New Method for Loubère, Méziriac and Méziriac Type Magic Squares (Part III)

Rules for Construction

Loubère and Méziriac Squares-Background

The Siamese method which includes both the 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...

Equality of Squares of Part II and Part III

The squares in this section were found equivalent to those in Part II. Rotation of the squares in this section 180^o degrees about the main diagonal showed the squares to be identical to those in Part II. Thus placing the digit 1 south or the west of the central cell produce the same result. There is also some information in this section that is not covered in Part II.

The Loubère, Méziriac and Méziriac type Squares

The 5×5 regular Loubère ((X) on the left) and the Méziriac ((Y) on the right) squares are shown below with both the starting number 1 at row 3, column 1 and column 2, respectively. Only (Y) is magic. Square (X) and we may state here, all other Loubère squares are non magic whether n is prime or composite. The 7×7 (Z) and 9×9 (AA) Méziriac squares are also shown, although (Z) is magic and (AA) is not:

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

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

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

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

So that examining all three Méziriac squares (Y), (Z) and (AA) it is seen construction is performed using a shift of two cells left or similarly 3, 5 or 7 shifts right, respectively. The Méziriac (Z) is magic while the Méziriac (AA) in which n is composite, i.e., (3×3) is not.

As was shown in Part IA a table was generated for right cell shifts. For these squares, however, since the digit 1 is place to the west of the central cell C the shifts are the same but in the westward direction as in the following table:

Left Cell Shifts
C	2	4	6	8	10	12	14	16	18	20	22	24	26	28	...	W ←

Moreover, from the partial table (since the table is ∞) those n that are divisible by three 6,12,18; those by five 10, 20; and those by seven 14, 28 cannot be constructed. Construction of the other two 7×7 squares (AB) and (AC), are both shown below only (AB) is magic while the Loubère (T) is not.

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

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

The results for the 9×9, where n is composite, are different from those of Part IA. (AA) was shown to be non magic and so are the left cell shift of 6 and 8 (the Loubère). Only the (AD), a 4 down shift, square shown below is magic:

9×9
C	2	4	6	8	W ←

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

Moreover, other squares with composite n may or may not contain viable magic squares. It was found, thru actual construction, that for any of the 15×15 types no magic squares were possible but that for the 21×21 types, three were magic - the 4, the 10 and the 16:

Left Cell Shifts for 21×21
C	2	4	6	8	10	12	14	16	18	20	W ←

where 6, 12 and 18 are impossible to construct due to divisibility by three and the 14, impossible due to divisibility by seven. For the 25×25 types seven were magic - the 2, 6, 8, 12, 16, 18 and 22 with a definite 1,2,1,2,1 cell pattern. The Méziriac, to the immediate right of C, for the 25×25 is also magic contrary to those where n is equal to 9, 15 or 21 all of which are divisible by three. Note that the 10 and 20 squares being divisible by 5 are impossible to construct.

Left Cell Shifts for 25×25
C	2	4	6	8	10	12	14	16	18	20	22	24	W ←

Since these magic squares require a laborious manual construction a rule for their constructions has been developed as shown as follows:

(a) Construct the cell shift table for each square
(b) Divide the numbers into symmetry pairs
The last number in the table is not magic, not part of the symmetry.
(c) Run a plane of symmetry through the middle of the remaining values
(d) Cross out those numbers that are divisible by the composite parts
(e) By symmetry match up that crossed out values with uncrossed ones
(f) Cross out the matched number
(g) By symmetry match the uncrossed value with uncrossed ones

The uncrossed values remaining should be relatively prime to the prime numbers of the composite numbers, i.e., they are not divisible by the composite primes only by the number one. The uncrossed numbers in white are those squares which are magic. Confirmation via actual construction confirms the rule. In this case the rules were developed after the construction. Thus, once we have the rules it's easy to determine which squares are magic without having to go thru the actual construction.

Below is an example using and n order of 25:

I
C	2	4	6	8	10	12	14	16	18	20	22	24	N ↑

II
C	2	4	6	8	10	12	14	16	18	20	22	24	N ↑

II
C	2	4	6	8	10	12	14	16	18	20	22	24	N ↑

So by symmetry there are three values to the left of the symmetry plane and three to the right plus the middle 12 for a total of seven squares. The rules in this paper are similar to but not identical to those in Part IV where it can be compared.

