A New Procedure for Magic Squares (Part IB)

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. 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 9x9 Magic Square

Method: Sequential Readout - use of a mask

Construct the 9x9 Square 1 where 9 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 3 and filling every other the cell (square 6).
Upon reaching the numeral 5 perform a knight break enter 6 at cell 4 and continue filling cells in a normal fashion.
After performing exception (2) skip over center row and add 19 to the second cell in the 6th row and continue adding numerals in normal fashion in the last row of the square.
From 36 go to the center row and fill in the row consecutively (square 7).

6
4		5		1		2		3
	9		6		7		8
14		10		11		12		13
	18		17		16		15

	19		20		21		22
24		25		26		27		23
	29		30		31		28
34		35		36		32		33

⇒

7
4		5		1		2		3
	9		6		7		8
14		10		11		12		13
	18		17		16		15
37	38	39	40	41	42	43	44	45
	19		20		21		22
24		25		26		27		23
	29		30		31		28
34		35		36		32		33

⇒

Fill in the rest of the square, going backwards this time (square 8), starting at 45 → 46. At 63 go to 64 adjacent to the numeral 2 on the first row.
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 in pink (Square 9).

8
									188
4	66	5	67	1	64	2	65	3	277	92
71	9	72	6	68	7	69	8	70	380	-11
14	76	10	73	11	74	12	75	13	358	11
81	18	77	17	78	16	79	15	80	461	-92
37	38	39	40	41	42	43	44	45	369	0
47	19	48	20	49	21	50	22	46	322	47
24	52	25	53	26	54	27	51	23	335	34
57	29	58	30	59	31	55	28	56	403	-34
34	62	35	63	36	60	32	61	33	416	-47
369	369	369	369	369	369	369	369	369	190

⇒

9
									188
4	66	5	67	93	64	2	65	3	369
71	9	72	6	57	7	69	8	70	369
14	76	10	73	22	74	12	75	13	369
81	18	77	17	-14	16	79	15	80	369
37	38	39	40	41	42	43	44	45	369
47	19	48	20	96	21	50	22	46	369
24	52	25	53	60	54	27	51	23	369
57	29	58	30	25	31	55	28	56	369
34	62	35	63	-11	60	32	61	33	369
369	369	369	369	369	369	369	369	369	190

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 3 (as in the de la Hire method) that all sums will equal.

We start by subtracting the diagonals(188,190) from 369 to give 181 and 179, respectively and which will be used as what I call the "de la Hire constants".
Addition of the sum of these two numbers, 181 + 179 = 360 to 369 gives 729 a magic pre-sum. This sum is just right for our purpose.
The following equations are used such that the following conditions are obeyed:
The right diagonal: 729 = 188 + 2(181) + 179
The left diagonal: 729 = 190 + 181 + 2(179)
The rows and columns: 729 = 369 + 181 + 179
Generate the mask using the 181 and 179 factors and adding these factors to the appropriate cells in square 9 to generate square 10.
Square 10 has a magic sum equal to 729, i.e., S = 729 = ½(n³ + 81n).

Mask B
			181					179
	179			181
181		179
				179	181
						360
							179	181
		181	179
179							181
	181				179

⇒

10
									729
4	66	5	248	93	64	2	65	182	729
71	188	72	6	238	7	69	8	70	729
195	76	189	73	22	74	12	75	13	729
81	18	77	17	165	197	79	15	80	729
37	38	39	40	41	42	403	44	45	729
47	19	48	20	96	21	50	201	227	729
24	52	206	232	60	54	27	51	23	729
236	29	58	30	25	31	55	209	56	729
34	243	35	63	-11	239	32	61	33	729
729	729	729	729	729	729	729	729	729	729

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