A New Procedure for Magic Squares (Part IA)

Centered Zig Zag Sequential Mask-Generated 5x5 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 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

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).
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.
From 10 go to the center row and fill in the row consecutively (square 2).
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).
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
4		1		3
	5		2

	6		8
10		7		9

⇒

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

⇒

3
					34
4	24	1	21	3	53	12
23	5	25	2	22	77	-12
11	12	13	14	15	65	0
17	6	19	8	16	66	-1
10	18	7	20	9	64	1
65	65	65	65	65	39

⇒

4
					34
4	24	13	21	3	65
23	5	13	2	22	65
11	12	13	14	15	65
17	6	18	8	16	65
10	18	8	20	9	65
65	65	65	65	65	39

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(34,39) from 65 to give 31 and 26, respectively and which will be used as what I call the "de la Hire constants".
Addition of 31 and 26 to 65 gives 122 a magic pre-sum. However, attempts were unsuccessful to obtaining a working mask so instead 179 = 65 + 2(26) + 2(31) turned out to be a better choice.
The following equations are used such that the following conditions are obeyed:
The right diagonal: 179 = 34 + 2(26) +3(31)
The left diagonal: 179 = 36 + 3(29) + 2(31)
The rows and columns: 179 = 65 + 2(26) + 2(31).
Generate the mask using the 26 and 31 factors or sums thereof (57 and 88) and adding these factors to the appropriate cells in square 4 to generate square 5.
Square 5 has a magic sum equal to 179, i.e., S = 179 = ½(n³ + 46n + 3).

4
					34
4	24	13	21	3	65
23	5	13	2	22	65
11	12	13	14	15	65
17	6	18	8	16	65
10	18	8	20	9	65
65	65	65	65	65	39

Mask A
	26	31		31
57		26	31
		57	26	31
57			57
	88			26

⇒

5
					179
4	50	44	21	60	179
80	5	39	33	22	179
11	12	70	40	46	179
74	6	18	65	16	179
10	106	8	20	35	179
179	179	179	179	179	179

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.