New Method and Rules for Loubère Type Squares (Part II)

Loubère Square 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... It's also a fact that only one Loubère square per order n has been handed down thru the centuries. In addition, construction of the square requires a one down shift after filling of a diagonal to move to the next diagonal until the square is filled.

Loubère Type Squares of Order 15

This page is a continuation of Part I which contains the rules for construction of these Loubère type squares.

The K shift table for the 15^th order squares are listed below. Only three squares can be produced from these K values, the 4, 6 and the 1, the latter corresponding to the regular Loubère square. Since we know the Loubère is magic only those squares with the initial number 1 at the second and fourteenth positions will be constructed since all other squares with K values in green are either non magic or inconstructible. Again the K table is a table of knight moves for the square whose initial starting number 1 is place at position ½K in the square being constructed.

The K shift table is constructed out of two complementary strands connected at the hinge where the hinge is the regular Loubère square. Moreover, the table is divided into complementary pairs which are used to produce some white (magic) and some green (inconstructible or non magic constructible) squares. To use the table below, one first crosses out the K values which are divisible by 3 or 5 followed by crossing out their complements. Any K values not crossed out (in white) lead to the formation of magic squares. These non crossed values are relatively prime to the order n whether that order is prime or a product of primes. Note that K knight moves of 12 and 5, a complementary pair, result in inconstructible squares.

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

The first 15^th order square is the D4 below with the starting 1 at row 1 cell 2 of the 15×15 square. The headings above row one of the squares are explained in Part I:

D4 (4→ 13←)
2	4	6	8	10	12	14	1	3	5	7	9	11	13	15
173	1	69	122	190	18	86	139	207	35	103	156	224	52	120
15	68	121	189	17	85	138	206	34	102	155	223	51	119	172
67	135	188	16	84	137	205	33	101	154	222	50	118	171	14
134	187	30	83	136	204	32	100	153	221	49	117	170	13	66
186	29	82	150	203	31	99	152	220	48	116	169	12	65	133
28	81	149	202	45	98	151	219	47	115	168	11	64	132	185
80	148	201	44	97	165	218	46	114	167	10	63	131	184	27
147	200	43	96	164	217	60	113	166	9	62	130	183	26	79
199	42	95	163	216	59	112	180	8	61	129	182	25	78	146
41	94	162	215	58	111	179	7	75	128	181	24	77	145	198
93	161	214	57	110	178	6	74	127	195	23	76	144	197	40
160	213	56	109	177	5	73	126	194	22	90	143	196	39	92
212	55	108	176	4	72	125	193	21	89	142	210	38	91	159
54	107	175	3	71	124	192	20	88	141	209	37	105	158	211
106	174	2	70	123	191	19	87	140	208	36	104	157	225	53

The second 15^th order square is the D13 below with the starting 1 at row 1 cell 14 of the 15×15 square:

D13 (13→ 4←)
2	4	6	8	10	12	14	1	3	5	7	9	11	13	15
224	103	207	86	190	69	173	52	156	35	139	18	122	1	120
102	206	85	189	68	172	51	155	34	138	17	121	15	119	223
205	84	188	67	171	50	154	33	137	16	135	14	118	222	101
83	187	66	170	49	153	32	136	30	134	13	117	221	100	204
186	65	169	48	152	31	150	29	133	12	116	220	99	203	82
64	168	47	151	45	149	28	132	11	115	219	98	202	81	185
167	46	165	44	148	27	131	10	114	218	97	201	80	184	63
60	164	43	147	26	130	9	113	217	96	200	79	183	62	166
163	42	146	25	129	8	112	216	95	199	78	182	61	180	59
41	145	24	128	7	111	215	94	198	77	181	75	179	58	162
144	23	127	6	110	214	93	197	76	195	74	178	57	161	40
22	126	5	109	213	92	196	90	194	73	177	56	160	39	143
125	4	108	212	91	210	89	193	72	176	55	159	38	142	21
3	107	211	105	209	88	192	71	175	54	158	37	141	20	124
106	225	104	208	87	191	70	174	53	157	36	140	19	123	2

Two 11^th Order Squares

The next two squares are 11^th order and are complementary to each other. The E8 and E5 have right and left knight moves opposite to one another as seen in the caption above the square. From the K shift table it can be seen that nine of the squares that are generated from these K values are constructible and magic.

11×11 K moves
2	4	6	8	10
					1
11	9	7	5	3

E8 (8→ 5←)
2	4	6	8	10	1	3	5	7	9	11
14	94	53	1	81	40	120	68	27	107	66
93	52	11	80	39	119	67	26	106	65	13
51	10	79	38	118	77	25	105	64	12	92
9	78	37	117	76	24	104	63	22	91	50
88	36	116	75	23	103	62	21	90	49	8
35	115	74	33	102	61	20	89	48	7	87
114	73	32	101	60	19	99	47	6	86	34
72	31	100	59	18	98	46	5	85	44	13
30	110	58	17	97	45	4	84	43	112	71
109	57	16	96	55	3	83	42	111	70	29
56	15	95	54	2	82	41	121	69	28	108

E5 (5→ 8←)
2	4	6	8	10	1	3	5	7	9	11
40	14	120	94	68	53	27	1	107	81	66
13	119	93	67	52	26	11	106	80	65	39
118	92	77	51	25	10	105	79	64	38	12
91	76	50	24	9	104	78	63	37	22	117
75	49	23	8	103	88	62	36	21	116	90
48	33	7	102	87	61	35	20	115	89	74
32	6	101	86	60	34	19	114	99	73	47
5	100	85	59	44	18	113	98	72	46	31
110	84	58	43	17	112	97	71	45	30	4
83	57	42	16	111	96	70	55	29	3	109
56	41	15	121	95	69	54	28	2	108	82

This completes this section (Part II). To return to Part I or to return to homepage.

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

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

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

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

New Method and Rules for Loubère Type Squares (Part II)

Loubère Square Background

Loubère Type Squares of Order 15

Two 11th Order Squares

Two 11^th Order Squares

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

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