A New Procedure for Magic Squares (Part IV)

7x7 and 11x11 Cross Squares

A Discussion of the New Method

This follows as a continuation of new 9x9 and 13x13 Cross Squares (Part III). This page, however, will treat only the cross squares of two 4n + 3 squares as was shown 5x5 cross squares. These squares have non-colored squares whose sum is equal to ½(n² + 1)  ½(n - 3)² .

Construction of 7x7 and 11x11 Cross Magic Squares

1. Produce square 1 with the sum and row columns (in gray).
2. As shown below this square is not magic because all the columns and rows don't sum to 175. The last column shows numerically how far this sum is from 175.
3. Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 175 and then...
4. Adjust the center row by adding and subtracting numbers from the selected cells so that the sums become 175 to generate 2.
5. The center three row and columns are colored yellow and light green in the cross.
6. The four white squares have sums of 25x4 = 100 and the center 3x3 light green square is magic.
7. The sum of three colored cells, vertically or horizontally, is equal to 75, e.g 15 + 31 + 29 or 3 + 67 + 5 = 75.

1. Produce square 3 with the sum and row columns (in gray).
2. As shown below this square is not magic because all the columns and rows don't sum to 671. The last column shows numerically how far this sum is from 671.

3 671 1 120 3 118 5 116 7 114 9 112 11 616 -55 110 13 108 15 106 17 104 19 102 21 100 715 44 23 98 25 96 27 94 29 92 31 90 33 638 -33 88 35 86 37 84 39 82 41 80 43 78 693 22 45 76 47 74 49 72 51 70 53 68 55 660 -11 66 57 64 59 62 61 60 63 58 65 56 671 0 67 54 69 52 71 50 73 48 75 46 77 682 11 44 79 42 81 40 83 38 85 36 87 34 649 -22 89 32 91 30 93 28 95 26 97 24 99 704 33 22 101 20 103 18 105 16 107 14 109 12 627 -44 111 10 113 8 115 6 117 4 119 2 121 726 55 666 675 668 673 670 671 672 669 674 667 676 671     -5 4 -3 2 -1 0 1 -2 3 -4 5

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3. Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 671 and then...
4. Adjust the center row by adding and subtracting numbers from the selected cells so that the sums become 671 to generate square 4.
5. The four white squares have sums of 61x16 = 976 and the center 3x3 light green square is magic with S = 183 or S = ½(n³ + 113n).
6. The sum of three colored cells, vertically or horizontally, is equal to 183 which equal 3 times the center cell.

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