A New Procedure for Magic Squares (Part IA)

Consecutive Boustrophedonic 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 first leftmost column and entered across every other cell until the end of the row is reached. A 2 down, 1 left knight break is used to get to the next line.

The next line is then added in reverse order to the previous, i.e. , boustrophedonically. 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 follow the sum equation shown in the New block Loubère Method. :

S = ½(n³ ± an)

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Construction of a 5x5 Magic Square

Method: Reading boustrophedonically (like a sidewinder snake) - 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 1, on reaching the end of the row knight break (2 down, 1 left) and continue adding numbers in the reverse mode (squares 1 and 2).
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 no duplicates has been generated (Square 3).

1
1		2		3

	5		4

6		7		8

⇒

2
					60
1	15	2	14	3	35	30
21	10	22	9	23	85	-20
16	5	17	4	18	60	5
11	25	12	24	13	85	-20
6	20	7	19	8	60	5
55	75	60	70	65	60
10	-10	5	-5	0

⇒

3
					70
1	15	32	14	3	65
21	10	2	9	23	65
26	-5	27	-1	18	65
11	25	-8	24	13	65
6	20	12	19	8	65
65	65	65	65	65	70

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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 the diagonals(60,60) from 65 to give both 5 and which will be used as what I call the "de la Hire constants".

However, this constant is too small and will generate too many duplicates. The constant to use are both 20 and 5 which will produce 90 as the magic pre-sum.

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

The right diagonal and left diagonals: 90 = 70 + 20

The rows and columns: 90 = 65 + 20 + 5.

Generate the mask using the 20 and 5 factors or sums thereof (55) adding these factors to the appropriate cells in square 3 to generate square 4.

Square 4 has a magic sum equal to 90, i.e., S = 90 = ½(n³ + 11n).

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3
					70
1	15	32	14	3	65
21	10	2	9	23	65
26	-5	27	-1	18	65
11	25	-8	24	13	65
6	20	12	19	8	65
65	65	65	65	65	70

Mask A
			5	20
	20			5
20	5
5		20
		5	20

⇒

4
					90
1	15	32	19	23	90
21	30	2	9	28	90
46	0	27	-1	18	90
16	25	12	24	13	90
6	20	17	39	8	90
90	90	90	90	90	90

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Construction of a 7x7 Magic Square

Method: Reading consecutive from left to right boustrophedonically - 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 1, on reaching the end of the row knight break (2 down, 1 left) and continue adding numbers in the reverse mode (squares 5 and 6).
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 one duplicate has been generated (Square 7).

5
1		2		3		4
15		16		17		18
	7		6		5
	21		20		19
8		9		10		11
22		23		24		25
	14		13		12

⇒

6
							190
1	28	2	27	3	26	4	91	84
15	42	16	41	17	40	18	189	-14
29	7	30	6	31	5	32	140	35
43	21	44	20	45	19	46	238	-63
8	35	9	34	10	33	11	140	35
22	49	23	48	24	47	25	238	-63
36	14	37	13	38	12	39	189	-14
154	196	161	189	168	182	175	189
21	-21	14	-14	7	-7	0

⇒

7
							112
1	28	2	111	3	26	4	175
15	42	16	27	17	40	18	175
29	7	30	41	31	5	32	175
64	0	58	-57	52	12	46	175
8	35	9	69	10	33	11	175
22	49	23	-15	24	47	25	175
36	14	37	-1	38	12	39	175
175	175	175	175	175	175	175	112

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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(112,112) from 175 to give 63, and which will be used as what I call the "de la Hire constant".

Addition of 63 to 175 gives 238 a magic pre-sum.

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

The right and left diagonals: 238 = 112 + 2(63)

The rows and columns: 238 = 175 + 63

Generate the mask using the 63 factor, adding this factor to the appropriate cells in square 7 generates square 8.

Square 8 has a magic sum equal to 238, i.e., S = 238 = ½(n³ + 19n).

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Mask B
						63
		63
	63
			63
63
					63
				63

⇒

8
							238
1	28	2	111	3	26	67	238
15	42	79	27	17	40	18	238
29	70	30	41	31	5	32	238
64	0	58	6	52	12	46	238
71	35	9	69	10	33	11	238
22	49	23	-15	24	110	25	238
36	14	37	-1	101	12	39	238
238	238	238	238	238	238	238	238

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