Magic Squares Wheel Method-Redux Part IV

New Variant of Order 9

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 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 order 44 → 37 → 39 → 40 → 41 → 42 → 43 → 45 → 38. 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 will produce a square that is not a border. A particular border square is produced in the next page Part V using the methods outlined below.

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 {20,62}, {22,60}, {24,58} 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 {26,56} and {22,60} for row two; reversed {25,57} and {21,61} for row eight; {32,50} and {16,66} for column two; reversed {31,51} and {15,57} for column eight. All employing the Summation table to input the correct number parity (Square VI). Note that R/C 1 and 9 may also be broken down into the sums {81+83+83} and {83+81+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 44 37 39 40 41 42 43 45 38

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

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

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

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

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

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8. Place {53,30} in row one, followed by {29,52} in the last row in reverse order. Place {49,33} in row two, followed by {34,48} in reverse order in row eight (Square VII).
9. Finally fill in row one and row nine with the two adjacent pairs {54,28} and the other {27,55} in reverse order. Then finish by filling in row three with {46,35} and row seven with {36,47} (Square VIII).

A regular 9^th order square along with its Summation and complement tables are listed below. The left diagonal is in sequential 37 → 38 → 39 → 40 → 41 → 42 → 43 → 44 → 45 and the Summation table while different from square VIII is similar in that a symmetrical structure is present in the parity column, i.e, both top and bottom sections have the same parity ranging from E+E+E, E+E+O, E+O+O and O+O+O. Parities that don't follow this structure are doomed to be non magic. In addition while R/C line 1 can also be specified as 82+81+83 this violates parity since it places the value of 83 above the R/C line 5 and changes the parity to E+O+O a value already present on R/C line 3.

To go to next section Part V.
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