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

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 Parts (IA and IB) and Part IV

The squares in this section were found to be equivalent to those in Parts (IA and IB). Rotation of the squares in this section 180^o degrees about the main diagonal gave squares identical to those in Parts (IA and IB). The 11×11 squares at the end of this section, however, are an addition to those shown in Parts (IA and IB).

The Loubère, Méziriac and Méziriac type Squares

The 5×5 regular Loubère (AG on the left) and the Méziriac (AH on the right) squares are shown below . The 7×7 (AI) and 9×9 (AJ) Méziriac squares are also shown:

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

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

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

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

Both the Méziriac and the Loubère are magic for every odd number n. In addition, all three Méziriac squares (AH), (AI) and (AK) use a shift of two cells up or a complementary shift of 3, 5 or 7 down, respectively. These squares are what are known as the regular Méziriac, though the digit 1 is positioned over the central cell instead of to the right as it is normally published. Both sites are equivalent, however.

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

Up Cell Shifts
C	2	4	6	8	10	12	14	16	18	20	22	24	26	28	...	N ↑

Note that the first entry, 2, corresponds to a Méziriac and the last cell entry to a Loubère (depending on what n is) while all other entries between these two to Méziriac types.

Moreover, the partial table above (since the table is ∞) has the following characteristics: 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 (AK) and (AL) are both are shown below. Besides the Loubère (AL) the Méziriac type (AK) is also magic.

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

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

The results for the 9×9, where n is composite, are identical to Part IA. (AM), a 4 up shift, as well as the 6 up shift square (not shown) are the only two squares that are not magic:

9×9
C	2	4	6	8	N ↑

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

In addition, other squares with composite n may or may not contain viable magic squares. It was found (by actual construction) that three magic squares were possible for the 15×15 types, two for the 21×21 types and eight for the 25×25 types (shown below in the shift cell table). Moreover, the cell shift tables below plus the 9×9 above appear to show a particular symmetrical cell pattern [(1,1), (1,1,1), (1,1) and (2,1,2,1,2)]. Note that all those in white are either non magic or inconstructible:

15×15
C	2	4	6	8	10	12	14	N ↑

21×21
C	2	4	6	8	10	12	14	16	18	20	N ↑

25×25
C	2	4	6	8	10	12	14	16	18	20	22	24	N ↑

The rule for forming these squares are as follows:

(a) Construct the cell shift table for each square
(b) Divide the numbers into symmetry pairs
(c) Cross out those numbers that are divisible by the composite parts
(d) By symmetry match up that crossed out values with uncrossed ones
(e) Cross out the matched number
(f) 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 ↑

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

So by symmetry there are four values to the left of the symmetry plane and four to the right.

(AN) and (AO) two out of the other five possible prime squares of 11×11, both of which are magic, are depicted below. The other square in the set (K) shown in Part IB was presented but with a 180^o degree rotation about the main diagonal.

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

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

To finish off this discussion: below is the 15×15 square (NM) with the digit 1 in position two cells north of the central cell, thus requiring a four cell shift available from the shift cell table. This square, though constructible, is non magic (first and second row do not sum to 1695).

NM (4↑ 11↓)
57	219	156	93	45	207	144	81	18	195	132	69	6	168	120
218	155	92	44	206	143	80	17	194	131	68	5	167	119	56
154	91	43	205	142	79	16	193	130	67	4	166	118	55	217
105	42	204	141	78	30	192	129	66	3	180	117	54	216	153
41	203	140	77	29	191	128	65	2	179	116	53	215	152	104
202	139	76	28	190	127	64	1	178	115	52	214	151	103	40
138	90	27	189	126	63	15	177	114	51	213	165	102	39	201
89	26	188	125	62	14	176	113	50	212	164	101	38	200	137
25	187	124	61	13	175	112	49	211	163	100	37	199	136	88
186	123	75	12	174	111	48	225	162	99	36	198	150	87	24
122	74	11	173	110	47	224	161	98	35	197	149	86	23	185
73	10	172	109	46	223	160	97	34	196	148	85	22	184	121
9	171	108	60	222	159	96	33	210	147	84	21	183	135	72
170	107	59	221	158	95	32	209	146	83	20	182	134	71	8
106	58	220	157	94	31	208	145	82	19	181	133	70	7	169

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

57	219	156	93	45	207	144	81	18	195	132	69	6	168	120
218	155	92	44	206	143	80	17	194	131	68	5	167	119	56
154	91	43	205	142	79	16	193	130	67	4	166	118	55	217
105	42	204	141	78	30	192	129	66	3	180	117	54	216	153
41	203	140	77	29	191	128	65	2	179	116	53	215	152	104
202	139	76	28	190	127	64	1	178	115	52	214	151	103	40
138	90	27	189	126	63	15	177	114	51	213	165	102	39	201
89	26	188	125	62	14	176	113	50	212	164	101	38	200	137
25	187	124	61	13	175	112	49	211	163	100	37	199	136	88
186	123	75	12	174	111	48	225	162	99	36	198	150	87	24
122	74	11	173	110	47	224	161	98	35	197	149	86	23	185
73	10	172	109	46	223	160	97	34	196	148	85	22	184	121
9	171	108	60	222	159	96	33	210	147	84	21	183	135	72
170	107	59	221	158	95	32	209	146	83	20	182	134	71	8
106	58	220	157	94	31	208	145	82	19	181	133	70	7	169

57	219	156	93	45	207	144	81	18	195	132	69	6	168	120
218	155	92	44	206	143	80	17	194	131	68	5	167	119	56
154	91	43	205	142	79	16	193	130	67	4	166	118	55	217
105	42	204	141	78	30	192	129	66	3	180	117	54	216	153
41	203	140	77	29	191	128	65	2	179	116	53	215	152	104
202	139	76	28	190	127	64	1	178	115	52	214	151	103	40
138	90	27	189	126	63	15	177	114	51	213	165	102	39	201
89	26	188	125	62	14	176	113	50	212	164	101	38	200	137
25	187	124	61	13	175	112	49	211	163	100	37	199	136	88
186	123	75	12	174	111	48	225	162	99	36	198	150	87	24
122	74	11	173	110	47	224	161	98	35	197	149	86	23	185
73	10	172	109	46	223	160	97	34	196	148	85	22	184	121
9	171	108	60	222	159	96	33	210	147	84	21	183	135	72
170	107	59	221	158	95	32	209	146	83	20	182	134	71	8
106	58	220	157	94	31	208	145	82	19	181	133	70	7	169

57	219	156	93	45	207	144	81	18	195	132	69	6	168	120
218	155	92	44	206	143	80	17	194	131	68	5	167	119	56
154	91	43	205	142	79	16	193	130	67	4	166	118	55	217
105	42	204	141	78	30	192	129	66	3	180	117	54	216	153
41	203	140	77	29	191	128	65	2	179	116	53	215	152	104
202	139	76	28	190	127	64	1	178	115	52	214	151	103	40
138	90	27	189	126	63	15	177	114	51	213	165	102	39	201
89	26	188	125	62	14	176	113	50	212	164	101	38	200	137
25	187	124	61	13	175	112	49	211	163	100	37	199	136	88
186	123	75	12	174	111	48	225	162	99	36	198	150	87	24
122	74	11	173	110	47	224	161	98	35	197	149	86	23	185
73	10	172	109	46	223	160	97	34	196	148	85	22	184	121
9	171	108	60	222	159	96	33	210	147	84	21	183	135	72
170	107	59	221	158	95	32	209	146	83	20	182	134	71	8
106	58	220	157	94	31	208	145	82	19	181	133	70	7	169