A New Procedure for Magic Squares (Part ID)

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 and entered across every other cell until the partial row is filled. A (2 right, 1 down) knight break is used to get to the next line. (This knight break is also placing half of the numbers going down and the other half going up). 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 in New block Loubère Method and New Consecutive Mask-Generated Method.

S = ½(n³ ± an)
S = ½(n³ ± an + b)

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 right, 1 down) 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
2		1		3
	4		5
6		7		8
	9		10
11		12		13

⇒

2
					35
2	14	1	15	3	35	30
16	4	17	5	18	60	5
6	19	7	20	8	60	5
21	9	22	10	23	85	-20
11	24	12	25	13	85	-20
56	70	59	75	65	36
9	-5	6	-10	0

⇒

3
					46
2	14	31	15	3	65
16	4	22	5	18	65
15	14	18	10	8	65
21	9	2	10	23	65
11	24	-8	25	13	65
65	65	65	65	65	47

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(46,47)from 65 to obtain 19 and 18, 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 are used such that the following conditions are obeyed: The right diagonal: 120 = 46 + 2(19) + 2(18) The left diagonal: 120 = 47 + 19 + 3(18) The rows and columns: 120 = 65 + 19 + 3(18).
Generate the mask using the 18 and/or 19 factors or a sum thereof (18 + 19) = 37 and 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 120, i.e., S = 120 = ½(n³ + 23n).

3
					46
2	14	31	15	3	65
16	4	22	5	18	65
15	14	18	10	8	65
21	9	2	10	23	65
11	24	-8	25	13	65
65	65	65	65	65	47

Mask A
37	18
		18	37
18				37
	37		18
		37		18

⇒

4
					120
39	32	31	15	3	120
16	4	40	42	18	120
33	14	18	10	45	120
21	46	2	28	23	120
11	24	29	25	31	120
120	120	120	120	120	120

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

⇒

6
							259
25	2	26	1	27	3	28	112	63
4	29	5	30	6	31	7	112	63
32	8	33	9	34	10	35	161	14
11	36	12	37	13	38	14	161	14
39	15	40	16	41	17	42	210	-35
18	43	19	44	20	45	21	210	-34
46	22	47	23	48	24	49	259	-84
175	155	182	160	189	168	196
0	20	-8	15	-13	7	-18

⇒

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 175 from the diagonals(259,259) from 175 to give 113, and which will be used as what I call the "de la Hire constant". Addition of 113 to 175 gives 288 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: No change The rows and columns: 288 = 175 + 113
Generate the mask using the 113 factor, 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 8.
Square 8 has a magic sum equal to 288, i.e., S = 288 = ½(n³ + 33n + 2).

7
							288
25	2	26	64	27	3	28	175
4	29	5	93	6	31	7	175
32	8	33	23	34	10	35	175
11	56	5	66	-1	45	-7	175
39	15	40	-19	41	17	42	175
18	43	19	9	20	45	21	175
46	22	47	-61	48	24	49	175
175	175	175	175	175	175	175	288

Mask B
	113
		113
113
					113
						113
			113
				113

⇒

8
							288
25	115	26	64	27	3	28	288
4	29	118	93	6	31	7	288
145	8	33	23	34	10	35	288
11	56	5	66	-1	158	-7	288
39	15	40	-19	41	17	155	288
18	43	19	22	20	45	21	288
46	22	47	-61	161	24	49	288
288	288	288	288	288	288	288	288

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