A New Procedure for Magic Squares (Part IC)

Consecutive Knight Break Mask-Generated Squares

A Discussion of the New Method

Magic squares such as the Loubère have a center cell which must always contain the middle number of a series of consecutive numbers, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n² + 1). The properties of these regular or associated Loubère squares are:

That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n² + 1, i.e., or twice the number in the center cell and are complementary to each other.

In this method the numbers on the square are placed consecutively starting from the center cell of the first row column and entered across every other cell until the partial row is filled. A 2 down, 1 right knight break is used to get to the next line. (This knight break is also equivalent to a slant 2 break in the right down direction). The next line is then added in the same manner.

The square obtained, which is not magic, is modified into a form which can be converted into a magic one 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 follo either of the two sum equation shown: New block Loubère Method and New Consecutive Mask-Generated Method. :

S = ½(n³ ± an)
S = ½(n³ ± an + b)

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Construction of a 5x5 Magic Square

Method: Reading consecutive from left to right and use of a mask

Construct the 5x5 Square 1 where 5 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 3, on all the requisite numbers to the row, knight break (2 down, 1 right) and continue adding numbers in regular mode.

Since all sums of all the columns or rows are not equal to 65 add or subtract the numbers in the last row from those numbers in the center row. Then add or subtract the numbers in the last column to those in the center column. At this point four duplicates has been generated (in pink, Square 3).

1
3		1		2

	4		5

7		8		6

⇒

2
					60
3	14	1	15	2	35	30
23	10	21	9	22	85	-20
17	4	18	5	16	60	5
13	24	11	25	12	85	-20
7	20	8	19	6	60	5
63	72	57	73	58	62
2	-7	6	-8	7

⇒

3
					71
3	14	31	15	2	65
23	10	1	9	22	65
19	-3	29	-3	23	65
13	24	-9	25	12	65
7	20	13	19	6	65
65	65	65	65	65	73

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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 equal a magic sum.

We start by subtracting 65 from the diagonals(71,73) 6 and 8, respectively and which will be used as what I call the "de la Hire constants".

Since four duplicates are generated (square 3) the following equations were found useful:

The following equations are used such that the following conditions are obeyed:

The right diagonal: 101 = 71 + 3(8) + 6

The left diagonal: 101 = 71 + 2(8) + 2(6)

The rows and columns: 101 = 65 + 3(8) + 2(6).

Generate the mask using the 6 and 8 factors or sums thereof adding these factors to the appropriate cells in square 3 to generate square 4 (not an easy task).

Square 4 has a magic sum equal to 101, i.e., S = 101 = ½(n³ + 11n +2).

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3
					71
3	14	31	15	2	65
23	10	1	9	22	65
19	-3	29	-3	23	65
13	24	-9	25	12	65
7	20	13	19	6	65
65	65	65	65	65	73

Mask A
14	8		6	8
22	6			8
	8	8	14	6
	14	8		14
		20	16

⇒

4
					101
17	22	31	21	10	101
45	16	1	9	30	101
19	5	37	11	29	101
13	38	-1	25	26	101
7	20	33	35	6	101
101	101	101	101	101	101

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Construction of a 7x7 Magic Square

Method: Reading consecutive from left to right and 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, on reaching the last number in the set (3), knight break (2 down, 1 right) and continue adding numbers using this break (squares 5, 6 and 7).
Since all sums of all the columns or rows are not equal to 175 add or subtract the numbers in the last row from those numbers in the center row. Then add or subtract the numbers in the last column to those in the center column. At this point four duplicats have been generated (Square 8).

5
	3		1		2

7		4		5		6

	8		9		10

12		13		14		11

⇒

6
	3		1		2
	16		17		15
7		4		5		6
20		21		18		19
	8		9		10
	24		22		23
12		13		14		11

7
							160
28	3	25	1	26	2	27	112	63
41	16	42	17	39	15	40	210	-35
7	29	4	30	5	31	6	112	63
20	45	21	43	18	44	19	210	-35
33	8	34	9	35	10	32	161	14
49	24	46	22	47	23	48	259	-84
12	37	13	38	14	36	11	161	14
190	162	185	160	184	161	183	160
-15	13	-10	15	-9	14	-8

⇒

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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 7 (as in the de la Hire method) that all sums will equal to a magic sum.

We start by subtracting the diagonals(140,140) from 175 to give 35, and which will be used as what I call the "de la Hire constant".

Addition of 35 to 175 gives 210 a magic pre-sum. However, because of the number of duplicates involved a much higher number was picked so as to avoid any duplication.

The following equations are used such that the following conditions are obeyed:

The right and left diagonals: 585 = 140 + 7(35)

The rows and columns: 585 = 175 + 6(35)

Generate the mask using the 35 and multiples of 35 as the factors, adding these factors to the appropriate cells in square 7 to generate square 8, where the colors of the cells in the mask match those of square 9.

Square 8 has a magic sum equal to 385, i.e., S = 385 = ½(n³ + 61n).

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8
							140
28	3	25	64	26	2	27	175
41	16	42	-18	39	15	40	175
7	29	4	93	5	31	6	175
5	58	11	23	9	58	11	175
33	8	34	23	35	10	32	175
49	24	46	-62	47	23	48	175
12	37	13	52	14	36	11	175
175	175	175	175	175	175	175	140

Mask B
105				35		70
				105	105
			140	70
	105	35			70
105			70			35
		175			35
	105					105

⇒

9
							385
133	3	25	64	61	2	97	385
41	16	42	-18	144	120	40	385
7	29	4	233	75	31	6	385
5	163	46	23	9	128	11	385
138	8	34	93	35	10	67	385
49	24	221	-62	47	58	48	385
12	142	13	52	14	36	116	385
385	385	385	385	385	385	385	385

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