General Method for Large Order Staircase 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.

This page is a continuation of Part I which contains the general rules for construction of the staircase squares. Also previous methods have shown how to construct these squares but in a specific manner for each of the methods as was depicted for first row, middle column and left diagonal generated squares. Two types of squares will be covered, the prime 11 and the composite 15 orders.

Four Staircase Squares of Order 11

Four 11th order squares were constructed according to the new general method where the initial starting 1's were chosen at random as depicted in H1, H2, H3 and H4. All are magic which is typical for a prime order square, except where the 1 is on the main diagonal, the first column or the last row. In addition, a knight move may be shown to proceed in several manners. For example, one may also consider going via an alternate knight route in order to lower the amount of cell counting during knight breaks, especially when tackling large squares. Thus, H2 (3→ 9↓) may also proceed via (3→ 2↑) by subtracting 9 from 11 and H4 (8→ 4↓) via (3← 4↓) by subtracting 8 by 11 where the direction of one of the arrows in the ordered pair is reversed.

H1 (3→ 6↓)
1	3	5	7	9	0	2	4	6	8	10	k/j
112	48	105	41	98	23	80	16	73	9	66	1
47	104	40	97	33	79	15	72	8	65	111	3
103	39	96	32	78	14	71	7	64	121	46	5
38	95	31	88	13	70	6	63	120	45	102	7
94	30	87	12	69	5	62	119	55	101	37	9
29	86	22	68	4	61	118	54	100	36	93	0
85	21	67	3	60	117	53	110	35	92	28	2
20	77	2	59	116	52	109	34	91	27	84	4
76	1	58	115	51	108	44	90	26	83	19	6
11	57	114	50	107	43	89	25	82	18	75	8
56	113	49	106	42	99	24	81	17	74	10	10

H2 (3→ 9↓)
1	3	5	7	9	0	2	4	6	8	10	k/j
76	86	96	106	116	5	15	25	35	45	66	1
85	95	105	115	4	14	24	34	55	65	75	3
94	104	114	3	13	23	44	54	64	74	84	5
103	113	2	12	33	73	53	63	73	83	93	7
112	1	22	32	42	52	62	72	82	92	102	9
11	21	31	41	51	61	71	81	91	101	111	0
20	30	40	50	60	70	80	90	100	121	10	2
29	39	49	59	69	79	89	110	120	9	19	4
38	48	58	68	78	99	109	119	8	18	28	6
47	57	67	88	98	108	118	7	17	27	37	8
56	77	87	97	107	117	6	16	26	36	46	10

H3 (4→ 2↓)
1	3	5	7	9	0	2	4	6	8	10	k/j
83	100	7	24	52	69	97	114	21	38	66	1
110	6	23	51	68	96	113	20	37	65	82	3
5	33	50	67	95	112	19	36	64	81	109	5
32	49	77	94	111	18	35	63	80	108	4	7
48	76	93	121	17	34	62	79	107	3	31	9
75	92	120	16	44	61	78	106	2	30	47	0
91	119	15	43	60	88	105	1	29	46	74	2
118	14	42	59	87	104	11	28	45	73	90	4
13	41	58	86	103	10	27	55	72	89	117	6
40	57	85	102	9	26	54	71	99	116	12	8
56	84	101	8	25	53	74	98	115	22	39	10

H4 (8→ 4↓)
1	3	5	7	9	0	2	4	6	8	10	k/j
71	87	92	108	113	8	13	29	34	50	66	1
86	91	107	112	7	12	28	44	49	65	70	3
90	106	111	6	22	27	43	48	64	69	85	5
105	121	5	21	26	42	47	63	68	84	89	7
120	4	20	25	41	46	62	67	83	99	104	9
3	19	24	40	45	61	77	82	98	103	119	0
18	23	39	55	60	76	81	97	102	118	2	2
33	38	54	59	75	80	96	101	117	1	17	4
37	53	58	74	79	95	100	116	11	16	32	6
52	57	73	78	94	110	115	10	15	31	36	8
56	72	88	93	109	114	9	14	30	35	51	10

Two Staircase Squares of Order 15

The following two squares of order 15 are L1 and L2 both magic.

