NON-CONSECUTIVE MAGIC SQUARES WHEEL METHOD (Part I)

A Discussion of the non-consecutive 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 complimentary table as a guide. The cells in the same color are associated with one another; for example, only 3 → 4 → 22 → 23 are still pair-associated consecutively together, while the other colored pairs are not. Because of the way these pairs are grouped the 5x5 square produced will not be initially magic but must be converted into one by a series of steps.

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 {3,4,22,23} (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 {2,5,21,26} (Square A3) in regular order.
4. Then by the central row from the pairs {1,6,20,25} in reverse order (Square A4). We now have a partial square with not all sums equal to 65.
5. 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, with the non-consecutive pairs in different colors.

A1 11 12 13 14 15

⇒

A2 11 23 12 22 13 4 14 3 15

⇒

A3 11 2 23 12 5 22 13 4 21 14 3 24 15

⇒

A4 65 11 2 23 36 12 5 22 39 25 20 13 6 1 65 4 21 14 39 3 24 15 42 39 36 65 42 39 65

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

6. 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

7. At this point all sums are not equal to 65 so modifications are carried out to generate a magic square. Convert 20 and 6, respectively, to 24 and 2. This converts all sums in the last grey row to 65 (Square A7).

A5 65 11 7 2 19 23 62 12 5 22 39 25 20 13 6 1 65 4 21 14 39 3 18 24 8 15 68 39 61 65 69 39 65

⇒

A6 65 11 7 2 19 23 62 9 12 5 22 16 64 25 20 13 6 1 65 17 4 21 14 10 66 3 18 24 8 15 68 65 61 65 69 65 65

⇒

A7 65 11 7 2 19 23 62 9 12 5 22 16 64 25 24 13 2 1 65 17 4 21 14 10 66 3 18 24 8 15 68 65 65 65 65 65 65

⇒

8. Convert 5 and 21 to 6 and 20, respectively, which converts all sums in rows 2 and 4 to 65 (Square A8).
9. Convert 2 and 24 in rows 1 and 5 to 5 and 21, respectively. All are now 65 (Square A9). A9 is also the square obtained by the regular wheel method. It will be shown that this is typical even for larger squares.

This completes Part I of the non-consecutive Magic Square Wheel method. Part II contains a second variation. To go back to homepage.