Magic Squares Wheel Method-Redux Part I

A Discussion of the Magic Square Wheel Method

This is a redux of the original web pages Method I and Method II which discusses the wheel method using a rather more complicated approach. This page gives a better simplifying approach. Thw wheel method is an entrely new approach invented 20 years ago but different from the traditional step methods such as the Loubère and Méziriac. These methods may be considered sequential approach while the wheel is random access approach. To start a wheel magic square is constructed using a complementary table as guide. The table is a pair of complementary strands connected at a hinge with the central cell of the square as the hinge. And the numbers of this table are added to an empty square using certain rules which are listed below. This complementary table approach is also currently in use in a new method for the generation of Loubère type squares.

1. The left diagonal is added 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 n = 5 table described above. The group of numbers corresponding to this left diagonal are 11 → 12 → 13 → 14 → 15 (Square A1). Note that the left diagonal is shown differently as it is normally portrayed (it always is the right diagonal) but the squares differ only by rotation.
2. The right diagonal is added from bottom left corner to the right upper corner from the comlpementary pairs {3,4}, i.e., numbers 3,4,22,23, to give Square A2.
3. Enter the central column with the complementary pairs {5,6} (Square A3) in regular order.
4. Enter the central row with the complementary pairs {1,2} in reverse order, i.e., left to right (Square A4).
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.

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 11 5 23 12 6 22 25 24 13 2 1 4 20 14 3 21 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

6. A quick aside: the complementary table consists of pairs of complements having complementary sum = n+1. Threre are also two other non complementary sums between adjacent pairs and these are n and n+2. All three sums are useful in determining what numbers to place in the white non spoke positions. Square A4 we see has the sum of the first and last rows as well as the first and last columns are equal to the complementary sum = 26. In addition, the sum of the sum of rows (2 and 4) and columns (2 and 4) is equal to the non complementary sum between two adjacent complementary pairs = 25. The numbers inserted into these white cells must adhere to this rule in order for the square to be magic.
7. The non spoke portions of the square (those in white) are filled with the four pairs that are left over (Square A5 and A6).
8. The complement {7,19} is placed in the first row as shown. The second complement {8, 18} is added to the last row in inverse order so that the number 8 is opposite and at a symmetric distance from the number 7 while the number 18 is opposite and at a symmetric distance to the number 19 giving the partial square A5.
9. Similarly the {9,17} pair is addded to the first column the pair {10,16} is added in inverse order as shown in A6. This completes the square.

A5 11 7 5 19 23 12 6 22 25 24 13 2 1 4 20 14 3 18 21 8 15

⇒

A6 11 7 5 19 23 9 12 6 22 16 25 24 13 2 1 17 4 20 14 10 3 18 21 8 15

10. An alternative is to switch the non spoke numbers to give the two variants:

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 adjacent 5x5 conformations wheel magic squares and their accompanying complementary tables showing that the numbers must be taken in groups of pairs for the square to be magic. (Note that a 5x5 square requires one of these conformations).

One example of a 7x7 magic square

Since each comformation of a 7x7 wheel magic square can produce 7 adjacent wheel comformations, we'll take the first subset {1,2,3,4,5,6,7,8,9} and their complements as an example. Note that unadjacent conformations are also possible. Also these squares have internal non spokes cells which will be colored in blue.

1. The magic square is first constructed by filling in the left diagonal with a group of numbers from the 7x7 complementary table below. For a 7x7 square the numbers in the left diagonal correspond to 22 → 23 → 24 → 25 → 26 → 28 → 28. (Square B1)
2. The right diagonal is added from the bottom left corner to the right upper corner choosing the pairs {4,5,6} plus their complements {44,45,46} to give Square B2.
3. This is followed by addition of the central column pairs {7,8,9) and their complements {41,42,43} to give Square B3.
4. Then followed by addition of the central column pairs {1,2,3) plus their complement {47,48,49} in reverse order to give Square B4.
5. If we sum up the numbers in each row (R) and column (C) of square IV we get the results in the Summation table where the Sums can be broken down into the requisite sum pairs. If we know what the sums are in each of the rows and columns, all that is required placing in the complement pairs. If we look at the parity column we can see that the parity of the two row or column numbers in parenthesis (1,7), (2,6) and (3,5) must be the same (where O is odd and E is even) in order for the square to be magic. The summation table is a necessity in order to facilitate the placement of the complement numbers. It's also good to have the accompanying complement table so as to add up and cross out the numbers as they are picked.

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

Summation Table R/C Sum Pair Sum Parity 1 100 50+50 E+E 2 99 49+50 O+E 3 98 49+49 O+O 4 - - - 5 102 51+51 O+O 6 101 50+51 E+O 7 100 50+50 E+E

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

6. The numbers remaining are the non spokes (in white). Pair up these numbers, using the Summation table as a guide, as follows: {10,11}, {12,13}, {14,15}, {16,17}, {18,19} and {20,21} along with their complements. First the number 10 is placed in row 1, cell 2 and its complement 40 in the sixth cell (Square B5).
7. Next the pair {11,39} is placed in reverse order just opposite to the {10,40} pair (Square B5). By opposite is meant opposite and across via an imaginary line running through the center cell.
8. Next the number 12 is placed after the 10 and 38 its complement before the 40. The (11,37} pair is place in reverse order to the {12,38} pair as was done above (Square B6).
9. Repeat the process with the next eight numbers {16,34} pair in column one and {33,17} pair in reverse order in column seven. Similar placement of the pair {14,36} in column one and {35,15} in column seven completes the partial square (Square B7).

B5 22 10 7 40 46 23 8 45 24 9 44 49 48 47 25 3 2 1 6 41 26 5 42 27 4 39 43 11 28

⇒

B6 22 10 12 7 38 40 46 23 8 45 24 9 44 49 48 47 25 3 2 1 6 41 26 5 42 27 4 39 37 43 13 11 28

⇒

B7 22 10 12 7 38 40 46 16 23 8 45 33 14 24 9 44 35 49 48 47 25 3 2 1 36 6 41 26 15 34 5 42 27 17 4 39 37 43 13 11 28

10. Almost there. The square the pair {18,32} are placed on row two while {31,19} are placed in reverse order on row six (Square B8).
11. We finish filling up the square like a 2 dimensional Chinese wooden puzzle. The {20,30} pair is placed on column three; the {29,21} pair is placed in reverse order on column five (Square B9).

This completes the Magic Square Wheel Redux Part I. To go to next Part II which discusses three other variants of 5^th order squares.
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