A New Procedure for Magic Squares (Part V)

9x9 Mask-Generated Squares

A mask

A Discussion of the New Method

This page discusses converting a regular read square into its boustrophedonical twin and then using a symmetrical C2 mask (symmetrical by 180° rotation) of n2 numbers to convert it into a magic square as was shown in for generating magic squares using numerical masks.
The magic squares produced have only partially sequential order and not all the numbers from 1 to n2 are present. Because of the manipulation numbers greater than n2or less than 1 are generated.

In addition, it will also be shown that the sums of these squares follow the new sum equation as was shown in the New block Loubère Method:

S = ½(n3 ± an)

Construction of a 9x9 Magic Square

Method: Reading Boustrophedonically
  1. Generate square A with the sum and row columns (in gray).The last column and row give the amounts that the sums differ from 369.
  2. As shown below this square is not magic because all the columns and rows don't sum to 369.
  3. Reverse rows 2, 4, 6 and 8 so that the read-out is boustrophedonically, i.e., right then left (Square 2).
  4. 1
    369
    1 80 378 576 7749333-36
    7211701368 15 6617 6439627
    1962216023 58 2556 27351-18
    5429523150 3348 3546378 9
    37 44 394241 40 4338 453690
    36 47 344932 51 3053 28360-9
    55 26 572459 22 6120 63387 18
    18 65 166714 6912 7110342 -27
    73 8 75677 4 792 81405 36
    365372367 370 369368 371366373 369   
    -4 3 -210 -1 2-3 4-5
    2
    369
    1 80 378 576 7749333-36
    6417661568 13 7011 7239627
    1962216023 58 2556 27351-18
    4635483350 3152 2954378 9
    37 44 394241 40 4338 453690
    28 53 305132 49 3447 36360-9
    55 26 572459 22 6120 63387 18
    10 71 126914 6716 6518342 -27
    73 8 75677 4 792 81405 36
    333396351 378 369360 387342405 369   
    -36 27 -1890 -9 -18-27 36
  5. Adjust the left diagonal, in blue, by adding and subtracting numbers from the selected cells to get a sum of 369 (Square 3).
  6. Adjust the center row by adding and subtracting numbers from the selected blue cells.
  7. Use the mask A consisting of the numbers 1, 3 and 5 to remove duplicates. Where the numbers intersect with the numbers of square 2 add the appropriate factor of n2, 81 for "1", 243 for "3" and 405 for "5".
  8. 3
    369
    37 80 378 576 7749369
    64-10661568 13 7011 72369
    1962396023 58 2556 27369
    4635482450 3152 2954369
    37 44 394241 40 4338 45369
    28 53 305132 58 3447 36369
    55 26 572459 22 4320 63369
    10 71 126914 6716 9218369
    73 8 75677 4 792 45369
    369369369 369 369369 369369369 369
    +
    Mask A
    1 5
    33
    1 5
    5 1
    3 3
    1 5
    5 1
    3 3
    5 1
  9. Addition of the right factor gives square 3 with S = ½(n3 + 109n).
3
855
118 80 378 576 412749855
64-106615311 13 70254 72855
19621206023 463 2556 27855
451354810550 3152 2954855
37 287 394241 40 43281 45855
28 53 305132 139 3447 441855
55 26 5742959 22 12420 63855
10 314 1269257 6716 9218855
73 8 480677 4 792 126855
855855855 855 855855 855855855 855

This completes this section on the new 9x9 Mask-Generated Methods (Part V). The next section contains New Zig Zag Cross and Mask Squares (Part I). To return to homepage.


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