A New Procedure for Magic Squares (Part IC)

Centered Sequential Mask-Generated Squares

A Discussion of the New Method

This method differs from the Centered Sequential Mask-Generated Squares (Part IA) which employs 4n + 1 squares. The 4n + 3 squares are produced via an alternative novel route using a sequence of numbers (either positive or negative) which tells us the direction and number of steps to move during a break.

We begin by placing the numeral 1 in the center cell of the top row and consecutively entering the next numeral into every other cell as was shown in the previous page. After partially filling in the first row (1 down,1 right) or the equivalent (1 right,1 down) knight move is used to get to the next row (see the examples below).
Upon reaching the row adjacent to the center row, the numbers are filled in a zig zag pattern as was shown in Zig Zag Consecutive 7x7 Mask-Generated Squares leaving the center row unfilled in the first pass. After filling in the zig zag rows continue as before filling bottom of the square as was previously done employing (1,down,1,right) knight moves as breaks. From the last row go to the center row and fill in the center row, followed by the bottom rows. (Best seen in the examples). From the bottom row we jump back to the zig zag rows and fill these in in reverse order, followed by inserting the next number into the last cell in the first row.
The first move is to add the numbers in reverse order until the first row is filled in. This ensures that the square is primed to use the new break sequences which may be either either positive (move in the right direction) or negative numbers (move in the left direction). In addition the size of the numeral tells how many steps to travel in that direction including the (1 down) break. The rest of the square is filled in using these numerical sequences and are filled in the direction shown in the last column of the sequence table 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 the Break Sequence Table

Table S is generated by using the following general sequence:

{ ½(n + 1), [½(n - 3), - ½(n + 1)]_r }

where n is an odd number of the type 4m + 3 and r is a repeating sequence equal to ¼(n - 7).


n	r
7
11	1
15	2
19	3
23	4

Sequence Table
Number of cells to move per break									Direction of row fillings
(4)									R
(6, 4, -6)									R, L, R
(8, 6, -8, 6, -8)									R, L, R, L, R
(10, 8, -10, 8, -10, 8, -10)									R, L, R, L, R, L, R
(12, 10, -12, 10, -12, 10, -12, 10, -12)									R, L, R, L, R, L, R, L, R

Construction of a 7x7 Magic Square

Method: Sequential Readout - use of a mask

Construct the 7x7 Square 1 where 7 = 4n + 3 by adding numbers in a consecutive manner starting at row 1 cell 4 and filling every other the cell (square 1).
Upon reaching the numeral 3 generate a (1 right, 1 down) knight break move and continue filling cells in a normal fashion.
Upon reaching 7 go down one cell to 8 and add numbers sequentially in a zig zag pattern skipping over center row.
From 14 go down three cells to 15 and continue adding numbers form right to left. From 21 go to the center row and fill in the row consecutively (square 2). Drop down to 29 fill in the last two rows using a (1 right, 1 down) knight break move.

1
	3		1		2
7		4		5		6
8		10		12		14

	9		11		13
16		17		18		15
	20		21		19

⇒

2
	3		1		2
7		4		5		6
8		10		12		14
22	23	24	25	26	27	28
	9		11		13
16	30	17	31	18	29	15
34	20	35	21	32	19	33

⇒

From 35 go to 36 and repeat the zig zag in a reverse manner.
From 42 go to the last cell in the first row and insert 43. Continue filling in the first row in reverse order (Square 4). This primes the square for the new break sequences.
At 46 go a distance of four units ( 3 right, 1 down) as directed in the first line of the sequence table.
Continue filling in the second row moving to the right as shown in the last column of the sequence table. At this point all columns at this point sum to 175, while the row sums are to be adjusted (by + or - values) according to the last cell in square 4 in order to sum to 175.
Square 5 shows the result of the adjustment with the generation of 1 pink duplicate.

3
	3		1		2
7		4		5		6
8	41	10	39	12	37	14
22	23	24	25	26	27	28
42	9	40	11	38	13	36
16	30	17	31	18	29	15
34	20	35	21	32	19	33

⇒

4
							232
46	3	45	1	44	2	43	184	-9
7	49	4	47	5	48	6	166	9
8	41	10	39	12	37	14	161	14
22	23	24	25	26	27	28	175	0
42	9	40	11	38	13	36	189	-24
16	30	17	31	18	29	15	156	19
34	20	35	21	32	19	33	194	-19
175	175	175	175	175	175	175	230

⇒

5
							232
46	3	45	-8	44	2	43	175
7	49	4	56	5	48	6	175
8	41	10	53	12	37	14	175
22	23	24	25	26	27	28	175
42	9	40	-3	38	13	36	175
16	30	17	50	18	29	15	175
34	20	35	2	32	19	33	175
175	175	175	175	175	175	175	230

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 175 from the diagonals(230,232) to give 55 and 57, respectively, and which will be used as what I call the "de la Hire constants".
Addition of 55 and 37 to 175 gives 287 a magic pre-sum. .
The following equations are used such that the following conditions are obeyed:
The right diagonal: 287 = 232 + 55
The left diagonal: 287 = 280 + 57
The rows and columns: 287 = 175 + 55 + 57.
Generate the mask using the 55 and 57 factors or sums thereof (112) adding these factors to the appropriate cells in square 5 to generate square 6.
Square 6 has a magic sum equal to 287, i.e., S = 287 = ½(n³ + 33n).

Mask A
	57				55
		112
			55			57
55				57
57						55
	55				57
			57	55

⇒

6
							154
46	60	45	-8	44	57	43	154
7	49	116	56	5	48	6	154
8	41	10	108	12	37	71	154
77	23	24	25	83	27	28	154
99	9	40	-3	38	13	91	154
16	85	17	50	18	86	15	154
34	20	35	59	87	19	33	154
154	154	154	154	154	154	154	154

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