A New Procedure for Magic Squares (Part IA)

Centered Zig Zag Sequential Mask-Generated 5x5 Squares

A mask

A Discussion of the New Method

Skip the discussion go to examples

In this method the numbers on a 4n + 1 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 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 th 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 5x5 Magic Square

Method: Sequential Readout - use of a mask
  1. Construct the 5x5 Square 1 where 5 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 3 and filling the first top rows in a zigzag manner (square 1).
  2. Upon reaching the numeral 5 skip over center row and add 6 two cells down and continue adding numerals in normal zigzag fashion stopping at 10.
  3. From 10 go to the center row and fill in the row consecutively (square 2).
  4. Fill in the rest of the square, starting at 16 and zigzagging to 20 then going to cell 4 line 1 snd zigzagging to 25 (square 3).
  5. Since only the columns are equal to the magic sum while the row and diagonal sums are not, add or subtract the numbers in the last column to those of the center column. At this point four duplicates have been generated (Square 4).
  6. 1
    4 1 3
    5 2
     
    6 8
    107 9
    2
    4 1 3
    5 2
    111213 14 15
    6 8
    107 9
    3
    34
    4 24 1 21 35312
    23525 2 2277-12
    111213 14 15 650
    17619 8 16 66-1
    10187 20 9641
    656565 65 6539
    4
    34
    4 24 13 21 365
    23513 2 2265
    111213 14 15 65
    17618 8 16 65
    10188 20 965
    656565 65 6539
  7. 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.
4
34
4 24 13 21 365
23513 2 2265
111213 14 15 65
17618 8 16 65
10188 20 965
656565 65 6539
+
Mask A
26 31 31
57 2631
57 26 31
57 57
88 26
5
179
4 50 44 2160179
80539 33 22179
111270 40 46 179
74618 65 16 179
101068 20 35179
179179179 179 179179

This completes this section on a new Centered Zig Zag Sequential Mask-Generated Squares (Part IA). The next section deals with Centered Zig Zag Sequential Mask-Generated 9x9 Squares (Part IB). To return to homepage.


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