A New Procedure for Magic Squares (Part II)

9x9 Zig Zag Cross Squares and Mask-Generated Squares

A Discussion of the New Method

This follows as a continuation of new 5x5 mask-generated and cross squares and will treat only the 9x9 4n + 1 squares. As before not all the cross numbers are 41 only those in the center row. The center column has top and bottom numbers whose sum averages, however, to ½(45 + 37) = 41. See square 4 below.

The second square generated using the mask method uses a symmetrical mask with C₂ rotation along the center column (see Method II below). This mask besides having this property also has a sum of 6 for each row, column and diagonal. This is done so that all sums at the end are the same. In addition, it will also be shown that the sums of this square follows the new sum equation as was shown in the New block Loubère Method:

Construction of 9x9 Cross Magic Squares

1. Construct square 1 according to previous method (Squares 1, 2 and 3) in new 5x5 mask-generated and cross squares.
2. Square 2 shows the path in green of the even numbers 36 → 38 ... → 44 → 46.



1 1 5 9 13 17 3 7 11 15 19 23 27 31 35 21 25 29 33 37 39 41 43 45 49 53 57 61 47 51 55 59 63 67 71 75 79 65 69 73 77 81

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2 1 5 9 13 17 3 7 11 15 19 23 27 31 35 21 25 29 33 37 38 39 40 41 42 43 44 45 36 49 32 53 28 57 24 61 20 47 34 51 30 55 26 59 22 63 18 67 14 71 10 75 6 79 2 65 16 69 12 73 8 77 4 81



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3. Square 3 is constructed with the row and column sums (in grey) as well as differences (in white).
4. As shown below square 3 is not magic because all the columns and rows don't sum to 369. The last column shows numerically how far this sum is from 369.
5. Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 369 and then...
6. Adjust the center row by adding and subtracting numbers from the selected cells so that the sums become 369 to generate 3, containing yellow duplicate cells only in the cross.
7. Two white squares have sums of 81x8 = 648 and two 83x8 = 664.

1. Generate square 5 with odd numbers and half of the even numbers in reverse order.
2. Continue starting at 44 and jumping to the first cell in row 4. This is different from square 2 with 46 at the last cell of row 4.
3. Square 5 shows the path in green of the even numbers 36 → 38 ... → 44 → 46.
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5 1 5 9 13 17 3 7 11 15 19 23 27 31 35 21 25 29 33 37 38 39 40 41 42 43 44 45 36 49 32 53 28 57 24 61 20 47 34 51 30 55 26 59 22 63 18 67 14 71 10 75 6 79 2 65 16 69 12 73 8 77 4 81

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

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4. Where identical sums intersect a cell add or subtract the difference from the last row or column. This produces square 7 with the duplicates (20,41,45,62).
5. Using a symmetrical mask such as B remove all the duplicates to generate square 8 having a magic sum of 855 and with S = ½(n³ + 109n) and the modification of 37 numbers.

This completes this section on the new 9x9 Cross Squares Method and Mask (Part II).
The next section deals with new 7x7 and 11x11 Cross Squares Methods (Part III). To return to homepage.