A New Procedure for Magic Squares (Part IC)

Centered Sequential 13x13 Zig Zag 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. As was shown for the 9x9 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 one 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, i.e., 2 cells to the right, 4 cells to the right and 3 cells to the left as shown in the sequence table derived from the sequence equation. Again this is modified from the 9x9 method where the row adjacent to the center row was initially filled in a reverse manner instead of to the right as in this new method.

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 13x13 Magic Square

Method: Sequential Readout - use of a mask

Construct the 13x13 Square in a consecutive zigzag manner starting at row 1 cell 7 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 the numeral 12 a 2 down cell break is performed to numeral 13 and addition to the cells in a normal fashion.
From 39 go down two cells skipping over the center row and add 40 then continuing adding numerals in normal fashion in the last row of the square.
From 78 go to the center row and fill in the row consecutively.

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

	51		40		42		44		46		48
50		52		41		43		45		47		49
	65		54		56		58		60		62
64		53		55		57		59		61		63
	66		68		70		72		74		76
78		67		69		71		73		75		77

⇒

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

⇒

Fill in the bottom of the square starting at 91 and continuing to 130.

3
8		10		12		1		3		5		7
	9		11		13		2		4		6
22		24		26		15		17		19		21
	23		25		14		16		18		20
36		38		27		29		31		33		35
	37		39		28		30		32		34
79	80	81	82	83	84	85	86	87	88	89	90	91
93	51	95	40	97	42	99	44	101	46	103	48	92
50	94	52	96	41	98	43	100	45	102	47	104	49
107	65	109	54	111	56	113	58	115	60	117	62	106
64	108	53	110	55	112	57	114	59	116	61	105	63
121	66	123	68	125	70	127	72	129	74	118	76	120
78	122	67	124	69	126	71	128	73	130	75	119	77

⇒

From 130 go to row 1 cell 10 and fill in 131.
Continue filling in the square in a zigzag manner.

4
													530
8	136	10	138	12	140	1	142	3	131	5	133	7	866	239
135	9	137	11	139	13	141	2	143	4	132	6	134	1006	99
22	150	24	152	26	154	15	156	17	145	19	147	21	1048	57
149	23	151	25	153	14	155	16	144	18	146	20	148	1162	-57
36	164	38	166	27	168	29	157	31	159	33	161	35	1204	-99
163	37	165	39	167	28	169	30	158	32	160	34	162	1344	-239
79	80	81	82	83	84	85	86	87	88	89	90	91	1105	0
93	51	95	40	97	42	99	44	101	46	103	48	92	951	154
50	94	52	96	41	98	43	100	45	102	47	104	49	921	184
107	65	109	54	111	56	113	58	115	60	117	62	106	1133	-28
64	108	53	110	55	112	57	114	59	116	61	105	63	1077	28
121	66	123	68	125	70	127	72	129	74	118	76	120	1289	-184
78	122	67	124	69	126	71	128	73	130	75	119	77	1259	-154
1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	569

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 of Square 4 to generate Square 5.
At this point four duplicates have been generated in pink.

5
													530
8	136	10	138	12	140	240	142	3	131	5	133	7	1105
135	9	137	11	139	13	240	2	143	4	132	6	134	1105
22	150	24	152	26	154	72	156	17	145	19	147	21	1105
149	23	151	25	153	14	98	16	144	18	146	20	148	1105
36	164	38	166	27	168	-70	157	31	159	33	161	35	1105
163	37	165	39	167	28	-70	30	158	32	160	34	162	1105
79	80	81	82	83	84	85	86	87	88	89	90	91	1105
93	51	95	40	97	42	253	44	101	46	103	48	92	1105
50	94	52	96	41	98	227	100	45	102	47	104	49	1105
107	65	109	54	111	56	85	58	115	60	117	62	106	1105
64	108	53	110	55	112	85	114	59	116	61	105	63	1105
121	66	123	68	125	70	-57	72	129	74	118	76	120	1105
78	122	67	124	69	126	-83	128	73	130	75	119	77	1105
1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	569

⇒

Generate a mask below 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 5 (as in the de la Hire method) that all sums will equal.

We start by subtracting the diagonals(530,569) from 1105 to give 575 and 536, respectively and which will be used as what I call the "de la Hire constants".
Addition of the sum of these two numbers, 575 + 536 = 1111 to 1105 gives 2216 a magic pre-sum. Because of the number of duplicates in the center row a new sum was recalculated to give 3326.
The following equations are therefore used with the new sum such that the following conditions are obeyed:
The right diagonal: 3327 = 530 + 2(536) + 3(575)
The left diagonal: 3327 = 569 + 3(536) + 2(575)
The rows and columns: 3327 = 1105 + 2(536) + 2(575)
Generate the mask using the 536 and 575 factors and adding these factors to the appropriate cells in square 5 to generate square 6.
Square 6 has a magic sum equal to 3327, i.e., S = 3327 = ½(n³ + 342n + 11).

Mask A
		575				575		536		536
	575		575				536				536
				536	536					575		575
536	575		536					575
536		536						575	575
			575			536	575					536
				575	575		536		536
				536		536					575	575
		536			575			536			575
575					536	575				536
	536			575					536	575
575	536		536				575
		575							575		536	536

⇒

6
													3327
8	136	585	138	12	140	815	142	539	131	541	133	7	3327
135	584	137	586	139	13	240	538	143	4	132	542	134	3327
22	150	24	152	562	690	72	156	17	145	574	147	576	3327
685	598	151	561	153	14	98	16	719	18	146	20	148	3327
572	164	574	166	27	168	-70	157	606	734	33	161	35	3327
163	37	165	614	167	28	466	605	158	32	160	34	698	3327
79	80	81	82	658	659	85	622	87	624	89	90	91	3327
93	51	95	40	633	42	789	44	101	46	103	623	667	3327
50	94	588	96	41	673	227	100	581	102	47	679	49	3327
682	65	109	54	111	592	660	58	115	60	653	62	106	3327
64	644	53	110	630	112	85	114	59	652	636	105	63	3327
696	602	123	604	125	70	-57	647	129	74	118	76	120	3327
78	122	642	124	69	126	-83	128	73	705	75	655	613	3327
3327	3327	3327	3327	3327	3327	3327	3327	3327	3327	3327	3327	3327	3327

This completes this section on a new Centered Sequential 13x13 Zig Zag Mask-Generated Squares (Part IC). The next section deals with Centered 7x7 Sequential Mask-Generated Zig Zag Squares (Part ID). To return to homepage.