A New Procedure for Magic Squares (Part II)

9x9 Zig Zag Cross Squares and Mask-Generated Squares

A mask

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:

S = ½(n3 ± an)

Construction of 9x9 Cross Magic Squares

Method I: Reading from left to right
  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.

  3. 1
    1 5 9 13 17
    37 11 15
    192327 31 35
    2125 29 33
    373941 43 45
    4953 57 61
    475155 59 63
    6771 75 79
    656973 77 81
    2
    1 5 9 13 17
    37 11 15
    192327 31 35
    2125 29 33
    3738394041 42 4344 45
    3649325328 57 2461 20
    4734513055 26 5922 63
    1867147110 75 679 2
    6516691273 8 774 81
  4. Square 3 is constructed with the row and column sums (in grey) as well as differences (in white).
  5. 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.
  6. Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 369 and then...
  7. 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.
  8. Two white squares have sums of 81x8 = 648 and two 83x8 = 664.
3
369
1 78 5 74 970 1366 17 33336
80376772 11 6815 64396-27
1960235627 52 3148 3535118
6221582554 29 5033 46378-9
3738394041 42 4344 453690
3649325328 57 2461 203609
4734513055 26 5922 63387-18
1867147110 75 679 234227
6516691273 8 774 81405-36
365366367 368 369370 371372373 369   
-4 -3 -2-10 1 23 4
4
1 78 5 74 4570 1366 17 369
80376745 11 6815 64369
1960235645 52 3148 35369
6221582545 29 5033 46369
414141 414141 4141 41369
3649325337 57 2461 20369
4734513037 26 5922 63369
1867147137 75 679 2369
6516691237 8 774 81369
369369369 369 369369 369369369 369
Method II: Reading from left to right - Use of a Mask
  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.
  4. 5
    1 5 9 13 17
    37 11 15
    192327 31 35
    2125 29 33
    3738394041 42 4344 45
    3649325328 57 2461 20
    4734513055 26 5922 63
    1867147110 75 679 2
    6516691273 8 774 81
    6
    369
    1 66 5 70 974 1378 17 33336
    64368772 11 7615 80396-27
    1948235227 56 3160 3535118
    4621502554 29 5833 62378-9
    3738394041 42 4344 453690
    3649325328 57 2461 203609
    4734513055 26 5922 63387-18
    1867147110 75 679 234227
    6516691273 8 774 81405-36
    333342351 360 369378 387396405 369   
    36 27 1890 -9 -18-27 -36
  5. 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).
  6. 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.
<
7
369
37 66 5 70 974 1378 17 369
64368772 11 76-12 80369
1948415227 56 3160 35369
4621502554 20 5833 62369
3738394041 42 4344 45369
3649326228 57 2461 20369
4734513055 26 4122 63369
1894147110 75 679 2369
6516691273 8 774 45369
369369369 369 369369 369369369 369
+
Mask B
1 1 2 11
12 2 1
22 2
21 1 2
111 11 1
12 2 1
12 2 1
22 2
21 1 2
8
855
37 147 86 70 17174 94 159 17 855
64846816972 173 7669 80855
194820352189 56 19360 35855
208102502554 20 58114 224855
1181191204041 42 124125 126855
117211326228 57 24223 101855
473413219255 188 12222 63855
189414233172 237 679 2855
227161501273 8 1584 207855
855855855 855 855855 855855855 855

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.


Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com