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

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...

The Loubère, Méziriac and Méziriac type Squares

The 5×5 regular Loubère ((O) on the left) and the Méziriac ((P) on the right) squares are shown below with both the starting digit 1 at column 3, row 5 and row 4, respectively. Of these two, only (P) is magic. The square (O), a Loubère type but not the original one contructed by Loubère is non magic and we may state here, so are all other Loubère squares constructible in this section whether n is prime or composite. The 7×7 (Q) and 9×9 (R) Méziriac squares are also shown, where (Q) is magic and (R) is not:

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

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

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

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

We can state here that the reason for non magic is most probably due to the composite nature of n (3×3) as will be shown with other composite n squares below. In addition, all three Méziriac squares (P), (Q) and (R) use a shift of two cells down or a complementary shift of 3, 5 or 5 up, respectively. These squares, however, are not the ones envisioned by Méziriac but are of a new type.

As was shown in Part IA a table was generated for right cell shifts. For these squares, however, the digit 1 is being place south of the central cell and the shifts are the same but in the southward direction as in the following table:

...

S ↓

Note that the first entry, 2, corresponds to a Méziriac and the last entry to a Loubère while all other entries between these two to Méziriac types.

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. If we construct the other two 7×7 squares (S) and (T), only (S) is magic while the Loubère (T) is not.

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

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

The results for the 9×9, where n is composite, are different from those of Part IA. (R) above, as well as those listed in the down shift table below, viz. 6 and 8 are non magic. Only the (U), a 4 down shift, square shown below is magic:

15×15
C	2	4	6	8	S ↓

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

In addition, other squares with composite n may or may not contain viable magic squares. It was found (by actual construction) that for any of the 15×15 types no magic squares was possible but that for the 21×21 types, three were magic - the 4, the 10 and the 16 downward cell shift:

S ↓

where 6, 12 and 18 are impossible to construct due to divisibility by three and the 14, impossible due to divisibility by seven. In other words those cells in white are either non magic or inconstructible. Thus, for composite n there is no way of knowing whether any of its squares are magic unless they are constructed. However, the rules for these squares have been constructed and are available in Part IV.

The following depicts (V) and (W) two out of the five possible prime squares of 11×11 both of which are magic. The Loubère, the digit 1 in the bottom 11^th row (shown rotated by 180^o degrees along the main diagonal at the end of Part III), is the only non magic square in the set.

V (2↓ 9↑)
7	69	21	83	24	97	38	100	52	114	66
68	20	82	23	96	37	110	51	113	65	6
19	81	33	95	36	109	50	112	64	5	67
80	32	94	35	108	49	111	63	4	77	18
31	93	34	107	48	121	62	3	76	17	39
92	44	106	47	120	61	2	75	16	78	30
43	195	46	119	60	1	74	15	88	29	81
104	45	118	59	11	73	14	87	28	90	42
55	117	58	10	72	13	86	27	89	41	103
116	57	9	71	12	85	26	99	40	102	54
56	8	70	22	84	25	98	39	101	53	115

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

The 17^th order Luo-Shu formatted square (known as the Méziriac on this page) is shown in Wikipedia on the subject of water retention on mathematica subjects. The square is of the regular Méziriac type after rotation to conform to the way it is depicted on this page. This would place the starting 1 north of the central cell. The other Méziriac (X) with the starting 1 just south of the central cell is shown below with the shift two cells down instead of two cells up as in the regular Méziriac :

X (2↓ 15↑)
10	156	30	176	50	196	53	216	73	236	93	239	113	259	133	279	153
155	29	175	49	195	52	215	72	235	92	255	112	258	132	278	152	9
28	174	48	194	68	214	71	234	91	254	111	257	131	277	151	8	154
173	47	193	67	213	70	233	90	253	110	256	130	276	150	7	170	27
46	192	66	212	69	232	89	252	109	272	129	275	149	6	169	26	172
191	65	211	85	231	88	251	108	271	128	274	148	5	168	25	171	45
64	210	84	230	87	250	107	270	127	273	147	4	167	24	187	44	190
209	83	229	86	249	106	269	126	289	146	3	166	23	186	43	189	63
82	228	102	248	105	268	125	288	145	2	165	22	185	42	188	62	208
227	101	247	104	267	124	287	144	1	164	21	184	41	204	61	207	81
100	246	103	266	123	286	143	17	163	20	183	40	203	60	206	80	226
245	119	265	122	285	142	16	162	19	182	39	202	59	205	79	225	99
118	264	121	284	141	15	161	18	181	38	201	58	221	78	224	98	244
263	120	283	140	14	160	34	180	37	200	57	220	77	223	97	243	117
136	282	139	13	159	33	179	36	199	56	219	76	222	96	242	116	262
281	138	12	158	32	178	35	198	55	218	75	238	95	241	115	261	135
137	11	157	31	177	51	197	54	217	74	237	94	240	114	260	134	280

