A New Procedure for Magic Squares (Part ID)

Centered Sequential Zig Zag 7x7 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 7x7 Magic Square

Method: Sequential Zig ZagReadout - use of a mask
  1. Construct the 7x7 Square 1 by adding numbers in a consecutive zigzag manner starting at row 1 cell 4 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 7 go down one cell to 8 and add numbers same manner eventually skipping over the center row.
  3. At 14 perform move down two cells (a down break of 2) and continue the addition to the cells in a zigzag fashion.
  4. From 21 go to the center row and fill in the row consecutively (square 2).
  5. 1
    6 1 3
    57 2 4
    138 10 12
     
    149 11
    2116 18 20
    1517 19
    2
    6 1 3
    57 2 4
    138 10 12
    22232425 26 2728
    149 11
    2116 18 20
    1517 19
  6. From 28 go to 29 and repeat the zig zag in a reverse manner.
  7. From 35 break 2 up to 36 and add numbers zigzag across the center row but in a right hand fashion.
  8. At 42 go a distance of four units up to 43 and continue adding in zigzag fashion to fill in square 4.
  9. At this point all columns at this point sum to 175, while the row sums are to be adjusted (by + or - values) according to the last cell in square 4 in order to sum to 175.
  10. Square 5 shows the result of the adjustment with the generation of 2 pink duplicates.
  11. 3
    6 1 3
    57 2 4
    1339841 10 3612
    22232425 26 2728
    3814409 42 1137
    213116 33 18 3520
    30153217 34 1929
    4
    225
    46 6 48 143 345192-17
    547749 2 44415817
    1339841 10 3612 15916
    22232425 26 27281750
    3814409 42 1137191 -16
    213116 33 18 35201741
    30153217 34 1929176 -1
    175175175 175175175 175232
    5
    225
    46 6 48 -1643 345175
    547766 2 444175
    1339857 10 3612 175
    22232425 26 2728175
    381440-7 42 1137175
    213116 34 18 3520175
    30153216 34 1929175
    175175175 175175175 175230
  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.
Mask A
107
50 57
50 57
57 50
50 57
57 50
57 50
+ Square 5
6
282
46 6 48 9143345282
55477 66 21014282
133958 57 673612 282
792324 25 267728 282
386440 -7 421194282
213173 34 183570 282
307232 16 841929 282
282282282 282 282282 282282

This completes this section on a new Centered Sequential Mask-Generated Squares (Part IC). The next section deals with Centered 11x11 Sequential Mask-Generated Zig Zag Squares (Part IE). To return to homepage.


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