Magic Squares Wheel Method-Redux Part V

Two Variants of Order 9 - Border Squares

The wheel method is a means of constructing magic squares by a random access means instead of sequentially like the Loubère and Méziriac methods which has been rewritten in a more simplified form. The method patially fills up a square to form a wheel structure using numbers chosen from a complementary table of order n then randomly fills up the rest of the square with numbers chosen from whatever is left in the complementary table. This paper is a simplification of the original paper taking a more facile and explanatory approach. Filling up the non spoke numbers can be compared to the filling up a large Soduku square but using complementary table and Summation tables to aid in the filling of the square which will be discussed below.

One 9^th order variant (out of a total of 192 combinations), is constructed so that the group of numbers in the left diagonal ½(n²-n+2) to ½(n²+n) line up according to the following reverse order 40 → 39 → 38 → 37 → 41 → 45 → 44 → 43 → 42. The rest of the spoke numbers come from the first 12 pairs (in light brown) of the complementary table shown at the end. A pair corresponds to a number and its complement which are positioned on the square according to symmetrical considerations, i.e., each value from a pair of complements are equidistant from the center cell. In this case the pairs were chosen from the complementary table as follows: the first 4 pairs placed in the central row, the second 4 pairs placed on the right diagonal and the third 4 pairs in the central column all placed in random order, not sequentially, as was done in Part I. This section is a continuation of Part IV which produced a non border square as opposed to this section where the square generated is a border type.

1. To start the the wheel is constructed according to the method employed in Part I (Squares I-IV).
2. Aside I: 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. These pairs can only take on the values of 81, 82 or 83, values which correspond to adjacent complement pairs. We then take these values, go over to the complement table and choose the complement pairs that fulfill this sum requirement.
3. Aside II: If we look at the parity column we can see that the parity of the two row or column numbers (R/C) in (1,9), (2,8), (3,7) and (4,6) must be the same (where O is odd and E is even) in order for the square to be magic. Therefore, having the Summation table as a guide is a necessity in order to facilitate the placement of the complement numbers. It's also good to have the accompanying complement table on hand so as to add up and cross out the pair of numbers as they are chosen.
4. Continuing we add the three complement pairs {14,68}, {16,66}, {18,64} to rows four and six as shown then add the three complements {17,65}, {15,67}, {13,69} in reverse order as shown in square V.
5. Second add the three complement pairs {20,60}, {24,58}, {18,64} to columns four and six as shown then add the three complements {23,59}, {21,61}, {19,63} in reverse order as shown in square V.
6. Note that the large numbers decrease by two as they go into the square while the low numbers increase by two. The colors of adjacent pairs on the complement table should be the same and therefore are seen to be centrally opposite from one another. In addition, the numbers in blue are the non spokes of the central 5^th order square (not magic).
7. The central 7x7 square is filled in as was done in Part I using the summation table as a guide employing the pairs {23,48} and {21,61} for row two; reversed {34,49} and {22,61} for row eight; {46,35} and {15,66} for column two; reversed {47,36} and {16,67} for column eight. (Square VI). Note that R/C 3 and 7 may also be broken down into the sums {81+81+83} and {83+83+81} However, these numbers of O+O+O parity are disallowed for this table since it will be difficult if not impossible to produce a magic square via this approach.

Square I 40 39 38 37 41 45 44 43 42

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Square II 40 77 39 76 38 75 37 74 41 8 45 7 44 6 43 5 42

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Square III 40 9 77 39 10 76 38 11 75 37 12 74 41 8 70 45 7 71 44 6 72 43 5 73 42

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Square IV 40 9 77 39 10 76 38 11 75 37 12 74 81 80 79 78 41 4 3 2 1 8 70 45 7 71 44 6 72 43 5 73 42

Summation Table R/C Sum Pair Sum Parity 1 243 81+81+81 O+O+O 2 244 81+81+82 O+O+E 3 245 81+82+82 O+E+E 4 246 82+82+82 E+E+E 5 - - - 6 246 82+82+82 E+E+E 7 247 82+82+83 E+E+O 8 248 83+83+82 O+O+E 9 249 83+83+83 O+O+O

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Square V 40 62 9 19 77 39 60 10 21 76 38 58 11 23 75 68 66 64 37 12 74 18 16 14 81 80 79 78 41 4 3 2 1 13 15 17 8 70 45 65 67 69 7 24 71 59 44 6 22 72 61 43 5 20 73 63 42

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8. Place {25,56} in row one, followed by {57,26} in the last row in reverse order. Place {29,53} in row two, followed by {52,30} in reverse order in row eight (Square VII).
9. Finally fill in row one and row nine with the two adjacent pairs {27,54} and the other {55,28} in reverse order, then finish by filling in row three with {31,51} and row seven with {50,32} (Square VIII). VIII is a border square with the magic sum values of 123, 205, 287 and 369 for the 3^rd, 5^th, 7^th and the 9^th order squares of which it is composed.

Square VI 40 62 9 19 77 39 48 60 10 21 33 76 35 38 58 11 23 75 47 68 66 64 37 12 74 18 16 14 81 80 79 78 41 4 3 2 1 13 15 17 8 70 45 65 67 69 46 7 24 71 59 44 36 6 34 22 72 61 49 43 5 20 10 63 42

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Square VII 40 25 62 9 19 56 77 29 39 48 60 10 21 33 76 53 35 38 58 11 23 75 47 68 66 64 37 12 74 18 16 14 81 80 79 78 41 4 3 2 1 13 15 17 8 70 45 65 67 69 46 7 24 71 59 44 36 52 6 34 22 72 61 49 43 30 5 57 20 73 63 26 42

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Border Square VIII 40 25 27 62 9 19 54 56 77 29 39 48 60 10 21 33 76 53 31 35 38 58 11 23 75 47 51 68 66 64 37 12 74 18 16 14 81 80 79 78 41 4 3 2 1 13 15 17 8 70 45 65 67 69 50 46 7 24 71 59 44 36 32 52 6 34 22 72 61 49 43 30 5 57 55 20 73 63 28 26 42

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	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
																																								41
81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62	61	60	59	58	57	56	55	54	53	52	51	50	49	48	47	46	45	44	43	42

10. Square VIII can also be modified by placing all the spoke numbers in reverse direction and retaining all of the non spoke numbers. The Summation and complement tables are identical for this new square IXa and the color formatted IXb to show the internal squares. A partial order square X is included to show that changing the order of two numbers in each of the spokes of the wheel structure can produce at least one less square, the 5x5 being not magic. The only difference in the Summation table is that the sum and parity of R/C (2 and 3) and (7 and 8) are reversed leading to one less magic square.

Go to Part VI where two border 11^th order squares are displayed.
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