L1 (0→ 7↓)
1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
89	58	27	221	190	159	128	97	66	35	4	198	167	136	120	1
57	26	220	189	158	127	96	65	34	3	197	166	150	119	88	3
25	219	188	157	126	95	64	33	2	196	180	149	118	87	56	5
218	187	156	125	94	63	32	1	210	179	148	117	86	55	24	7
186	155	124	93	62	31	15	209	178	147	116	85	54	23	217	9
154	123	92	61	45	14	208	177	146	115	84	53	22	216	185	11
122	91	75	44	13	207	176	145	114	83	52	21	215	184	153	13
105	74	43	12	206	175	144	113	82	51	20	214	183	152	121	0
73	42	11	205	174	143	112	81	50	19	213	182	151	135	104	2
41	10	204	173	142	111	80	49	18	212	181	165	134	103	72	4
9	203	172	141	110	79	48	17	211	195	164	133	102	71	40	6
202	171	140	109	78	47	16	225	194	163	132	101	70	39	8	8
170	139	108	77	46	30	224	193	162	131	100	69	38	7	201	10
138	107	76	60	29	223	192	161	130	99	68	37	6	200	169	12
106	90	59	28	222	191	160	129	98	67	36	5	199	168	137	14

L2 (3→ 10↓) or (3→ 5↑)
1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
212	94	201	83	190	72	179	46	153	35	142	24	131	13	120	1
93	200	82	189	71	178	60	152	34	141	23	130	12	119	211	3
199	81	188	70	177	59	151	33	140	22	129	11	118	225	92	5
80	187	69	176	58	165	32	139	21	128	10	117	224	91	198	7
186	68	175	57	164	31	138	20	127	9	116	223	105	197	79	9
67	174	56	163	45	137	19	126	8	115	222	104	196	78	185	11
173	55	162	44	136	18	125	7	114	221	103	210	77	184	66	13
54	161	43	150	17	124	6	113	220	102	209	76	183	65	172	0
160	422	149	16	123	5	112	219	101	208	90	182	64	171	53	2
41	148	30	122	4	111	218	100	207	89	181	63	170	52	159	4
147	29	121	3	110	217	99	206	88	195	62	169	51	158	40	6
28	135	2	109	216	98	205	87	194	61	168	50	157	39	146	8
134	1	108	215	97	204	86	193	75	167	49	156	38	145	27	10
15	107	214	96	203	85	192	74	166	48	155	37	144	26	133	12
106	213	95	202	84	191	73	180	47	154	36	143	25	132	14	14

As stated previously all the composite squares generate a lower number of magic squares. Two 15x15 squares that are not magic (not shown) have their initial 1 at k/j position (7,1) and (0,4). The former has an M=31 in column 1, and the latter an M=16 on row 15.

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

1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
89	58	27	221	190	159	128	97	66	35	4	198	167	136	120	1
57	26	220	189	158	127	96	65	34	3	197	166	150	119	88	3
25	219	188	157	126	95	64	33	2	196	180	149	118	87	56	5
218	187	156	125	94	63	32	1	210	179	148	117	86	55	24	7
186	155	124	93	62	31	15	209	178	147	116	85	54	23	217	9
154	123	92	61	45	14	208	177	146	115	84	53	22	216	185	11
122	91	75	44	13	207	176	145	114	83	52	21	215	184	153	13
105	74	43	12	206	175	144	113	82	51	20	214	183	152	121	0
73	42	11	205	174	143	112	81	50	19	213	182	151	135	104	2
41	10	204	173	142	111	80	49	18	212	181	165	134	103	72	4
9	203	172	141	110	79	48	17	211	195	164	133	102	71	40	6
202	171	140	109	78	47	16	225	194	163	132	101	70	39	8	8
170	139	108	77	46	30	224	193	162	131	100	69	38	7	201	10
138	107	76	60	29	223	192	161	130	99	68	37	6	200	169	12
106	90	59	28	222	191	160	129	98	67	36	5	199	168	137	14

1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
212	94	201	83	190	72	179	46	153	35	142	24	131	13	120	1
93	200	82	189	71	178	60	152	34	141	23	130	12	119	211	3
199	81	188	70	177	59	151	33	140	22	129	11	118	225	92	5
80	187	69	176	58	165	32	139	21	128	10	117	224	91	198	7
186	68	175	57	164	31	138	20	127	9	116	223	105	197	79	9
67	174	56	163	45	137	19	126	8	115	222	104	196	78	185	11
173	55	162	44	136	18	125	7	114	221	103	210	77	184	66	13
54	161	43	150	17	124	6	113	220	102	209	76	183	65	172	0
160	422	149	16	123	5	112	219	101	208	90	182	64	171	53	2
41	148	30	122	4	111	218	100	207	89	181	63	170	52	159	4
147	29	121	3	110	217	99	206	88	195	62	169	51	158	40	6
28	135	2	109	216	98	205	87	194	61	168	50	157	39	146	8
134	1	108	215	97	204	86	193	75	167	49	156	38	145	27	10
15	107	214	96	203	85	192	74	166	48	155	37	144	26	133	12
106	213	95	202	84	191	73	180	47	154	36	143	25	132	14	14

