A New Procedure for Magic Squares (Part ID)

Centered Sequential Mask-Generated Squares

A mask

A Discussion of the New Method

Skip the discussion go to examples

This method differs from the Centered Sequential Mask-Generated Squares (Part IA) which employs 4n + 1 squares. The 4n + 3 squares are produced via an alternative novel route using a sequence of numbers (either positive or negative) which tells us the direction and number of steps to move during a break.

  1. We begin by placing the numeral 1 in the center cell of the top row and consecutively entering the next numeral into every other cell as was shown in the previous page. After partially filling in the first row (1 down,1 right) or the equivalent (1 right,1 down) knight move is used to get to the next row (see the examples below).
  2. Upon reaching the row adjacent to the center row, the numbers are filled in a zig zag pattern as was shown in Zig Zag Consecutive 7x7 Mask-Generated Squares leaving the center row unfilled in the first pass. After filling in the zig zag rows continue as before filling bottom of the square as was previously done employing (1,down,1,right) knight moves as breaks. From the last row go to the center row and fill in the center row, followed by the bottom rows. (Best seen in the examples). From the bottom row we jump back to the zig zag rows and fill these in in reverse order, followed by inserting the next number into the last cell in the first row.
  3. The first move is to add the numbers in reverse order until the first row is filled in. This ensures that the square is primed to use the new break sequences which may be either either positive (move in the right direction) or negative numbers (move in the left direction). In addition the size of the numeral tells how many steps to travel in that direction including the (1 down) break. The rest of the square is filled in using these numerical sequences and are filled in the direction shown in the last column of the sequence table shown below.

After converting the squares into semi-magic ones the square are converted into magic ones by the use of a mask. This mask generates numbers which are added to certain cells in the square to produce a final square composed of numbers which may not be in serial order. For example, negative numbers or numbers greater than n2 may be present in the square.

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 the Break Sequence Table

Table S is generated by using the following general sequence:

{ ½(n + 1), [½(n - 3), - ½(n + 1)]r }

where n is an odd number of the type 4m + 3 and r is a repeating sequence equal to ¼(n - 7).

 
nr
7 
111
152
193
234
Sequence Table
Number of cells to move per breakDirection of row fillings
(4)R
(6, 4, -6)R, L, R
(8, 6, -8, 6, -8)R, L, R, L, R
(10, 8, -10, 8, -10, 8, -10)R, L, R, L, R, L, R
(12, 10, -12, 10, -12, 10, -12, 10, -12)R, L, R, L, R, L, R, L, R

Construction of a 11x11 Magic Square

Method: Sequential Readout - use of a mask
  1. Construct the 11x11 Square 7 where 11 = 4n + 3 by adding numbers in a consecutive manner starting at row 1 cell 6 and filling every other the cell (square 1).
  2. Upon reaching the numeral 5 generate a (1 down, 1 right) knight break move and continue filling cells in a normal fashion.
  3. Conrinue filling in rows adding numbers sequentially using the (1 down, 1 right) knight break moves until 23 is reached.
  4. Go down one cell fill in 23 (in light brown color) and add the next sequential numbers in a zig zag pattern as was shown above for the 7x7 square, continuing from 33 to 34 (in light green color).
  5. Partially fill all the last rows, followed by the center row (square 8) .
  6. 7
    451 2 3
    101167 8 9
    161213 14 15
    22 17 18 192021
    23 2527 29 3133
     
    24 26 28 30 32
    35 3637 38 39 34
    41 4243 44 40
    47 4849 50 45 46
    53 5455 51 52
    8
    451 2 3
    101167 8 9
    161213 14 15
    22 17 18 192021
    23 2527 29 3133
    56 57 58596061 62 6364 6566
    24 26 28 30 32
    35 3637 38 39 34
    41 4243 44 40
    47 4849 50 45 46
    53 5455 51 52
  7. From 66 fill go to 67 and fill in all the empty cells to the numeral 88 (square 9).
  8. From 88 go to 89 and fill zig zag in a reverse manner to the number 99 (square 10). From 99 go to 100 and add the numbers in a reverse manner to prime the square for the new reverse sequences (Square 10).
  9. 9
    451 2 3
    101167 8 9
    161213 14 15
    22 17 18 192021
    23 2527 29 3133
    56 57 58596061 62 6364 6566
    24 26 28 30 32
    35 68 36693770 38 7139 6734
    74 41 75427643 77 44724073
    47 80 48814982 50 7845 7946
    8653 87548855 83 5184 5285
    10
    105410451031 102 2101 3100
    101167 8 9
    161213 14 15
    22 17 18 192021
    23 98 259627 94 2992 319033
    56 57 58596061 62 6364 6566
    99 24 972695 28 9330 913289
    35 68 36693770 38 7139 6734
    74 41 75427643 77 44724073
    47 80 48814982 50 7845 7946
    8653 87548855 83 5184 5285
  10. At 105 go a distance of six units (5 right, 1 down) as directed in the second line of the sequence table and fill in the row in the right direction.
  11. At 110 go a distance of four units (1 down, 3 right) as directed in the second line of the sequence table and fill in the row in the left direction.
  12. At 116 go a distance of 6 units(1 down, 5 left) as directed in the second line of the sequence table and fill in the row in the right direction according to the last row of the sequence table. At this point all columns at this point sum to 1695, while the row sums are to be adjusted according to the last cell in square 10 in order to sum to 1695.
  13. Square 11 shows the result of the adjustment with the generation of 4 pink duplicates.
  14. 10
    938
    1054104 51031 102 2101 3100 63041
    101091111061067 107 8108 959180
    1141611312112 13 111 14116 15 115 751-80
    22 121 17117 18 118 191192012021 71241
    23 98 259627 94 2992 319033 63833
    56 57 58596061 62 6364 65666710
    99 24 972695 28 9330 913289704-33
    35 68 36693770 38 7139 6734 564 107
    74 41 75427643 77 44724073657 14
    47 80 48814982 50 7845 7946 685 -14
    8653 87548855 83 5184 5285778 -107
    369369369 369 369369 369369369 369369932
    11
    938
    1054104 510342 102 2101 3100 671
    101091111061867 107 8108 9671
    1141611312112 -67111 14116 15 115 671
    22 121 17117 18 77191192012021 671
    23 98 259627 127 2992 319033 671
    56 57 58596061 62 6364 6566671
    99 24 972695 -5 9330 913289671
    35 68 366937177 38 7139 6734 671
    74 41 75427657 77 44724073671
    47 80 48814968 50 7845 7946 671
    8653 875488-52 83 5184 5285671
    671671671 671 671671 671671671 671671932
    +
  15. 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 3 (as in the de la Hire method) that all sums will equal.
Mask B
261 267
261267
267261
267261
267261
267 261
261267
267261
261 267
261 267
261267
12
1199
1054104 510342 102 263368 3100 1199
27110927811061867 107 8108 91199
1141611312379 194111 14116 15 115 1199
289 121 17117 279 77191192012021 1199
23 98 259627 127 2992 31357294 1199
56 324 58596061 323 6364 65661199
99 24 972695 -5 9330 912933561199
35 68 366937177 38 338300 6734 1199
74 41 753037657 344 447240731199
47 341 488149335 50 7845 7946 1199
8653 34832188-52 83 5184 52851199
119911991199 1199 11991199 119911991199 119911991199

This completes this section on a new Centered Sequential Mask-Generated Squares (Part IB). The next section deals with Centered Sequential Mask-Generated Squares (Part IE). To return to homepage.


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