A New Procedure for Magic Squares (Part IC)

Equation Generated 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. 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)

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Construction of a 13x13 Magic Square

Method: Sequential Readout - use of a mask

Construct the 13x13 Square 1 where 13 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 7 and filling every other the cell (square 1).
Upon reaching the numeral 7 perform a knight break enter 8 at row 2 cell 6 and continue filling 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 gp to the center row and fill in the row consecutively.

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

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

⇒

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

⇒

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

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

⇒

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From 130 go to row 1 cell 12 and fill in 131 in a backwards manner. Fill in the rest of the square using the sequence table to generate the moves. For example, to go to the second row move 2 cells (knight), followed by filling the row to the right. Continue filling the next row using six moves, 1 down and 5 right then filling the row to the right.
The next three moves involve 2 knight moves to the right and a final 3 moves total to the left.

4
													565
5	136	6	135	7	134	1	133	2	132	3	131	4	829	276
143	12	137	13	138	8	139	9	140	10	141	11	142	1043	62
19	148	20	149	14	144	15	145	16	146	17	147	18	998	107
155	26	156	21	150	22	151	23	152	24	153	25	154	1212	-107
33	162	27	157	28	158	29	159	30	160	31	161	32	1167	-62
164	34	165	35	166	36	167	37	168	38	169	39	163	1381	-276
79	80	81	82	83	84	85	86	87	88	89	90	91	1105	0
93	41	94	42	95	43	96	44	97	45	98	40	92	920	185
48	100	49	101	50	102	51	103	52	104	46	99	47	952	153
107	55	108	56	109	57	110	58	111	53	105	54	106	1089	16
62	114	63	115	64	116	65	117	59	112	60	113	61	1121	-16
121	69	122	70	123	71	124	66	118	67	119	68	120	1258	-153
76	128	77	129	78	130	72	125	73	126	74	127	75	1290	-185
1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	559

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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
													565
5	136	6	135	7	134	277	133	2	132	3	131	4	1105
143	12	137	13	138	8	201	9	140	10	141	11	142	1105
19	148	20	149	14	144	122	145	16	146	17	147	18	1105
155	26	156	21	150	22	44	23	152	24	153	25	154	1105
33	162	27	157	28	158	-33	159	30	160	31	161	32	1105
164	34	165	35	166	36	-109	37	168	38	169	39	163	1105
79	80	81	82	83	84	85	86	87	88	89	90	91	1105
93	41	94	42	95	43	281	44	97	45	98	40	92	1105
48	100	49	101	50	102	204	103	52	104	46	99	47	1105
107	55	108	56	109	57	126	58	111	53	105	54	106	1105
62	114	63	115	64	116	49	117	59	112	60	113	61	1105
121	69	122	70	123	71	-29	66	118	67	119	68	120	1105
76	128	77	129	78	130	-113	125	73	126	74	127	75	1105
1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	559

⇒

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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(565,559) from 1105 to give 540 and 546, respectively and which will be used as what I call the "de la Hire constants".

Addition of the sum of these two numbers, 540 + 546 = 1086 to 1105 gives 2191 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: 2191 = 565 + 2(540) + 546

The left diagonal: 2191 = 559 + 540 + 2(546)

The rows and columns: 2191 = 1105 + 540 + 546

Generate the mask using the 540 and 546 factors and adding these factors to the appropriate cells in square 5 to generate square 6.

Square 6 has a magic sum equal to 2191, i.e., S = 2191 = ½(n³ + 168n + 1).

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Mask A
				546		540
546											540
			546		540
			540						546
						546				540
540											546
	546							540
							546					540
		540			546
				540						546
							540	546
		540	546
									540			546

Square 5

⇒

6
													2191
5	136	6	135	553	134	817	133	2	132	3	131	4	2191
689	12	137	13	138	8	201	9	140	10	141	551	142	2191
19	148	20	695	14	684	122	145	16	146	17	147	18	2191
155	26	156	561	150	22	44	23	152	570	153	25	154	2191
33	162	27	157	28	158	513	159	30	160	571	161	32	2191
704	34	165	35	166	36	-109	37	168	38	169	585	163	2191
79	626	81	82	83	84	85	86	627	88	89	90	91	2191
93	41	94	42	95	43	281	590	97	45	98	40	632	2191
48	100	589	101	50	648	204	103	52	104	46	99	47	2191
107	55	108	56	649	57	126	58	111	53	651	54	106	2191
62	114	63	115	64	116	49	657	605	112	60	113	61	2191
121	609	668	70	123	71	-29	66	118	67	119	68	120	2191
76	128	77	129	78	130	-113	125	73	666	74	127	621	2191
2191	2191	2191	2191	2191	2191	2191	2191	2191	2191	2191	2191	2191	2191

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This completes this section on a new Equation Generated Centered Sequential 13x13 Mask-Generated Squares (Part IC). To return to homepage.