1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
89	58	27	221	190	159	128	97	66	35	4	198	167	136	120	1
57	26	220	189	158	127	96	65	34	3	197	166	150	119	88	3
25	219	188	157	126	95	64	33	2	196	180	149	118	87	56	5
218	187	156	125	94	63	32	1	210	179	148	117	86	55	24	7
186	155	124	93	62	31	15	209	178	147	116	85	54	23	217	9
154	123	92	61	45	14	208	177	146	115	84	53	22	216	185	11
122	91	75	44	13	207	176	145	114	83	52	21	215	184	153	13
105	74	43	12	206	175	144	113	82	51	20	214	183	152	121	0
73	42	11	205	174	143	112	81	50	19	213	182	151	135	104	2
41	10	204	173	142	111	80	49	18	212	181	165	134	103	72	4
9	203	172	141	110	79	48	17	211	195	164	133	102	71	40	6
202	171	140	109	78	47	16	225	194	163	132	101	70	39	8	8
170	139	108	77	46	30	224	193	162	131	100	69	38	7	201	10
138	107	76	60	29	223	192	161	130	99	68	37	6	200	169	12
106	90	59	28	222	191	160	129	98	67	36	5	199	168	137	14

1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
212	94	201	83	190	72	179	46	153	35	142	24	131	13	120	1
93	200	82	189	71	178	60	152	34	141	23	130	12	119	211	3
199	81	188	70	177	59	151	33	140	22	129	11	118	225	92	5
80	187	69	176	58	165	32	139	21	128	10	117	224	91	198	7
186	68	175	57	164	31	138	20	127	9	116	223	105	197	79	9
67	174	56	163	45	137	19	126	8	115	222	104	196	78	185	11
173	55	162	44	136	18	125	7	114	221	103	210	77	184	66	13
54	161	43	150	17	124	6	113	220	102	209	76	183	65	172	0
160	422	149	16	123	5	112	219	101	208	90	182	64	171	53	2
41	148	30	122	4	111	218	100	207	89	181	63	170	52	159	4
147	29	121	3	110	217	99	206	88	195	62	169	51	158	40	6
28	135	2	109	216	98	205	87	194	61	168	50	157	39	146	8
134	1	108	215	97	204	86	193	75	167	49	156	38	145	27	10
15	107	214	96	203	85	192	74	166	48	155	37	144	26	133	12
106	213	95	202	84	191	73	180	47	154	36	143	25	132	14	14

1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
89	58	27	221	190	159	128	97	66	35	4	198	167	136	120	1
57	26	220	189	158	127	96	65	34	3	197	166	150	119	88	3
25	219	188	157	126	95	64	33	2	196	180	149	118	87	56	5
218	187	156	125	94	63	32	1	210	179	148	117	86	55	24	7
186	155	124	93	62	31	15	209	178	147	116	85	54	23	217	9
154	123	92	61	45	14	208	177	146	115	84	53	22	216	185	11
122	91	75	44	13	207	176	145	114	83	52	21	215	184	153	13
105	74	43	12	206	175	144	113	82	51	20	214	183	152	121	0
73	42	11	205	174	143	112	81	50	19	213	182	151	135	104	2
41	10	204	173	142	111	80	49	18	212	181	165	134	103	72	4
9	203	172	141	110	79	48	17	211	195	164	133	102	71	40	6
202	171	140	109	78	47	16	225	194	163	132	101	70	39	8	8
170	139	108	77	46	30	224	193	162	131	100	69	38	7	201	10
138	107	76	60	29	223	192	161	130	99	68	37	6	200	169	12
106	90	59	28	222	191	160	129	98	67	36	5	199	168	137	14

1	3	5	7	9	11	13	0	2	4	6	8	10	12	14	k/j
212	94	201	83	190	72	179	46	153	35	142	24	131	13	120	1
93	200	82	189	71	178	60	152	34	141	23	130	12	119	211	3
199	81	188	70	177	59	151	33	140	22	129	11	118	225	92	5
80	187	69	176	58	165	32	139	21	128	10	117	224	91	198	7
186	68	175	57	164	31	138	20	127	9	116	223	105	197	79	9
67	174	56	163	45	137	19	126	8	115	222	104	196	78	185	11
173	55	162	44	136	18	125	7	114	221	103	210	77	184	66	13
54	161	43	150	17	124	6	113	220	102	209	76	183	65	172	0
160	422	149	16	123	5	112	219	101	208	90	182	64	171	53	2
41	148	30	122	4	111	218	100	207	89	181	63	170	52	159	4
147	29	121	3	110	217	99	206	88	195	62	169	51	158	40	6
28	135	2	109	216	98	205	87	194	61	168	50	157	39	146	8
134	1	108	215	97	204	86	193	75	167	49	156	38	145	27	10
15	107	214	96	203	85	192	74	166	48	155	37	144	26	133	12
106	213	95	202	84	191	73	180	47	154	36	143	25	132	14	14