The following depicts (AE) and (AF) two out of the five possible prime squares of 11×11 both of which are magic.

AE (2← 9→)
115	54	103	42	91	30	79	18	67	6	66
53	102	41	90	29	78	17	77	5	65	114
101	40	89	28	88	16	76	4	64	113	52
39	99	27	87	15	75	3	63	112	51	100
98	26	86	14	74	2	62	111	50	110	38
25	85	13	73	1	61	121	49	109	37	97
84	12	72	11	60	120	48	108	36	96	24
22	71	10	59	119	47	107	35	95	23	83
70	9	58	118	46	106	34	94	33	82	21
8	57	117	45	105	44	93	32	81	20	69
56	116	55	104	43	92	31	80	19	68	7

AF (6← 5→)
42	18	115	91	67	54	30	6	103	79	66
17	114	90	77	53	29	5	102	78	65	41
113	89	76	52	28	4	101	88	64	40	16
99	75	51	27	3	100	87	63	39	15	112
74	50	26	2	110	86	62	38	14	111	98
49	25	1	109	85	61	37	13	121	97	73
24	11	108	84	60	36	12	120	96	72	48
10	107	83	59	35	22	119	95	71	47	23
106	82	58	34	21	118	94	70	46	33	9
81	57	44	20	117	93	69	45	32	8	105
56	43	19	116	92	68	55	31	7	104	80

The Loubère, (A5), the digit 1 in the first column shown below, is the only non magic square in the set (first and second columns not equal to 671).

A5 (10← 1→)
67	79	91	103	115	6	18	30	42	54	66
78	90	102	114	5	17	29	41	53	65	77
89	101	113	4	16	28	40	52	64	76	88
100	112	3	15	27	39	51	63	75	87	99
111	2	14	26	38	50	62	74	86	98	110
1	13	25	37	49	61	73	85	97	109	121
12	24	36	48	60	72	84	96	108	120	11
23	35	47	59	71	83	95	107	119	10	22
34	46	58	70	82	94	106	118	9	21	33
45	57	69	81	93	105	117	8	20	32	44
56	68	80	92	104	116	7	19	31	43	55

This completes this section (Part III). To go to Part IV. To return to Part II. To return to homepage.

67	79	91	103	115	6	18	30	42	54	66
78	90	102	114	5	17	29	41	53	65	77
89	101	113	4	16	28	40	52	64	76	88
100	112	3	15	27	39	51	63	75	87	99
111	2	14	26	38	50	62	74	86	98	110
1	13	25	37	49	61	73	85	97	109	121
12	24	36	48	60	72	84	96	108	120	11
23	35	47	59	71	83	95	107	119	10	22
34	46	58	70	82	94	106	118	9	21	33
45	57	69	81	93	105	117	8	20	32	44
56	68	80	92	104	116	7	19	31	43	55

67	79	91	103	115	6	18	30	42	54	66
78	90	102	114	5	17	29	41	53	65	77
89	101	113	4	16	28	40	52	64	76	88
100	112	3	15	27	39	51	63	75	87	99
111	2	14	26	38	50	62	74	86	98	110
1	13	25	37	49	61	73	85	97	109	121
12	24	36	48	60	72	84	96	108	120	11
23	35	47	59	71	83	95	107	119	10	22
34	46	58	70	82	94	106	118	9	21	33
45	57	69	81	93	105	117	8	20	32	44
56	68	80	92	104	116	7	19	31	43	55

67	79	91	103	115	6	18	30	42	54	66
78	90	102	114	5	17	29	41	53	65	77
89	101	113	4	16	28	40	52	64	76	88
100	112	3	15	27	39	51	63	75	87	99
111	2	14	26	38	50	62	74	86	98	110
1	13	25	37	49	61	73	85	97	109	121
12	24	36	48	60	72	84	96	108	120	11
23	35	47	59	71	83	95	107	119	10	22
34	46	58	70	82	94	106	118	9	21	33
45	57	69	81	93	105	117	8	20	32	44
56	68	80	92	104	116	7	19	31	43	55