A New Procedure for Magic Squares (Part IA)

Centered Sequential 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 + 1 square are placed consecutively starting from the center cell in the top row and entered until every other cell is filled. Consecutive numbers are then added to the next rows using a (1,down,1,right) knight move as shown below in the examples. However, there are two exceptions to the rule:

(1) the center row is not filled in until until all the other rows are partially filled in and
(2) the row adjacent to the center row (on going down the square) has its entries filled in a different order, i.e., not using the (1,down,1,right) knight move.

Additions to the square going up are performed, first filling in the center row then filling in the empty cells. Note that since two types of odd squares exists, i.e., the 4n + 1 and the 4n + 3, two methods for filling in the squares are possible. The former example using n = 5 and 9 are 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 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 every other the cell (square 1).
  2. Upon reaching the numeral 3 (this row is adjacent to the center row so no knight break is done) enter 4 at cell 4 and continue filling cells in a normal fashion.
  3. Skip over center row and add 6 two cells down from the 5 and continue adding numerals in normal fashion breaking at 7.
  4. From 10 go to the center row and fill in the row consecutively (square 2). Fill in the rest of the square, at 20 adding the next numeral to the right of 1 (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
    3 1 2
    5 4
     
    6 7
    910 8
    2
    3 1 2
    5 4
    111213 14 15
    6 7
    910 8
    3
    34
    3 22 1 2124916
    25523 4 2481-16
    111213 14 15 650
    17618 7 16 641
    92010 19 866-1
    656565 65 6536
    4
    34
    3 22 17 21265
    2557 4 2465
    111213 14 15 65
    17619 7 16 65
    9209 19 865
    656565 65 6536
  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
3 22 17 21265
2557 4 2465
111213 14 15 65
17619 7 16 65
9209 19 865
656565 65 6536
+
Mask A
29 60
2929 31
3129 29
60 29
60 29
5
154
3 51 77 212154
54347 4 55154
114342 14 44 154
17619 67 45 154
69209 48 8154
154154154 154 154154

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


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