Novel Four 11th 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.

Staircase Squares of Order 11

This page is a continuation of Part I which contains the rules for construction of the new Staircase squares.

The K shift table for the 11^th order squares are listed below and is constructed of two complementary strands connected at the hinge where the hinge is the K for the central cell. Moreover, the table is divided into complementary pairs of knight break values (K) which consists of white cells that give rise to magic squares and green cells that give rise to non magic squares. In addition, the K values in white are relatively prime to the order n.

The K values (knight breaks) of the four squares of 11^th order and their four identical complements are shown in the complementary table and the squares constructed from these values, D1, D2, D3 and D4, are shown below depict the first break move, 11 → 12, (green to blue). The next break moves are similar. :

11×11 K moves
1	3	5	7	9
					0
10	8	6	4	2

D1 (3→ 8←)
1	3	5	7	9	0	2	4	6	8	10
85	104	2	32	51	70	89	119	17	36	66
103	1	31	50	69	99	118	16	35	65	84
11	30	49	68	98	117	15	34	64	83	102
29	48	67	97	116	14	44	63	82	101	10
47	77	96	115	13	43	62	81	100	9	28
76	95	114	12	42	61	80	110	8	27	46
94	113	22	41	60	79	109	7	26	45	75
112	21	40	59	78	108	6	25	55	74	93
20	39	58	88	107	5	24	54	73	92	111
38	57	87	106	4	23	53	72	91	121	19
56	86	105	3	33	52	71	90	120	18	37

D2 (5→ 6←)
1	3	5	7	9	0	2	4	6	8	10
49	43	26	20	3	118	101	95	78	72	66
42	25	19	2	117	100	94	88	71	65	48
24	18	1	116	110	93	87	70	64	47	41
17	11	115	109	92	86	69	63	46	40	23
10	114	108	91	85	68	62	45	39	33	16
113	107	90	84	67	61	55	38	32	15	9
106	89	83	77	60	54	37	31	14	8	112
99	82	76	59	53	36	30	13	7	111	105
81	75	58	52	35	29	12	6	121	104	98
74	57	51	34	28	22	5	120	103	97	80
56	50	44	27	21	4	119	102	96	79	73

D3 (7→ 4←)
1	3	5	7	9	0	2	4	6	8	10
109	31	74	117	39	82	4	47	90	12	66
30	73	116	38	81	3	46	89	22	65	108
72	115	37	80	2	45	99	21	64	107	29
114	36	79	1	55	98	20	63	106	28	71
35	78	11	54	97	19	62	105	27	70	113
88	10	53	96	18	61	104	26	69	112	34
9	52	95	17	60	103	25	68	111	44	87
51	94	16	59	102	24	67	121	43	86	8
93	15	58	101	23	77	120	42	85	7	50
14	57	100	33	76	119	41	84	6	49	92
56	110	32	75	118	40	83	5	48	91	13

E4 (9→ 2←)
1	3	5	7	9	0	2	4	6	8	10
25	116	86	45	15	106	76	35	5	96	66
115	85	55	14	105	75	34	4	95	65	24
84	54	13	104	74	44	3	94	64	23	114
53	12	103	73	43	2	93	63	33	113	83
22	102	72	42	1	92	62	32	112	82	52
101	71	41	11	91	61	31	111	81	51	21
70	40	10	90	60	30	121	80	50	20	100
39	9	89	59	29	120	79	49	19	110	69
8	99	58	28	119	78	48	18	109	68	38
98	57	27	118	88	47	17	108	67	37	7
56	26	117	87	46	16	107	77	36	6	97

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