New Squares from a Wheel Spoke/Anti-spoke Method

Part IV

A Spinning wheel

A Discussion of New Wheel Spoke/Anti-spoke Squares

The previous page showed the Wheel methods using the L-leap and wheel approaches to fill in the boundaries of the squares. This page will show how to construct modified magic squares using spoke and anti-spoke addition methods to the same square. By spoke method I mean that pairs are added in the usual manner as shown:

  1. Left bottom corner cell to right upper corner cell.
  2. Top middle column to bottom middle column.
  3. Rightmost middle row to leftmost middle row.

By non-spoke method I mean that pairs are added in the reverse manner:

  1. Right upper corner cell to bottom corner corner cell.
  2. Bottom middle column to top middle column.
  3. Leftmost middle row to rightmost middle row.

To construct a mixed square using a 5x5 example we:

  1. Fill in the internal 3x3 spoke numbers into a 5x5 square.
  2. Fill in the two corner cells to complete the main diagonal.
  3. Fill in the boundary cells in a anti-spoke manner.
  4. Fill in the non-spoke cells on the perimeter to complete the square.
1
   
123 24
25 13 1
2 23 14
 
⇒
2
11
123 24
25 13 1
2 23 14
15
⇒
3
11 22 5
123 24
625 13 1 20
2 23 14
21 4 15
⇒
4
11 19 22 8 5
17 123 24 9
625 13 1 20
102 23 14 16
21 7 4 18 15
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

Three 7x7 Examples

The following examples show the construction of two 7x7 squares using the spoke/anti-spoke method. The first example will use the regular consecutive approach, the second will use a non-consecutive combination.

  1. Fill in the main diagonal with consecutive numbers 22-28.
  2. Fill in the spoke cells for the 3x3 internal square.
  3. Fill in the perimeter cells for the 5x5 internal square in an anti-spoke manner.
  4. Fill in the perimeter cells for the 7x7 external square in an spoke manner.
  5. Fill in the internal 5x5 non-spoke cells.
  6. Fill in the outer boundary non-spoke cells as shown in method A variant 1 first top/bottom rows then side left/right columns.
1
22
23
24
25
26
27
28
⇒
2
22
23
24 3 48
49 25 1
2 47 26
27
28
⇒
3
22
23 46 5
24 3 48
6 49 25 144
2 47 26
45 4 27
28
4
22 9 42
23 46 5
24 3 48
43 6 49 25 144 7
2 47 26
45 4 27
8 41 28
⇒
5
22 36 34 9 17 15 42
23 40 4611 5
38 24 3 48 12
43 6 49 25 144 7
13 2 47 26 37
45 10 4 39 27
8 14 16 41 33 35 28
⇒
6
22 36 34 9 17 15 42
32 23 40 4611 518
3038 24 3 48 12 20
43 6 49 25 144 7
21 13 2 47 26 37 29
19 45 10 4 39 27 31
8 14 16 41 33 35 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 following table summarizes the pairs of numbers and their parities required to fill in the empty rows or columns (spoke numbers) for square 4 above, where two numbers e.g., 36/15 or 32/18 correspond to a pair.

Parity/Pair Table for the 7x7 Square 4 above
ROWS/COLUMNSPAIR OF NUMBERSPARITY
151+51O+O
250+50E+E
350+49E+O
551+50O+E
650+50E+E
749+49O+O

To construct a a second non-consecutive mixed square using a 7x7 example we:

  1. Fill in the 7x7 square with a non-consecutive diagonal.
  2. Fill the first level spoke numbers, then the anti-spoke numbers followed by the last level spoke numbers.
  3. Fill in the non-spoke cells to complete the square.

1
27
22
24
25
26
28
23
⇒
2
27 7 42
22 46 5
24 348
41 6 49 25 1 44 9
2 47 26
45 4 31
8 43 23
⇒
3
27 34 35 7 14 16 42
38 2232 46 19 513
39 30 24 34820 11
41 6 49 25 144 9
10 21 2 47 26 29 40
12 45 18 4 31 28 37
8 17 15 43 36 33 23
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 following table summarizes the pairs of numbers and their parities required to fill in the empty rows or columns (spoke numbers) for square 2 above, where two numbers e.g., 34/16 or 38/13 correspond to a pair.

Parity/Pair Table for the 7x7 Square 2 above
ROWS/COLUMNSPAIR OF NUMBERSPARITY
149+50O+E
251+51O+O
350+50E+E
550+50E+E
649+49O+O
750+51E+O

This last section shows a D magic square filled in in either of two ways, starting with anti-spoke first:

  1. Fill in the wheel starting from the inside out anti-spoke, spoke, anti-spoke.
  2. Fill in the external top/bottom and side boundaries, either first or
  3. Fill in the internal square, first (partially in blue).
  4. Fill in the final square from either 2 or 3.
1
22 43 8
23 3 48
24 46 5
949 6 25 441 41
45 4 26
2 47 27
42 7 28
⇒
2
22 11 13 43 38 40 8
15 23 3 4835
17 24 46 5 33
949 6 25 441 41
34 45 4 26 16
36 2 47 27 14
42 39 37 7 12 10 28
⇓   ⇓
3
22 43 8
23 30 321 48
19 24 46 5 31
949 6 25 441 41
32 45 4 26 18
2 20 47 29 27
42 7 28
⇒
4
22 11 13 43 38 40 8
15 23 30 321 4835
1719 24 46 5 31 33
949 6 25 441 41
34 32 45 4 26 18 16
36 2 20 47 29 27 14
42 39 37 7 12 10 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 parity table for this set of squares may be broken up into those for the entire square whose S=175 or for the internal square whose S=125, as shown below.

Parity/Pair Table for the 7x7 Square 1 and 3 above
R/CPAIR OF NUMBERSPARITYNUMBERPARITY
151+51O+O--
250+51E+O51O
350+50E+E50E
550+50E+E50E
649+50O+E49O
749+49O+O--

The equation for the number of square groups for this method is identical to method B.
This completes this section on method D. To return to previous method (Part III) or to return to homepage.

Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com