A New Procedure for Magic Squares (Part V)

9x9 Mask-Generated Squares

A Discussion of the New Method

This page discusses converting a regular read square into its boustrophedonical twin and then using a symmetrical C₂ mask (symmetrical by 180° rotation) of n² numbers to convert it into a magic square as was shown in for generating magic squares using numerical masks. The magic squares produced have only partially sequential order and not all the numbers from 1 to n² are present. Because of the manipulation numbers greater than n²or less than 1 are generated.

In addition, it will also be shown that the sums of these squares follow the new sum equation as was shown in the New block Loubère Method:

Construction of a 9x9 Magic Square

1. Generate square A with the sum and row columns (in gray).The last column and row give the amounts that the sums differ from 369.
2. As shown below this square is not magic because all the columns and rows don't sum to 369.
3. Reverse rows 2, 4, 6 and 8 so that the read-out is boustrophedonically, i.e., right then left (Square 2).



1 369 1 80 3 78 5 76 7 74 9 333 -36 72 11 70 13 68 15 66 17 64 396 27 19 62 21 60 23 58 25 56 27 351 -18 54 29 52 31 50 33 48 35 46 378 9 37 44 39 42 41 40 43 38 45 369 0 36 47 34 49 32 51 30 53 28 360 -9 55 26 57 24 59 22 61 20 63 387 18 18 65 16 67 14 69 12 71 10 342 -27 73 8 75 6 77 4 79 2 81 405 36 365 372 367 370 369 368 371 366 373 369     -4 3 -2 1 0 -1 2 -3 4 -5

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2 369 1 80 3 78 5 76 7 74 9 333 -36 64 17 66 15 68 13 70 11 72 396 27 19 62 21 60 23 58 25 56 27 351 -18 46 35 48 33 50 31 52 29 54 378 9 37 44 39 42 41 40 43 38 45 369 0 28 53 30 51 32 49 34 47 36 360 -9 55 26 57 24 59 22 61 20 63 387 18 10 71 12 69 14 67 16 65 18 342 -27 73 8 75 6 77 4 79 2 81 405 36 333 396 351 378 369 360 387 342 405 369     -36 27 -18 9 0 -9 -18 -27 36



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4. Adjust the left diagonal, in blue, by adding and subtracting numbers from the selected cells to get a sum of 369 (Square 3).
5. Adjust the center row by adding and subtracting numbers from the selected blue cells.
6. Use the mask A consisting of the numbers 1, 3 and 5 to remove duplicates. Where the numbers intersect with the numbers of square 2 add the appropriate factor of n², 81 for "1", 243 for "3" and 405 for "5".

3 369 37 80 3 78 5 76 7 74 9 369 64 -10 66 15 68 13 70 11 72 369 19 62 39 60 23 58 25 56 27 369 46 35 48 24 50 31 52 29 54 369 37 44 39 42 41 40 43 38 45 369 28 53 30 51 32 58 34 47 36 369 55 26 57 24 59 22 43 20 63 369 10 71 12 69 14 67 16 92 18 369 73 8 75 6 77 4 79 2 45 369 369 369 369 369 369 369 369 369 369 369

+

Mask A 1 5 3 3 1 5 5 1 3 3 1 5 5 1 3 3 5 1

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7. Addition of the right factor gives square 3 with S = ½(n³ + 109n).

This completes this section on the new 9x9 Mask-Generated Methods (Part V). The next section contains New Zig Zag Cross and Mask Squares (Part I).
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