A New Procedure for Magic Squares (Part IA)

Equation Generated 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. (This is a modification of the 5x5 square constructed previously ). 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 is an exception to the rule:

(1) the center row is not filled in until until all the other rows are partially filled in.

Additions to the square going up are performed by filling in the center row followed by the empty cells below the center row. The cells above the filled centered row are filled according to the general sequence formula shown below. The only exception being that when n = 5 only one move is possible (-3), i.e., 3 cells to the left as shown in the sequence table.

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:

{ (2), [½(n - 1), (2)r,(-3)] }

where n is an odd number of the type 4m + 1 and r is a repeating sequence equal to ½(n - 9).

 
nr
5-
91
132
173
214
Sequence Table
Number of cells to move per breakDirection of row fillings
(-3)R
(2, 4, -3)R, R, R
(2, 6, 2, 2, -3)R, R, R, R, R
(2, 8, 2, 2, 2, 2, -3)R, R, R, R, R, R, R
(2, 10, 2, 2, 2, 2, 2, 2, -3)R, R, R, R, R, R, R, R, R

Construction of a 5x5 Magic Square

Method: Sequential Readout - use of a mask
  1. Construct the 5x5 Square 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 knight break (1 down, 1 right) and enter a 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
    4 5
     
    7 6
    108 9
    2
    3 1 2
    4 5
    111213 14 15
    7 6
    108 9
    3
    37
    3 22 1 2124916
    24425 5 2381-16
    111213 14 15 650
    17718 6 16 641
    10208 19 966-1
    656565 65 6535
    4
    37
    3 22 17 21265
    2449 5 2365
    111213 14 15 65
    17719 6 16 65
    10207 19 965
    656565 65 6535
  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
37
3 22 17 21265
2449 5 2365
111213 14 15 65
17719 6 16 65
10207 19 965
656565 65 6535
+
Mask A
30 28
58
2830
2830
58
5
123
33 22 45 212123
2449 63 23123
394213 14 15 123
173549 6 16 123
10207 19 67123
123123123 123 123123

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


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