COLOR CODED WHEEL METHOD for the VARIANT 5 (a 9x9 Square)

A Discussion of Variant 5

For this one 9x9 example, variant 5CC is set up so that the group of numbers in the left diagonal ½ (n²-n+2) to ½(n²+n) are set up according to the following order 44 → 37 → 39 → 40 → 41 → 42 → 43 → 45 → 38, as shown in the partial complementary template starting at 1:

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Using this template (normal) the other diagonal, column and row of the wheel are filled in followed by filling in of the "non-spoke" numbers using 81, 82 or 83 as the only sum pairs as shown in the parity table.

1. Fill in the wheel as was done the original non color-coded wheel method to arrive at Square I.
2. To Square I add two final columns and rows into which a running total of each of the columns/rows will be placed (grey column). The final column/row contains the running total of the Δ 369 sums.
3. Fill in row/columns 1 and 9 (using the complementary table at the end as a guide) placing the lowest number of the pair into the yellow cells and adding the complementary numbers (paired with these numbers) semi-associatively, e.g., 15 → 67 → 16 → 66. (Square II).


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Square I 369     44             72             6 122 247     37         9         77     123 246         39     11     75         125 244             40 12 74             126 243 2 81 79 78 41 4 3 1 80 369 0             8 70 42             120 249         7     71     43         121 248     5         73         45     123 246 76             10             38 124 245 122 123 125 126 369 120 121 123 124 369     247 246 244 243 0 249 248 246 245

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Square II 369     44 13 15 17 72 65 67 69 6 368 -1 19 37         9         77 62 204 165 21     39     11     75     60 206 163 23         40 12 74         58 207 162 2 81 79 78 41 4 3 1 80 369 0 59         8 70 42         24 203 166 61     7     71     43     22 204 165 63 5         73         45 20 206 163 76 68 66 64 10 18 16 14 38 370 1 368 204 206 207 369 203 204 206 370 369     -1 165 163 162 0 166 165 163 1

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4. Fill in row/columns 2 and 8 similarly to arrive at Square III.
5. Fill in row/columns 3 and 7 similarly to arrive at Square IV. Note that four sums differ by ±1 from 369. These numbers must be modified to make the square completely magic.



Square III 369     44 13 15 17 72 65 67 69 6 368 -1 19 37 25 27 9 55 57 77 62 368 -1 21 29 39     11     75 52 60 287 82 23 31     40 12 74     50 58 288 81 2 81 79 78 41 4 3 1 80 369 0 59 51     8 70 42     32 24 286 83 61 53 7     71     43 30 22 287 82 63 5 56 54 73 28 26 45 20 370 1 76 68 66 64 10 18 16 14 38 370 1 368 368 287 288 369 286 287 370 370 369     -1 -1 82 81 0 83 82 1 1

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Square IV 369     44 13 15 17 72 65 67 69 6 368 -1 19 37 25 27 9 55 57 77 62 368 -1 21 29 39 33 11 49 75 52 60 369 0 23 31 35 40 12 74 46 50 58 369 0 2 81 79 78 41 4 3 1 80 369 0 59 51 47 8 70 42 36 32 24 369 0 61 53 7 48 71 34 43 30 22 369 0 63 5 56 54 73 28 26 45 20 370 1 76 68 66 64 10 18 16 14 38 370 1 368 368 369 369 369 369 369 370 370 369     -1 -1 0 0 0 0 0 1 1



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6. To do this the second cells in square Va in column/row 1 are interchanged with the ninth cells of column/row 9. Thus interchange occurs as follows: 14↔13 and 20↔19.
7. After removal of the 10^th and 11^th columns/rows and color coding the internal cells for better vizualization gives square Vb.
8. Similarly the complementaty table numbers are color coded for comparison with the numbers in the magic square.

The next page contains Method A-2, which uses a template invert form.
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