A New Procedure for Magic Squares (Part IE)

Centered Sequential Zig Zag 11x11 Mask-Generated Squares

A mask

A Discussion of the New Method

Skip the discussion go to examples

In this method the numbers on a 4n + 3 square are placed consecutively starting from the center cell in the top row and entered in a zig zag manner. Additions to the square are done by leaving the center row unfilled and continuing the zig zag pattern below the center row. (The addition across the center row is different from previous additions which began on the first cell. The addition of numbers in this case may come elsewhere)). The center row is then filled in consecutively followed by the filling in of the bottom and top rows again in a zig zag manner. Breaking involves moving down 2 cells to the next row and continuing addition of numbers to the right.
After the square is filled, the square is converted into a semi-magic one by addition or subtraction i,e. by the differences of numbers in the last row. The square is then converted by means of a numerical mask into a magic square. Moreover, this new square may have negative numbers or numbers greater than n2.

In addition, it will also be shown that the sums of these squares follows either of the two sum equations shown in the New block Loubère Method and Consecutive 5x5 Mask Generated squares:

S = ½(n3 ± an)
S = ½(n3 ± an + b)

Construction of a 11x11 Magic Square

Method: Sequential Zig Zag Readout - use of a mask
  1. Construct the 11x11 Square 1 by adding numbers in a consecutive zigzag manner starting at row 1 cell 5 and filling every other cell (square 1). Note that on reaching the final cell in the row the addition is continued at the other end of the row.
  2. Upon reaching 11 move down two cells to 12 and continue in the same manner eventually skipping over the center row.
  3. From 55 go to the center row and fill in the row consecutively (square 2).

  4. 1
    8 10 1 35
    7911 2 46
    20 22 13 1517
    192112 14 1618
    32 23 25 2729
      
    31 3324 26 2830
    44 35 37 3941
    43 3436 38 4042
    45 47 49 5153
    55 4648 50 5254
    2
    8 10 1 35
    7911 2 46
    20 22 13 1517
    192112 14 1618
    32 23 25 2729
    56 57 585960 6162 63646566
    31 3324 26 2830
    44 35 37 3941
    43 3436 38 4042
    45 47 49 5153
    55 4648 50 5254
  5. From 66 go down two cells to 67 and repeat the zig zag pattern (Square 3).
  6. From 88 move up to 89 and add numbers zigzag across the center row to the right.
  7. At 99 go a distance of five units up to 100 and continue adding in zigzag fashion to fill in square 4.
  8. At this point all columns at this point sum to 671, while the row sums are to be adjusted (by + or - values) according to the last cell in square 4 in order to sum to 671.
  9. 3
    8 10 1 35
    7911 2 46
    20 22 13 1517
    192112 14 1618
    92 32 942396 2598 27892991
    56 57 585960 6162 63646566
    3193 339524 9726 99289030
    68 44 703572 3774 39764167
    43 69 347136 7338 75407742
    80 45 824784 4986 51885379
    55 81 468348 8550 87527854
    4
    782
    104 8 10610108 1110 3101510365923
    7105910711109 2 10041026562109
    116 20 11822120 13111 1511317115880-109
    191172111912121 14 1121611418683-12
    92 32 9423962598 27 892991 696 -25
    56 57 585960 6162 636465666710
    3193 339524 9726 9928903064625
    68 44 703572 3774 3976416762348
    43 69 347136 7338 7540774259873
    80 45 8247844986 51885379744-73
    55 81 468348 8550 87527854719-48
    671671671 671671671 671671671 671671815
  10. Square 5 shows the result of the adjustment with the generation of 0 duplicates.

  11. 5
    782
    104 8 10610108 13110 31015103671
    7105910711218 2 10041026671
    116 20 11822120 -96111 1511317115671
    191172111912109 14 1121611418671
    92 32 942396098 27 892991 671
    56 57 58596061 62 63646566671
    3193 339524 12226 99289030671
    68 44 703572 8574 39764167671
    43 69 347136 14638 75407742671
    80 45 824784-2486 51885379671
    55 81 468348 3750 87527854671
    671671671 671671671 671671671 671671815
  12. Generate a mask whereby the sums of the columns and rows are constructed as in the box below. This assures that when each of these values is added to the corresponding cell in square 4 (as in the de la Hire method) that all sums will be equal.
Square 5 +
Mask A
111144
111 144
111 144
111 144
144 111
144 111
144111
111 144
144 111
144111
144 111
6
926
104 8 21710108 13254 31015103926
7216910711218 2 1001481026926
116 20 118133120 -96111 1511317259926
191172111912220 14 2561611418926
92 32 94239614498 27 8914091 926
56 201 585917161 62 63646566926
3193 33952412226 9928234141926
179 44 2143572 8574 39764167926
187 69 347136 14638 186407742926
80 45 8219184-24197 51885379926
55 81 4683192 3750 871637854926
926926926 926926926 926926926 926926926

This completes this section on a new Centered Sequential Mask-Generated Squares (Part IC). To return to homepage.


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