New Procedure for Loubère, Méziriac and Méziriac Type Magic Squares (Part I)

A Discussion of the New Method

Loubère and Méziriac magic squares have the property 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). In addition, the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S. Both squares also require an upward stepwise addition of consecutive numbers, i.e., 1,2,3...

The 5x5 regular Loubère (A on the left) and the Méziriac (B on the right) squares are shown below with both the starting number 1 at column 4, row 3. The 7x7 (C) and 9x9 (D) Méziriac squares are also shown while the Loubère will be generated in the next section.

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

B
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

C
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

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

It will be shown that as the digit 1 is moved to the right a new magic Méziriac type square will be generated (for odd n > 5) up to the rightmost column where the square is transformed into a Loubère.

The Loubère, Méziriac and Méziriac type Squares

Examination of the 5x5 Méziriac square (B) shows that on reaching the number 5 a shift of two cells to the right (or similarly a shift of three cells to the left) places the 6 in the rightmost cell. Examination of the Loubère square (A), on the other hand, it is four shifts to the right (or one shift to the left) places the 6 in the middle column. In theory then an increase in odd n should cause the shift to increase by two for each column to the right of the central column. For example, for the 7x7 square the shifts would be 2, 4 and 6 to the right for the digit 1 in columns 5, 6 and 7, respectfully.

For the 9x9 square the shifts would be 2, 4, 6 and 8 to the right for the digit 1 in columns 6, 7, 8 and 9, respectfully. For the two 7x7 squares (E) and (F) below the first shows a right shift of 4 and the second a right shift of 6 for the 7 to 8 transition. Alternatively the 7 to 8 transitions may be visualized as a left shift of 3 for the left square and a left shift of 1 for the right square. The sum of the left and right shifts is, therefore, 7 the value of n.

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

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

So far so good. Now for the 9x9 squares. We've already seen the Méziriac square above. What is required then are the 4, 6 and 8 shifts of the 9x9 squares (G), (H), (I) which are shown below:

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

H
		5			76			45
	4			75			44
3			74			43
		73			42			2
	81			41			1
80			40			9
		39			8			79
	38			7			78
37			6			77

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

What is seen is that the magic square for the 6 right shift (H) cannot be produced because that woud entail the superimposition of the number 10 directly over the number 40. It will be shown that when the number is a multiple of 6, i.e., 12, 18... no magic square is possible. In addition, when n is divisible by 3 then the Méziriac type square nxn will no longer have the full complement of possible squares. Consequently, the sequence, 6n + 3 expanded below, can be employed to determine which odd ns are divisible by 3. The number 3 in the sequence for the 3x3 stands for a square which is both a Loubère and a Méziriac.

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81...

As a postscript the magic square with the digit 1 in column 8 can be constructed but is not a Méziriac type square since it does not follow the upward stepwise construction rules employed by Méziriac. Taking magic square G and switching columns 2 with 3 and 7 with 8 produces magic square J which has some of the properties of G and some of the new (J):

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

This completes this section (Part I). The next section Part IB will (a) verify that other higher odd n will give the full complement of Méziriac type square and (b) verify that those that follow the sequence pattern above will not. To return to homepage.

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

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

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