A New Procedure for Magic Squares (Part III)

9x9 and 13x13 Cross Squares

A mask

A Discussion of the New Method

This follows as a continuation of new 7x7 Mask generated Squares (Part II). This page, however, will treat only the cross squares of two 4n + 1 squares as was shown 5x5 cross squares. These squares have non-yellow squares whose sum is equal to ½(n2 + 1)  ½(n - 1)2 .

Construction of 9x9 and 13x13 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 369. The last column shows numerically how far this sum is from 369.
  3. Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 369 and then...
  4. 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.
  5. The four white squares have sums of 41x16 = 656.
1
369
1 80 3 78 576 774 9 333 36
7211701368 15 6617 64396-27
1962216023 58 2556 2735118
5429523150 33 4835 46378-9
3744394241 40 4338 453690
3647344932 51 3053 283609
5526572459 22 6120 63387-18
1865166714 69 1271 1034227
73875677 4 792 81405-36
365372367 370 369368 371366373 369   
-4 3 -210 -1 2-3 4
2
369
1 168 3 78 4176 7749 369
7211701341 15 6617 64369
1962216041 58 2556 27369
5429523141 33 4835 46369
414141 4141 41 4141 41369
3647344941 51 3053 28369
5526572441 22 6120 63369
1865166741 69 1271 10369
73875641 4 792 81369
369369369 369 369369 369369369 369
  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 1105. The last column shows numerically how far this sum is from 1105.
  3. Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 1105 and then...
  4. Adjust the center row by adding and subtracting numbers from the selected cells so that the sums become 1105 to generate 3, containing yellow duplicate cells only in the cross.
  5. The four white squares have sums of 85x36 = 3060 according to the sum equation from above.
3
1105
1 168 31665 164 7162 9160 11 158131027-78
156151541715219150 21 14823 14625144117065
27142291403113833 136 35134 37132391053-52
130411284312645124 47 12249 12051118114439
53116551145711259 110 61108 63106651079-26
10467102691007198 73 9675 947792111813
79908188838685 84 8782 89809111050
78937695749772 99 70101 68103661092-13
105641076210960111 58 11356 11554117113126
52119501214812346 125 44127 42129401066-39
131381333613534137 32 13930 14128143115752
26145241472214920 151 18153 16155141040-65
15712159101618163 6 1654 1672169118378
109911101101 1108 11031106 110511041107 110211091100 11111105   
-6 5 -43-210 -1 2-3 4-56
4
1105
1 168 31665 164 85162 9160 11 158131105
15615154171521985 21 14823 146251441105
27142291403113885 136 35134 37132391105
13041128431264585 47 12249 120511181105
53116551145711285 110 61108 63106651105
10467102691007185 73 9675 9477921105
858585 858585 8585 85 85 8585 851105
78937695749785 99 70101 68103661105
10564107621096085 58 11356 115541171105
52119501214812385 125 44127 42129401105
13138133361353485 32 13930 141281431105
26145241472214985 151 18153 16155141105
1571215910161885 6 1654 16721691105
110511051105 1105 11051105 110511051105 110511051105 11051105

This completes this section on the new 9x9 and 13x13 Cross Squares Methods (Part III).
The next section deals with new 7x7 and 11x11 Cross Squares Methods (Part III). To return to homepage.


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