Office Applications and Entertainment, Magic Squares

12.10   Magic Squares (15 x 15)

12.10.1 Composed Magic Squares, Overlapping Sub Squares

A classical Magic Square of order 15, with miscellaneous each other Overlapping Sub Squares is shown below:

Mc15 = 1695

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

The Magic Square shown above is composed out of:

• One 3^th order Magic Center Square C;
• Two each other overlapping 5^th order Eccentric Magic Squares A1 and A2;
• Two each other overlapping 7^th order Eccentric Magic Squares B1 and B2;
• Two 4^th order Pan Magic Squares PM1 and PM2;
• Two 6^th order Eccentric Magic Squares F1 and F2 with embedded PM1 and PM2;
• Two 9^th order Eccentric Magic Squares D1 and D2 with embedded B1 and B2;

The construction of order 15 Magic Squares with miscellaneous each other Overlapping Sub Squares has been described in detail in Section 11.2.2.

Attachment 14.9.8.2 shows a few order 15 Composed Magic Squares with Overlapping Sub Squares, as defined above.

Alternatively order 15 Magic Squares, with two each other overlapping 8^th order Sub Squares, can be constructed based on suitable selected Latin Sub Squares, as illustrated in Section 25.6.

Attachment 25.6.2 shows a few 15^th order Associated Magic Squares with 8^th order Overlapping Sub Squares (Composed).

Attachment 25.6.3 shows a few 15^th order Associated Magic Squares with 8^th order Overlapping Sub Squares (Composed, Magic Middle and Center Squares).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the Latin Sub Squares.

12.10.2 Composed Magic Squares, Square Inlays Order 7 and 8

Associated Magic Squares of order 15 with Square Inlays of order 7 and 8 can be obtained by means of transformation of order 15 Composed Magic Squares, as illustrated in Section 7.8.1 for order 7 Magic Squares.

Mc15 = 1695

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

The Composed Semi Magic Square shown above is composed out of:

• One 7^th order Ultra Magic Corner Square (s7 = 7 * s1 / 15)
  
• One 8^th order Associated (Compact) Magic Corner Square (s8 = 8 * s1 / 15),
• Two Associated Magic Rectangles order 7 x 8
  each with two Embedded order 4 Semi Magic Squares (s4 = 4 * s1 / 15).

The Magic Corner Squares can be constructed by means of suitable selected Latin Squares, based on resp. order 7 and 8 Magic Lines for the integers 0 ... 14 as shown in Attachment 12.10.1a.

Based on the definition above, a routine can be developed to complete subject Composed Magic Squares (ref. MgcSqr15c).

Attachment 12.10.1b shows miscellaneous order 15 Composed Semi Magic Square, which could be found with subject routine.

Attachment 12.10.2 shows the resulting order 15 Associated Magic Squares with order 7 and 8 Square Inlays.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the Latin Sub Squares.

12.10.3 Associated Magic Squares

Associated Center Square Order 7

Associated Magic Squares of order 15 with an Associated Center Square of order 7 can be obtained by means of transformation of order 15 Composed Magic Squares as illustrated in Section 9.7.5 for order 9 Magic Squares.

Mc15 = 1695

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

Attachment 12.10.4 shows the Associated Magic Squares with order 7 Associated Center Squares, corresponding with the Composed Semi Magic Squares as shown in Attachment 12.10.1b.

12.10.4 Associated Magic Squares

Square Inlays Order 6 and 7 (overlapping)

The 15^th order Associated Inlaid Magic Square shown below:

Mc15 = 1695 152 221 215 163 146 140 110 67 20 50 56 73 125 131 26 162 129 201 119 12 83 168 31 224 46 104 136 181 29 70 160 89 177 38 123 196 114 6 27 192 93 138 57 213 72 157 203 108 1 84 171 44 132 21 54 144 99 186 216 75 71 166 39 126 209 117 8 78 219 51 141 96 189 24 161 68 111 14 87 173 33 121 204 18 183 147 102 48 222 164 41 42 128 198 106 9 81 179 211 194 91 149 59 16 191 61 3 76 174 36 134 207 113 19 92 190 52 150 223 165 35 210 167 77 135 32 15 47 145 217 120 28 98 184 185 62 4 178 124 79 43 208 22 105 193 53 139 212 115 158 65 202 37 130 85 175 7 148 218 109 17 100 187 60 155 151 10 40 127 82 172 205 94 182 55 142 225 118 23 69 154 13 169 88 133 34 199 220 112 30 103 188 49 137 66 156 197 45 90 122 180 2 195 58 143 214 107 25 97 64 200 95 101 153 170 176 206 159 116 86 80 63 11 5 74

Mc's 743 720 636 839

contains following inlays:

• Two each 7^th order Simple Magic Squares - Magic Sums s(1) = 743 and s(4) = 839 - with the center element in common,
• Two each 6^th order Simple Magic Squares - Magic Sums s(2) = 720 and s(3) = 636 - with symmetrical diagonals.

