New De Méziriac Knight-step Method and Squares (Part II)

A Discussion of the New Methods

An important general principle for generating odd magic squares by the De Méziriac method is that the center cell must always contain the middle number of the series of numbers used, 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 Méziriac 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.

The 5x5 and 7x7 regular Méziriac squares are shown below as examples:

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

4	29	12	37	20	45	28
35	11	36	19	44	27	3
10	42	18	43	26	2	34
41	17	49	25	1	33	9
16	48	24	7	32	8	40
47	23	6	31	14	39	15
22	5	30	13	38	21	46

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Méziriac squares are normally contructed using a stepwise approach where each subsequent number is added consecutively one cell at a time. In this new method each subsequent number is added in a stepwise knight step manner, followed by a right or an up break. In addition, n odd squares may be constructed with the initial number 1 on either of two broken diagonals shown below in light blue or yellow separated by symmetry by a light grey diagonal for a 5x5 square.

The set of 5x5 Méziriac Broken Diagonals
1			1
		1		1
	1		1
1		1
	1			1

The set of 7x7 Méziriac Broken Diagonals
1					1
				1		1
			1		1
		1		1
	1		1
1		1
	1					1

These new Méziriac type squares, have a two cell break like the regular Méziriac square and is labeled, as in previous web pages, as follows:

KMn^* (center cell#) [(2D,1L),2R] where KMn^* signifies a Knight-step nxn Méziriac square with the center cell number of the square and breaking right.
KMn^* (center cell#) [(1D,2L),2U] where KMn^* signifies a Knight-step nxn Méziriac square with the center cell number of the square and breaking up.

depending whether the 1's lie on the yellow or blue diagonal.

Every number on the main diagonal is represented at least once in this type of square.
For Méziriac Knight-step where n is divisible by 3, i.e., 3(2n + 1) no squares are magic for the (2D,1L),2R or 2U series.

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Construction Knight-step Méziriac Magic Squares

The set of 5x5 Squares (Method I)

To generate the regular square, KM5^* 13 [(2D,1L),2R], place a 1 left of center on a 5x5 square and fill in cells by advancing in a knight fashion to the down to the left until blocked by a previous number.
Move two cells right.
Repeat the process until the square is filled, as shown below in squares 1-5.

1
	6			5
	3
			1
4
		2

⇒

2
	6			5
	3		9
7			1
4		10		11
		2		8

⇒

3
	6		12	5
15	3	16	9
7		13	1
4		10		11
	14	2		8

⇒

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

⇒

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

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The Other Four 5x5 Méziriac Knight-step Squares

A KM5^* 9 [(2D,1L),2R]
14	2	20	8	21
6	24	12	5	18
3	16	9	22	15
25	13	1	19	7
17	10	23	11	4

B KM5^* 17 [(2D,1L),2R]
22	15	3	16	9
19	7	25	13	1
11	4	17	10	23
8	21	14	2	20
5	18	6	24	12

C KM5^* 21 [(2D,1L),2R]
1	19	7	25	13
23	11	4	17	10
20	8	21	14	2
12	5	18	6	24
9	22	15	3	16

D KM5^* 5 [(2D,1L),2R]
10	23	11	4	17
2	20	8	21	14
24	12	5	18	6
16	9	22	15	3
13	1	19	7	25

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A 7x7 Méziriac Knight-step Square

The construction of a 7x7 Méziriac Knight-step square KM7^* 25 [(2D,1L),2R] from the broken yellow diagonal is shown below using the knight-step approach.

1
		3		11
8					7
	4		12
				1		9
5		13
			2		10
	14		15			6

⇒

2
	27	3		11		19
8		16		24	7
28	4	29	12		20
	17		25	1		9
5		13		21		22
18		26	2		10
	14		15		23	6

⇒

3
44	27	3	35	11	36	19
8	40	16	48	24	7	32
28	4	29	12	37	20	45
41	17	49	25	1	33	9
5	30	13	38	21	46	22
18	43	26	2	34	10	42
31	14	39	15	47	23	6

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The Plane of Méziriac Squares

At this point it may be said that alternatively these squares may be constructed using a plane of four squares. For example using the 7x7 square KM7^* 25 [(2D,1L),2R] one can move up the right diagonal on a plane of four KM7^* 25 [(2D,1L),2R] and generate the complete set of 7 squares as is shown in Part IV of the new Bachet de Méziriac method.

This completes this section on regular and non-regular De La Méziriac two step squares (Part I). The next section deals with a new Méziriac Knight-step square method (Part III). To return to homepage.