A New Procedure for Magic Squares (Part III)

7x7 and 11x11 Zig Zag Cross Squares

A stairs

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

Method I: Reading from left to right
  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.

  3. 1
    1 5 9 13
    37 11
    1519
    17
    2
    1 5 9 13
    37 11
    1519
    2325 27
    1721
    3
    1 5 9 13
    37 11
    151929 33
    2325 27
    172131 35
  4. Continue with 37 at the lower right and zig zag across to 49 (Square 4).
  5. Next start adding the even numbers zigzagging backwards until 14 is reached.Follow the green cells to 20 (Square 5).
  6. The next number 22 is placed in the center row and all the numbers in that row are filled in (Square 6).
  7. 4
    1 5 9 13
    37 11
    151929 33
    2325 27
    172131 35
    3943 47
    374145 49
    5
    1 5 9 13
    37 11
    15181929 33
    2325 27
    1716212031 35
    143910436 47 2
    371241845 4 49
    6
    1 5 9 13
    37 11
    15181929 33
    2223242526 27 28
    1716212031 35
    143910436 47 2
    371241845 4 49
  8. The next number is 30 placed above 25, followed by the numbers 32 and 34.
  9. 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.
  10. Adjust the center column and rows by adding or subtracting the differences from the last column and rows (Square 9).
  11. 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.
  12. This square is partially magic since 41 appears twice.

7
1 5 9 13
37 11
1518193029 34 33
2223242526 27 28
1716212031 32 35
143910436 47 2
371241845 4 49
8
175
1 46 5 42 938 13154 21
48344740 11 36189 -14
1518193029 34 33178-3
2223242526 27 28 1750
1716212031 32 35172 3
143910436 47 216114
371241845 4 49196-21
154157164 175 186193 196175
21 18 110-11 -18 -21
9
1 46 5 63 938 13
48344-740 11 36
151819 2729 34 33
434135 2515 9 7
171621 2331 32 35
143910576 47 2
371241-1345 4 49

Construction of a 11x11 Cross Magic Square

Method I: Reading from left to right
  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.

  3. 10
    1 5 9 13 1721
    371115 19
    23273135 3943
    25293337 41
    45495367 71 75
    57596163 65
    47515569 73 77
    81858993 97
    79838791 95 99
    103107111115 119
    101105109113 117 121
    11
    671
    1 118 5114 9 11013106 1710221 616 55
    120311671121110815 104 19100715 -44
    23962792318835 84 398043638 33
    9825942990338637 82 4178693 -22
    4548495253686772 71 7675 676 -5
    565758 596061 626364 6566 671 0
    4746515055546970 73 7477666 5
    4481408536893293 28 9724649 22
    7942833887349130 95 2699704 -33
    22103181071411110115 6 1192 627 44
    1012010516109121138 117 4121 726 -55
    636639646 649 656671 686693696 703706671
    35 32 2522150 -15 -22-25-32-35
  4. Adjust the center row and column with the differences from the last row and columns.
  5. 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.
12
1 118 5114 9 16513106 1710221 671
12031167112 -3310815 104 19100671
2396279231 12135 84 398043671
9825942990 118637 82 4178671
454849 5253 636772 71 7675 671
918983 8175 614741 39 3331 671
474651 5055 596970 73 7477 671
4481408536 1113293 28 9724671
7942833887 19130 95 2699671
221031810714 15510115 6 1192 671
1012010516109 -431138 117 4121 671
671671671 671 671671 671671671 671671671

This completes this section on the new 7x7 and 11x11 Cross Squares Method (Part III).
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Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com