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

 s(1) = 14 * s1 / 15 - s(4)
 s(2) = 12 * s1 / 15 - s(3)

With s1 = 1695 the Magic Sum of the 15^th order Inlaid Magic Square.

The Associated Border can be described by following linear equations:

a(217) =    - s1 / 15 + a(219) - s(3) + s(4)
a(216) =    - s1 / 15 + a(220) - s(3) + s(4)
a(215) =    - s1 / 15 + a(221) - s(3) + s(4)
a(214) =    - s1 / 15 + a(222) - s(3) + s(4)
a(213) =    - s1 / 15 + a(223) - s(3) + s(4) 
a(212) =    - s1 / 15 + a(224) - s(3) + s(4)
a(211) =      s1      - a(212) - a(213) - a(214) - a(215) - a(216) - a(217) - a(218) - a(219) +
                                                          - a(220) - a(221) - a(222) - a(223) - a(224) - a(225)
a(196) =      s1      - a(210) - s(3) - s(4)
a(181) =      s1      - a(195) - s(3) - s(4)
a(166) =      s1      - a(180) - s(3) - s(4)
a(151) =      s1      - a(165) - s(3) - s(4)
a(136) =      s1      - a(150) - s(3) - s(4)
a(121) =      s1      - a(135) - s(3) - s(4)
a(120) =      s1      - a( 15) - a(30) - a(45) - a(60) - a(75) - a( 90) - a(105) - a(135) - a(150) +
                                                               - a(165) - a(180) - a(195) - a(210) - a(225)

Which can be incorporated in an optimised guessing routine MgcSqr15k1.

The Magic Center Squares can be constructed by means of suitable selected (Semi-) Latin Squares, based on resp. order 6 and 7 Magic Lines for the integers 0 ... 14 as shown in Attachment 12.10.5a.

Attachment 12.10.5b shows a few 15^th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

12.10.5 Composed Magic Squares, Associated

Diamond Inlays Order 5

The 15^th order Composed Inlaid Magic Square shown below, contains nine each order 5 Sub Squares with Diamond Inlay.

The nine Sub Squares contain each 25 consecutive integers, with corresponding Magic Sum (shown right).

Mc15 = 1695 150 143 135 132 130 128 140 131 142 149 139 129 138 147 137 127 134 145 136 148 146 144 141 133 126 25 18 10 7 5 3 15 6 17 24 14 4 13 22 12 2 9 20 11 23 21 19 16 8 1 200 193 185 182 180 178 190 181 192 199 189 179 188 197 187 177 184 195 186 198 196 194 191 183 176 175 168 160 157 155 153 165 156 167 174 164 154 163 172 162 152 159 170 161 173 171 169 166 158 151 125 118 110 107 105 103 115 106 117 124 114 104 113 122 112 102 109 120 111 123 121 119 116 108 101 75 68 60 57 55 53 65 56 67 74 64 54 63 72 62 52 59 70 61 73 71 69 66 58 51 50 43 35 32 30 28 40 31 42 49 39 29 38 47 37 27 34 45 36 48 46 44 41 33 26 225 218 210 207 205 203 215 206 217 224 214 204 213 222 212 202 209 220 211 223 221 219 216 208 201 100 93 85 82 80 78 90 81 92 99 89 79 88 97 87 77 84 95 86 98 96 94 91 83 76

MC5 690 65 940 815 565 315 190 1065 440

Methods to construct Magic Squares composed of Magic Sub Squares have been discussed in detail in Section 9.9.1.

Att 18.4.01 Sht. 2, provides some additional examples of order 15 Magic Squares, composed of nine order 5 Sub Squares for miscellaneous types.

12.10.6 Concentric Magic Squares, Diamond Inlay Order 8

The 15^th order Concentric Inlaid Magic Square shown below, contains one each 8^th order Diamond Inlay (s8 = 904).

Mc15 = 1695

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

As the order 8 Diamond Inlay contain only odd numbers, the Concentric Inlaid Magic Square is a Lozenge Square.

The method to generate order 15 Concentric Lozenge Squares with order 8 Diamond Inlays has been discussed in detail in Section 18.8.3.

Attachment Lozenge 15.2 shows a few more order 15 Concentric Lozenge Squares with order 8 Diamond Inlays.

12.10.8 Summary

The obtained results regarding the miscellaneous types of order 15 Magic Squares as deducted and discussed in previous sections are summarised in following table:

Comparable routines as listed above, can be used to generate alternative types of order 17 Magic Squares.