Magic Squares Wheel Method(A1):Variant

A Discussion of the Magic Square Wheel Method

Construction of these magic squares requires is an approach which differs from the traditional step methods such as the Loubère and Méziriac. The magic square is constructed as follows using a complementary table as a guide.

1. The left diagonal is filled with the group of numbers ½ (n²-n+2) to ½(n²+n) in consecutive order (top left corner to the right lower corner) from the numbers listed in the complementary table described above, for example using n = 5. For a 5x5 square the numbers in the left diagonal correspond to 11 → 12 → 13 → 14 → 15 (Square A1).
2. Add the right diagonal in reverse order from bottom left corner to the right upper corner choosing only from the pair {3,4} (as in the little square below) to give Square A2.

3	→	4
		↓
23	←	22

3. This is followed by the central column from the pairs {5,6} (Square A3) in regular order.
4. Then by the central row from the pairs {1,2} in reverse order (Square A4). We now have a partial square with not all sums equal to 65.
5. To fill up the rest of the square we can also use a color scheme whereby the difference of the partial sums is subtracted from S = 65. This gives the last column/row which when crossed togethr gives a yellow cell. It is into this cell that the initial number is placed. See below for more information.
6. The result of these operations figuratively speaking resembles the hub and spokes of a wheel where the cells in color correspond to the spoke and hub of the wheel.

A1 11 12 13 14 15

⇒

A2 11 23 12 22 13 4 14 3 15

⇒

A3 11 5 23 12 6 22 13 4 20 14 3 21 15

⇒

A4 65 11 5 23 39 26 12 6 22 40 25 25 24 13 2 1 65 0 4 20 14 38 27 3 21 15 39 26 39 40 65 38 39 65 26 25 0 27 26

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

7. Fill in the spoke portions of the square with the four pairs that are left over (Square A5 and A6). This is done as follows:
   Use the small squares (from the complementary table) as an aid in in how to place the adjacent pair of numbers (in white), which may be placed in either a clockwise or anticlockwise manner as shown below:

7		8		7		8
↓	↗	↓		↑	↙	↑
19		18		19		18

A "spoke" (as used above) consists of n - ½(n + 1) pairs of numbers including their complements but not the hub. For example, the set {(1,2)(3,4)(5,6)} and their complements forms a triad of consecutive adjacent numbers, which are grouped together, as shown in the complementary table above, to form the three other "spokes".

The remainder of the cells (the "non-spoke" numbers) are then subsequently filled using adjacent complementary pairs and are added semi-associatively as opposed to the "spoke" pairs which are associative, i.e diametrically equidistant from the center cell and are filled in as in A5 and A6.

The other two examples of 5x5 magic squares

The following contains the other two 5x5 conformations wheel magic squares (by analysis using of the ad hoc equation ¼ (n²-4n + 7)) and their accompanying complementary tables to show that the numbers must be taken in groups of pairs for the square to be magic. (Note that each combination of a 5x5 square requires three of these conformations).

One example of a 7x7 magic square

Since each comformation of a 7x7 wheel magic square can produce 7 wheel comformations, we'll take the first subset {1,2,3,4,5,6,7,8,9} and their complements as an example.

1. The magic square is first constructed by filling in the left diagonal with a group of numbers from the 7x7 complentary table below. For a 7x7 square the numbers in the left diagonal correspond to 22 → 23 → 24 → 25 → 26 → 28 → 28. (Square B1)
2. Add the right diagonal in reverse order from bottom left corner to the right upper corner choosing the pairs {4,5,6} to give Square B2.
3. This is followed by addition of the central column pairs {7,8,9) to give Square B3.
4. Then by addition of the central column pairs {1,2,3) in reverse order to give Square B4.

B1 22 23 24 25 26 27 28

⇒

B2 22 46 23 45 24 44 25 6 26 5 27 4 28

⇒

B3 22 7 46 23 8 45 24 9 44 25 6 41 26 5 42 27 4 43 28

⇒

B4 22 7 46 23 8 45 24 9 44 49 48 47 25 3 2 1 6 41 26 5 42 27 4 43 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

Parity Table

ROW or COLUMN	SUM	Δ 175	PAIR OF NUMBERS	PARITY (odd or even)
1	75	100	50+50 or 51+49	E + E
2	76	99	49+50	O + E
3	77	98	49+49	O + O
5	73	102	51+51	O + O
6	74	101	50+51	O + E
7	75	100	50+50 or 51+49	E + E

5. Do a summation of each column, row and diagonal on B4. The magic sum S is 175.
6. Set up a parity table (above) on square B4. The difference (Δ) of column 2 from 175 corresponds to column 3. Each row or column with empty cells on B4 requires 2 sets of pairs that sum to the number on column 3 of the parity table. Each pair of numbers for the 7x7 square is either n², n²+1 or n²+2, i.e., 49, 50 or 51. Also we see that we can classify these numbers under two groups, one in light blue (column 3) and one pink.
7. In order for the pairs to conform to parity, those pairs required are listed in column 4 of the parity table and compared to parity in column 5. Those that do not conform are struck out.
8. Fill in the empty cells with pairs of numbers from the 7x7 complementary table. For example, line 1 having 10+40=50 (E Parity), requires the pair 12+38=50 (E Parity), as listed in the first line of the parity table along with their complements.
9. To fill up the magic square we notice that Square B4 may be filled in a simple manner. Here is where we modify the method somewhat. Where the last entries (in light blue) of the columns coincide with the last entries of the rows also in light blue the cell is colored yellow. It is into these cells that the first number from the complementary pairs is placed. See Square B4.
10. The reason we follow this route is that as the squares get larger it gets more and more difficult to assign numbers to cells. This method removes that ambiguity and produces consistent results, since we now force the assignment. It is still possible to assign numbers to cells without using this method and arrive at different squares, but several other s steps must be taken as is done in The Reverse Wheel Method XI.

B4 175 22 7 46 75 100 23 8 45 76 99 24 9 44 77 98 49 48 47 25 3 2 1 175 6 41 26 73 102 5 42 27 74 101 4 43 28 75 100 75 76 77 175 73 74 75 175 100 99 98 102 101 100

⇒

B5 175 22 10 12 7 38 40 46 175 23 8 45 76 24 9 44 77 49 48 47 25 3 2 1 175 6 41 26 73 5 42 27 74 4 39 37 43 13 11 28 175 75 125 126 175 124 125 75 175

⇒

B6 175 22 10 12 7 38 40 46 175 16 23 8 45 33 125 14 24 9 44 35 126 49 48 47 25 3 2 1 175 36 6 41 26 15 124 34 5 42 27 17 125 4 39 37 43 13 11 28 175 175 125 126 175 124 125 175 175

11. After filling in the first and last rows and columns (Square B6), fill in the internal cells using parity and the color code. This produces first B7 and finally B8.

This completes the Magic Square Wheel method. The next web page contains variants 2, 3 and 4. A new Reverse Wheel Method is now listed online.
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