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

10	156	30	176	50	196	53	216	73	236	93	239	113	259	133	279	153
155	29	175	49	195	52	215	72	235	92	255	112	258	132	278	152	9
28	174	48	194	68	214	71	234	91	254	111	257	131	277	151	8	154
173	47	193	67	213	70	233	90	253	110	256	130	276	150	7	170	27
46	192	66	212	69	232	89	252	109	272	129	275	149	6	169	26	172
191	65	211	85	231	88	251	108	271	128	274	148	5	168	25	171	45
64	210	84	230	87	250	107	270	127	273	147	4	167	24	187	44	190
209	83	229	86	249	106	269	126	289	146	3	166	23	186	43	189	63
82	228	102	248	105	268	125	288	145	2	165	22	185	42	188	62	208
227	101	247	104	267	124	287	144	1	164	21	184	41	204	61	207	81
100	246	103	266	123	286	143	17	163	20	183	40	203	60	206	80	226
245	119	265	122	285	142	16	162	19	182	39	202	59	205	79	225	99
118	264	121	284	141	15	161	18	181	38	201	58	221	78	224	98	244
263	120	283	140	14	160	34	180	37	200	57	220	77	223	97	243	117
136	282	139	13	159	33	179	36	199	56	219	76	222	96	242	116	262
281	138	12	158	32	178	35	198	55	218	75	238	95	241	115	261	135
137	11	157	31	177	51	197	54	217	74	237	94	240	114	260	134	280

10	156	30	176	50	196	53	216	73	236	93	239	113	259	133	279	153
155	29	175	49	195	52	215	72	235	92	255	112	258	132	278	152	9
28	174	48	194	68	214	71	234	91	254	111	257	131	277	151	8	154
173	47	193	67	213	70	233	90	253	110	256	130	276	150	7	170	27
46	192	66	212	69	232	89	252	109	272	129	275	149	6	169	26	172
191	65	211	85	231	88	251	108	271	128	274	148	5	168	25	171	45
64	210	84	230	87	250	107	270	127	273	147	4	167	24	187	44	190
209	83	229	86	249	106	269	126	289	146	3	166	23	186	43	189	63
82	228	102	248	105	268	125	288	145	2	165	22	185	42	188	62	208
227	101	247	104	267	124	287	144	1	164	21	184	41	204	61	207	81
100	246	103	266	123	286	143	17	163	20	183	40	203	60	206	80	226
245	119	265	122	285	142	16	162	19	182	39	202	59	205	79	225	99
118	264	121	284	141	15	161	18	181	38	201	58	221	78	224	98	244
263	120	283	140	14	160	34	180	37	200	57	220	77	223	97	243	117
136	282	139	13	159	33	179	36	199	56	219	76	222	96	242	116	262
281	138	12	158	32	178	35	198	55	218	75	238	95	241	115	261	135
137	11	157	31	177	51	197	54	217	74	237	94	240	114	260	134	280

10	156	30	176	50	196	53	216	73	236	93	239	113	259	133	279	153
155	29	175	49	195	52	215	72	235	92	255	112	258	132	278	152	9
28	174	48	194	68	214	71	234	91	254	111	257	131	277	151	8	154
173	47	193	67	213	70	233	90	253	110	256	130	276	150	7	170	27
46	192	66	212	69	232	89	252	109	272	129	275	149	6	169	26	172
191	65	211	85	231	88	251	108	271	128	274	148	5	168	25	171	45
64	210	84	230	87	250	107	270	127	273	147	4	167	24	187	44	190
209	83	229	86	249	106	269	126	289	146	3	166	23	186	43	189	63
82	228	102	248	105	268	125	288	145	2	165	22	185	42	188	62	208
227	101	247	104	267	124	287	144	1	164	21	184	41	204	61	207	81
100	246	103	266	123	286	143	17	163	20	183	40	203	60	206	80	226
245	119	265	122	285	142	16	162	19	182	39	202	59	205	79	225	99
118	264	121	284	141	15	161	18	181	38	201	58	221	78	224	98	244
263	120	283	140	14	160	34	180	37	200	57	220	77	223	97	243	117
136	282	139	13	159	33	179	36	199	56	219	76	222	96	242	116	262
281	138	12	158	32	178	35	198	55	218	75	238	95	241	115	261	135
137	11	157	31	177	51	197	54	217	74	237	94	240	114	260	134	280