Office Applications and Entertainment, Magic Squares

20.3   Construction Methods, Inlaid Magic Squares

20.3.1 Simple Magic Squares (14 x 14)
       Generator Method

Simple Magic Squares of order 14 can be constructed very efficiently with the Generator Principle, as applied for the construction of Bimagic Squares (ref. Section 15).

The Generator Method, as applied for Simple Magic Squares of order 14 can be summarised as follows:

• Generate Magic Series for the consecutive integers {1 ... 196} and the related Magic Sum (1379);
• Construct Generators with 14 Magic Rows, based on the Magic Series obtained above;
• Construct Semi Magic Squares, by permutating the numbers within the rows of the Generators;
• Permutate the rows and columns within the Semi Magic Squares, in order to obtain Magic Squares.

Suitable Generators for order 14 Magic Squares can be constructed semi-automatically (ref. CnstrGen14).

An example of a Simple Magic Square, constructed with the generator method described above, is shown below:

Mc14 = 1379

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

The Generator Method, as applied for the Simple Magic Square shown above, is illustrated in Attachment 20.3.11.

The applied Semi Magic Square results in numerous Essential Different Magic Squares. A few sets of potential diagonals are shown in Attachment 20.3.12.

Each Essential Different Simple Magic Square corresponds with 322560 transformations, as described below.

• Any line n can be interchanged with line (15 - n). The possible number of transformations is 2⁷ = 128.
  It should be noted that for each square the 180^o rotated aspect is included in this collection.
  
  
• Any permutation can be applied to the lines 1, 2, ... 7, provided that the same permutation is applied to the lines 14, 13, ... 8. The possible number of transformations is 7! = 5040.

The resulting number of transformations, excluding the 180^o rotated aspects, is 128/2 * 5040 = 322560.

20.3.2 Simple Magic Squares (14 x 14)
       Order 5 Magic Square Inlay

Order 14 Simple Magic Squares with order 5 Square Inlay(s) can be constructed with the generator method.

An example is shown below:

Mc14 = 1379, Mc5 = 685

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

The Generator Method, as applied for the Inlaid Magic Square shown above, is illustrated and described in detail in Attachment 20.3.2.

The applied Square Inlay is an order 5 Associated Magic Square with order 3 Diamond Inlay.

Potential Order 5 Magic Squares with Diamond Inlays might be constructed for the integers {1 ... 196} with routine MgcSqr2032.

Alternatively Order 5 Simple Magic, Associated, Pan Magic or Ultra Magic Squares can be used as Square Inlay(s).

20.3.3 Simple Magic Squares (14 x 14)
       Order 6 Magic Square Inlay

Order 14 Simple Magic Squares with order 6 Square Inlay(s) can be constructed with the generator method.

An example is shown below:

Mc14 = 1379, Mc6 = 684

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

The Generator Method, as applied for the Inlaid Magic Square shown above, is illustrated and described in detail in Attachment 20.3.3.

The applied Square Inlay is an order 6 Most Perfect Magic Square.

Potential Order 6 Most Perfect Magic Squares might be constructed for the integers {1 ... 196} with routine MgcSqr2033.

Alternatively other types of Order 6 Magic Squares (ref. Sections 6) can be used as Square Inlay(s).

20.3.4 Simple Magic Squares (14 x 14)
       Order 7 Magic Square Inlay

Order 14 Simple Magic Squares with order 7 Square Inlay(s) can be constructed with the generator method.

An example is shown below:

Mc14 = 1379, Mc7 = 798

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

The Generator Method, as applied for the Inlaid Magic Square shown above, is illustrated and described in detail in Attachment 20.3.4.

The applied Square Inlay is an order 7 Associated Magic Square with order 3 and 4 Diamond Inlays.

Potential Order 7 Associated Magic Squares - with order 3 and 4 Diamond Inlays - might be constructed for the integers {1 ... 196} with routine MgcSqr2034.

Alternatively other types of Order 7 Magic Squares (ref. Sections 7) can be used as Square Inlay(s).

20.3.5 Simple Magic Squares (14 x 14)
       Order 9 Magic Square Inlay

Order 14 Simple Magic Squares with order 9 Square Inlay(s) can be constructed with the generator method.

An example is shown below:

Mc14 = 1379, Mc9 = 1026

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

The Generator Method, as applied for the Inlaid Magic Square shown above, is illustrated and described in detail in Attachment 20.3.5.

The applied Square Inlay is an order 9 Associated Magic Square with order 4 and 5 Diamond Inlays.

Potential Order 9 Associated Magic Squares - with order 4 and 5 Diamond Inlays - might be constructed as described in Secion 18.4.3.

Also for following example:

the Generator Method, as applied for the Inlaid Magic Square shown above, is illustrated and described in detail in Attachment 20.3.5.

The applied Square Inlay is an order 9 Composed Magic Square with order 3 Square Inlays.

The resulting square shown above corresponds with 8⁶ * 2³ * (5!) * (4!)² = 1,45 10¹¹ solutions, which can be obtained by selecting other aspects of the nine inlays and variation of the (eccentric) border.

Potential Order 9 Composed Magic Squares - with order 3 Square Inlays - might be constructed as described in Secion 9.9.2.

Alternatively other types of Order 9 Magic Squares (ref. Sections 9) can be used as Square Inlay(s).

20.3.6 Simple Magic Squares (14 x 14)
       Order 10 Magic Square Inlay

Order 14 Simple Magic Squares with order 10 Square Inlay(s) can be constructed with a variation on the generator method.

An example is shown below:

Mc14 = 1379, Mc10 = 990

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

the Generator Method, as applied for the Inlaid Magic Square shown above, is illustrated and described in detail in Attachment 20.3.6.

The applied Square Inlay is an order 10 Associated, Compact Pan Magic Square (Non Consecutive Integers) as described in Secion 10.5.2.

The resulting square shown above corresponds with 100 * 1152 * 4 * (10!)² = 6,068 10¹⁸ solutions, which can be obtained by selecting other aspects of the order 10 inlay and variation of the border.

Alternatively other types of Order 10 Magic Squares (ref. Sections 10) can be used as Square Inlay(s).