A New Procedure for Magic Squares (Part IE)

Centered Sequential Zig Zag 11x11 Mask-Generated Squares

A Discussion of the New Method

In this method the numbers on a 4n + 3 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 addition across the center row is different from previous additions which began on the first cell. The addition of numbers in this case may come elsewhere)). 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 the 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 11x11 Magic Square

Method: Sequential Zig Zag Readout - use of a mask

Construct the 11x11 Square 1 by adding numbers in a consecutive zigzag manner starting at row 1 cell 5 and filling every other cell (square 1). Note that on reaching the final cell in the row the addition is continued at the other end of the row.
Upon reaching 11 move down two cells to 12 and continue in the same manner eventually skipping over the center row.
From 55 go to the center row and fill in the row consecutively (square 2).

1
	8		10		1		3		5
7		9		11		2		4		6
	20		22		13		15		17
19		21		12		14		16		18
	32		23		25		27		29

31		33		24		26		28		30
	44		35		37		39		41
43		34		36		38		40		42
	45		47		49		51		53
55		46		48		50		52		54

⇒

2
	8		10		1		3		5
7		9		11		2		4		6
	20		22		13		15		17
19		21		12		14		16		18
	32		23		25		27		29
56	57	58	59	60	61	62	63	64	65	66
31		33		24		26		28		30
	44		35		37		39		41
43		34		36		38		40		42
	45		47		49		51		53
55		46		48		50		52		54

⇒

From 66 go down two cells to 67 and repeat the zig zag pattern (Square 3).
From 88 move up to 89 and add numbers zigzag across the center row to the right.
At 99 go a distance of five units up to 100 and continue adding in zigzag fashion to fill in square 4.
At this point all columns at this point sum to 671, while the row sums are to be adjusted (by + or - values) according to the last cell in square 4 in order to sum to 671.

3
	8		10		1		3		5
7		9		11		2		4		6
	20		22		13		15		17
19		21		12		14		16		18
92	32	94	23	96	25	98	27	89	29	91
56	57	58	59	60	61	62	63	64	65	66
31	93	33	95	24	97	26	99	28	90	30
68	44	70	35	72	37	74	39	76	41	67
43	69	34	71	36	73	38	75	40	77	42
80	45	82	47	84	49	86	51	88	53	79
55	81	46	83	48	85	50	87	52	78	54

⇒

4
											782
104	8	106	10	108	1	110	3	101	5	103	659	23
7	105	9	107	11	109	2	100	4	102	6	562	109
116	20	118	22	120	13	111	15	113	17	115	880	-109
19	117	21	119	12	121	14	112	16	114	18	683	-12
92	32	94	23	96	25	98	27	89	29	91	696	-25
56	57	58	59	60	61	62	63	64	65	66	671	0
31	93	33	95	24	97	26	99	28	90	30	646	25
68	44	70	35	72	37	74	39	76	41	67	623	48
43	69	34	71	36	73	38	75	40	77	42	598	73
80	45	82	47	84	49	86	51	88	53	79	744	-73
55	81	46	83	48	85	50	87	52	78	54	719	-48
671	671	671	671	671	671	671	671	671	671	671	815

⇒

Square 5 shows the result of the adjustment with the generation of 0 duplicates.

5
											782
104	8	106	10	108	13	110	3	101	5	103	671
7	105	9	107	11	218	2	100	4	102	6	671
116	20	118	22	120	-96	111	15	113	17	115	671
19	117	21	119	12	109	14	112	16	114	18	671
92	32	94	23	96	0	98	27	89	29	91	671
56	57	58	59	60	61	62	63	64	65	66	671
31	93	33	95	24	122	26	99	28	90	30	671
68	44	70	35	72	85	74	39	76	41	67	671
43	69	34	71	36	146	38	75	40	77	42	671
80	45	82	47	84	-24	86	51	88	53	79	671
55	81	46	83	48	37	50	87	52	78	54	671
671	671	671	671	671	671	671	671	671	671	671	815

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 671 from the diagonals(782,815) to give 111 and 144, respectively, and which will be used as what I call the "de la Hire constants".
Addition of 111 and 144 to 671 gives 926 a magic pre-sum.
The following equations are used such that the following conditions are obeyed:
The right diagonal: 926 = 782 + 144
The left diagonal: 926 = 815 + 111
The rows and columns: 282 = 175 + 50 + 57.
Generate the mask using the 111 and 144 factors or sums thereof adding these factors to the appropriate cells in square 5 to generate square 6.
Square 6 has a magic sum equal to 926, i.e., S = 926 = ½(n³ + 47n + 4).

Square 5

Mask A
		111				144
	111							144
			111							144
					111		144
					144				111
	144			111
									144	111
111		144
144							111
			144			111
				144				111

⇒

6
											926
104	8	217	10	108	13	254	3	101	5	103	926
7	216	9	107	11	218	2	100	148	102	6	926
116	20	118	133	120	-96	111	15	113	17	259	926
19	117	21	119	12	220	14	256	16	114	18	926
92	32	94	23	96	144	98	27	89	140	91	926
56	201	58	59	171	61	62	63	64	65	66	926
31	93	33	95	24	122	26	99	28	234	141	926
179	44	214	35	72	85	74	39	76	41	67	926
187	69	34	71	36	146	38	186	40	77	42	926
80	45	82	191	84	-24	197	51	88	53	79	926
55	81	46	83	192	37	50	87	163	78	54	926
926	926	926	926	926	926	926	926	926	926	926	926

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

											782
104	8	106	10	108	13	110	3	101	5	103	671
7	105	9	107	11	218	2	100	4	102	6	671
116	20	118	22	120	-96	111	15	113	17	115	671
19	117	21	119	12	109	14	112	16	114	18	671
92	32	94	23	96	0	98	27	89	29	91	671
56	57	58	59	60	61	62	63	64	65	66	671
31	93	33	95	24	122	26	99	28	90	30	671
68	44	70	35	72	85	74	39	76	41	67	671
43	69	34	71	36	146	38	75	40	77	42	671
80	45	82	47	84	-24	86	51	88	53	79	671
55	81	46	83	48	37	50	87	52	78	54	671
671	671	671	671	671	671	671	671	671	671	671	815

											782
104	8	106	10	108	13	110	3	101	5	103	671
7	105	9	107	11	218	2	100	4	102	6	671
116	20	118	22	120	-96	111	15	113	17	115	671
19	117	21	119	12	109	14	112	16	114	18	671
92	32	94	23	96	0	98	27	89	29	91	671
56	57	58	59	60	61	62	63	64	65	66	671
31	93	33	95	24	122	26	99	28	90	30	671
68	44	70	35	72	85	74	39	76	41	67	671
43	69	34	71	36	146	38	75	40	77	42	671
80	45	82	47	84	-24	86	51	88	53	79	671
55	81	46	83	48	37	50	87	52	78	54	671
671	671	671	671	671	671	671	671	671	671	671	815

											782
104	8	106	10	108	13	110	3	101	5	103	671
7	105	9	107	11	218	2	100	4	102	6	671
116	20	118	22	120	-96	111	15	113	17	115	671
19	117	21	119	12	109	14	112	16	114	18	671
92	32	94	23	96	0	98	27	89	29	91	671
56	57	58	59	60	61	62	63	64	65	66	671
31	93	33	95	24	122	26	99	28	90	30	671
68	44	70	35	72	85	74	39	76	41	67	671
43	69	34	71	36	146	38	75	40	77	42	671
80	45	82	47	84	-24	86	51	88	53	79	671
55	81	46	83	48	37	50	87	52	78	54	671
671	671	671	671	671	671	671	671	671	671	671	815