A New Procedure for Magic Squares (Part IV)

7x7 and 11x11 Cross Squares

A mask

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

Method I: Reading from left to right
  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
175
1 48 3 46 544 7154 -21
42940 11 3813 3618914
153417 32 1930 21168-7
282326 25 2427 22175 0
29 20 31 18 3316 351827
14 37 12 39 1041 8161-14
43 6 45 4 472 49196 21
172177174 175 176173 178175   
-3 2 -1 0 1-2 3
2
175
1 48 3 67 544 7 175
42940 -3 3813 36175
153417 39 1930 21175
312127 25 2329 19175
29 20 31 11 3316 35175
14 37 12 53 1041 8175
43 6 45 -17 472 49175
175175175 175 175175 175175
  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. 3
    671
    1 120 31185 116 7114 9112 11 616-55
    110131081510617104 19 10221 10071544
    23982596279429 92 3190 33638-33
    88358637843982 41 8043 7869322
    45764774497251 70 5368 55660-11
    66576459626160 63 5865 566710
    67546952715073 48 7546 7768211
    44794281408338 85 3687 34649-22
    89329130932895 26 9724 9970433
    22101201031810516 107 14109 12627-44
    1111011381156117 4 1192 12172655
    666675668 673 670671 672669674 667676 671   
    -5 4 -32-101 -2 3-45
  4. Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 671 and then...
  5. Adjust the center row by adding and subtracting numbers from the selected cells so that the sums become 671 to generate square 4.
  6. 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).
  7. The sum of three colored cells, vertically or horizontally, is equal to 183 which equal 3 times the center cell.
4
671
1 120 31185 171 7114 9112 11671
1101310815106-27 104 19 10221 100671
2398259627 12729 92 3190 33671
8835863784 1782 41 8043 78671
457647 744983 5170 53 68 55671
715367 576361 5965 55 69 51671
675469 527139 7348 75 46 77671
4479428140 10538 85 3687 34671
8932913093 -595 26 9724 99671
221012010318 14916 107 14109 12671
111101138115 -49117 4 1192 121671
671671671 671 671671 671671671 671671 671

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.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com