A New Procedure for Magic Squares (Part IA)

Equation Generated Centered Sequential Mask-Generated Squares

A Discussion of the New Method

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). ********************************************************************************************************************************************************


n	r
5	-
9	1
13	2
17	3
21	4

Sequence Table
Number of cells to move per break									Direction 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

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).
Upon reaching the numeral 3 knight break (1 down, 1 right) and enter a 4 and continue filling cells in a normal fashion.
Skip over center row and add 6 two cells down from the 5 and continue adding numerals in normal fashion breaking at 7.
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).
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).

1
3		1		2
	4		5

	7		6
10		8		9

⇒

2
3		1		2
	4		5
11	12	13	14	15
	7		6
10		8		9

⇒

3
					37
3	22	1	21	2	49	16
24	4	25	5	23	81	-16
11	12	13	14	15	65	0
17	7	18	6	16	64	1
10	20	8	19	9	66	-1
65	65	65	65	65	35

⇒

4
					37
3	22	17	21	2	65
24	4	9	5	23	65
11	12	13	14	15	65
17	7	19	6	16	65
10	20	7	19	9	65
65	65	65	65	65	35

********************************************************************************************************************************************************

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.

We start by subtracting the diagonals(35,37) from 65 to give 30 and 28, respectively and which will be used as what I call the "de la Hire constants".

Addition of 30 and 28 to 65 gives 123 a magic pre-sum. However,

The following equations are used such that the following conditions are obeyed:

The right diagonal: 123 = 37 + 2(28) + 30

The left diagonal: 123 = 35 + 28 + 2(30)

The rows and columns: 123 = 65 + 28 + 30

Generate the mask using the 28 and 30 factors or sums thereof (58) adding these factors to the appropriate cells in square 4 to generate square 5.

Square 5 has a magic sum equal to 123, i.e., S = 123 = ½(n³ + 24n + 1).

********************************************************************************************************************************************************

4
					37
3	22	17	21	2	65
24	4	9	5	23	65
11	12	13	14	15	65
17	7	19	6	16	65
10	20	7	19	9	65
65	65	65	65	65	35

Mask A
30		28
			58
28	30
	28	30
				58

⇒

5
					123
33	22	45	21	2	123
24	4	9	63	23	123
39	42	13	14	15	123
17	35	49	6	16	123
10	20	7	19	67	123
123	123	123	123	123	123

********************************************************************************************************************************************************

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.