A New Procedure for Magic Squares (Part III)

7x7 and 11x11 Zig Zag Cross Squares

A Quickie Discussion of the New Method

This follows as a continuation of new 9x9 mask-generated and cross squares and will treat only the 7x7 and 11x11 (4n + 3) squares. As previously with these type of squares the sum of the middle three numbers (the center rows) is equal to 3(½)(n² + 1). For the three middle columns the sum is 3(½)(n² + 1) ± ½(n - 3) depending upon whether the three cells are above or below the center row.

Construction of a 7x7 Cross Magic Square

1. Construct square 1 according to previous method by zig zagging across every other column. At the number 15 move two rows down then zig zag across to 19. thus making this method different from the 4n + 1. Put the number 21 into the next cell two rows below.
2. At this point a discontinuity occurs after 21 since row 4 will contain the consecutive numbers from 22 to 28. Therefore, place 23 at cell 2 of row 4 followed by 25 and 27. This is then followed by 29 to 35 in zig zag order.



1 1 5 9 13 3 7 11 15 19 17

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2 1 5 9 13 3 7 11 15 19 23 25 27 17 21

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3 1 5 9 13 3 7 11 15 19 29 33 23 25 27 17 21 31 35

3. Continue with 37 at the lower right and zig zag across to 49 (Square 4).
4. Next start adding the even numbers zigzagging backwards until 14 is reached.Follow the green cells to 20 (Square 5).
5. The next number 22 is placed in the center row and all the numbers in that row are filled in (Square 6).

4 1 5 9 13 3 7 11 15 19 29 33 23 25 27 17 21 31 35 39 43 47 37 41 45 49

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5 1 5 9 13 3 7 11 15 18 19 29 33 23 25 27 17 16 21 20 31 35 14 39 10 43 6 47 2 37 12 41 8 45 4 49

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6 1 5 9 13 3 7 11 15 18 19 29 33 22 23 24 25 26 27 28 17 16 21 20 31 35 14 39 10 43 6 47 2 37 12 41 8 45 4 49

6. The next number is 30 placed above 25, followed by the numbers 32 and 34.
7. Jump to the next row and fill in the rest of the even numbers in zig zag order (Square 8). At this point all sums do not equal to 175 as shown by the next to last column and row.
8. Adjust the center column and rows by adding or subtracting the differences from the last column and rows (Square 9).
9. Square 9 contains a cross with an internal 3x3 square having a sum of 75. The sum of the 3x3 center column and rows follows the equations shown above giving S = 75 for the center three columns and S = 77 and 72, respectively, for the center three top and bottom rows.
10. This square is partially magic since 41 appears twice.

Construction of a 11x11 Cross Magic Square

1. Generate square 10 with all odd numbers as was shown above. There is a discontinuity at 55 going to 67 since the numbers 57 thru 66 go into the center row.
2. Fill in the even numbers in like manner but in reverse (Square 11). Again there is a discontinuity in 54 to 68. The center row and columns which are to be adjusted noted in light green. The pink cells show the route going into and coming out of the center rows.



10 1 5 9 13 17 21 3 7 11 15 19 23 27 31 35 39 43 25 29 33 37 41 45 49 53 67 71 75 57 59 61 63 65 47 51 55 69 73 77 81 85 89 93 97 79 83 87 91 95 99 103 107 111 115 119 101 105 109 113 117 121

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11 671 1 118 5 114 9 110 13 106 17 102 21 616 55 120 3 116 7 112 11 108 15 104 19 100 715 -44 23 96 27 92 31 88 35 84 39 80 43 638 33 98 25 94 29 90 33 86 37 82 41 78 693 -22 45 48 49 52 53 68 67 72 71 76 75 676 -5 56 57 58 59 60 61 62 63 64 65 66 671 0 47 46 51 50 55 54 69 70 73 74 77 666 5 44 81 40 85 36 89 32 93 28 97 24 649 22 79 42 83 38 87 34 91 30 95 26 99 704 -33 22 103 18 107 14 111 10 115 6 119 2 627 44 101 20 105 16 109 12 113 8 117 4 121 726 -55 636 639 646 649 656 671 686 693 696 703 706 671 35 32 25 22 15 0 -15 -22 -25 -32 -35

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3. Adjust the center row and column with the differences from the last row and columns.
4. Again the center 3x3 square is magic with sum of 183. The sum of the center three columns and rows are, respectively, 183 for the rows and 187 (for the top) and 179 (for the bottom) columns. Duplicate numbers are also present.

This completes this section on the new 7x7 and 11x11 Cross Squares Method (Part III).
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