29 28

Squares B3 and C1 in Border Form

B3
24 20 14 9 35 29 44
18 23 16 8 33 45 32
12 10 22 7 46 40 38
47 48 49 25 1 2 3
37 39 4 43 28 11 13
31 5 34 42 17 27 19
6 30 36 41 15 21 26
   
C1
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

This completes the Magic Square Wheel Transpose method. To see the Shift wheel type variant 5x5 square.
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Copyright © 2013 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com