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Magic squares and cubes of oddly even order, that is 4m+2, are notorious for being more difficult to construct. That undoubtedly accounts for the fact that my survey of published magic cubes revealed few cubes of order 6. Also there was not much variation of features in those I did discover. Following are seven from a total of 14 cubes in my collection.
| Violle - 1838 | Not magic |
| Kingery - 1909 | Not magic. Six simple magic squares. |
| Sayles - 1910 | Simple magic. Not associated. Magic cubelets. |
| Worthington - 1910 | Simple magic. Six simple magic squares. |
| Abe - 1948 | Pantriagonal (a rarity for order 6). Not associated. |
| Johnson - 1989 | Semi-pantriagonal magic. Associated. |
| Hendricks - 1999 | Not normal. Inlay for an order 10 magic cube. |
This cube is not considered magic by present definition. Even though the 4 triagonals are correct, rows, columns and pillars are not.
The total of the 36 cells in each of the 24 square arrays (the 3 x 6 orthogonal and the 6 oblique) sum to 6 x 651 = 3906. Both diagonals of these 24 arrays also sum correctly to 651.
All 4 triagonals sum correctly to 651. None of the 18 planar squares have any rows or columns that sum to 651! Two of the oblique squares have all rows summing correctly and four have all columns summing correctly. This cube has exactly the same features as the Violle order 4 except it is not associated.
I - Top
II
III
1 32
33 4 35 6 187 188 207 208 191 210 193 200 201 202 197
198
42 68 70 39 71 37 48 44 64 63 47 61 162 164 166
165 161 157
78 107 106 75 104 73 120 119 136 135 116 133 126 131 130
129 122 121
109 143 142 111 140 114 79 83 100 99 80 102 85 95 94
93 86 90
150 176 177 148 179 145 156 152 171 172 155 169 54 56 57
58 53 49
181 215 213 184 212 186 7 11 27 28 8 30 13 23 21
22 14 18
IV
V
VI -
Bottom
19 14
15 16 23 24 205 206 189 190 209 192 31 2 3 34 5
36
168 158 160 159 167 163 66 62 46 45 65 43 180 146 148
177 149 175
132 125 124 123 128 127 102 101 82 81 98 79 108 77 76
105 74 103
91 89 88 87 92 96 133 137 118 117 134 120 139 113 112
141 110 144
60 50 51 52 59 55 174 170 153 154 173 151 72 38 39
70 41 67
199 197 195 196 200 204 25 29 9 10 26 12 211 185 183
214 182 216
Par B. Violle, Traité complet des Carrés Magiques, 1837 (French), pp 539-542.
This cube is not magic by current definition
because no triagonals are correct.
6 horizontal planes are simple magic squares. All orthogonal lines (rows,
columns and pillars) sum correctly to 651.
The special feature of this cube is the bent triagonals
between corners of the cube. Start at 1 corner of a horizontal square
and move toward the center of the cube, then back to the opposite corner of the
starting square to get a bent diagonal.
However, unlike the bent triagonal cubes of order 4, this cube is not
semi-pantriagonal.
Examples starting with the top square:
1 + 191 + 51 + 166 + 26 + 216 = 651 (planes 1, 2, 3)
1 + 119 + 130 + 87 + 98 + 216 = 651 (planes 1, 6, 5)
There are 2 bent triagonals for each pair of corners for each plane.
I - Top
II
III
1 215
214 3 212 6 198 20 21 196 23 193 37 179 178 39 176
42
210 8 208 207 11 7 25 191 27 28 188 192 174 44 172
171 47 43
204 203 15 16 14 199 31 32 184 183 185 36 168 167 51
52 50 163
13 17 201 202 200 18 186 182 34 33 35 181 49 53 165
166 164 54
12 206 9 10 209 205 187 29 190 189 26 30 48 170 45
46 173 169
211 2 4 213 5 216 24 197 195 22 194 19 175 38 40
177 41 180
IV
V
VI-
Bottom
145 71
70 147 68 150 144 74 75 142 77 139 126 92 93 124 95
121
66 152 64 63 155 151 79 137 81 82 134 138 97 119 99
100 116 120
60 59 159 160 158 55 85 86 130 129 131 90 103 104 112
111 113 108
157 161 57 58 56 162 132 128 88 87 89 127 114 110 106
105 107 109
156 62 153 154 65 61 133 83 136 135 80 84 115 101 118
117 98 102
67 146 148 69 149 72 78 143 141 76 140 73 96 125 123
94 122 91
H. M. Kingery, A Magic Cube
of Six, The Monist, 19, 1909, pp434-441
W. S. Andrews, Magic Squares & Cubes, 2nd edition,
Dover Publ. 1960 (1917), pp 189-196.
This is a simple magic cube but has the unique feature that if the cube is divided into 27 2x2x2 cubelets, the six faces of each cubelet and 2 of the 6 diagonal planes will each sum the same value. These 27 values step from 382 to 486.
For example, the top left 2x2x2 cubelet
The six
faces: Two diagonal planes
4 4 4 193
139 85 4 85
85 85 139 112 166 166 166 139
166 112 58 31 31 31 31 58
139 193 193 58 58 112
193 112
394 394 394 394 394 394 394 394
I - Top
II
III
4 139
161 26 174 147 193 58 80 215 39 66 18 153 136 163 23
158
85 166 107 188 93 12 112 31 134 53 120 201 99 180 1
82 104 185
98 152 138 3 103 157 125 71 57 192 130 76 181 19 95
176 171 9
179 17 84 165 184 22 44 206 111 30 49 211 100 154 149
14 90 144
183 21 13 175 89 170 48 210 202 40 116 35 167 5 108
189 172 10
102 156 148 94 8 143 129 75 67 121 197 62 86 140 162
27 91 145
IV
V
VI -
Bottom
207 72
55 28 212 77 155 20 150 15 169 142 74 209 69 204 34
61
126 45 190 109 131 50 101 182 96 177 88 7 128 47 123
42 115 196
46 208 122 41 36 198 6 87 106 187 92 173 195 114 133
52 119 38
127 73 68 203 117 63 141 168 160 25 11 146 60 33 79
214 200 65
32 194 135 54 37 199 151 16 137 83 105 159 70 205 56
110 132 78
113 59 81 216 118 64 97 178 2 164 186 24 124 43 191
29 51 213
H. A. Sayles, A Magic Cube
of Six, The Monist, 20, 1910, pp 299-303
W. S. Andrews, Magic Squares & Cubes, 2nd edition,
Dover Publ. 1960 (1917), page 197.
A simple magic cube but the 2 central orthogonal planes in
each direction are simple magic. These magic squares may be transformed to
the surface of the cube by changing the order of the planes from 1, 2, 3, 4, 5,
6 to 3, 2, 1, 6, 5, 4. By Trumps definition, this would then be an s-magic
cube.
Four oblique arrays of this cube have all columns correct, 2 have rows correct.
This cube is not associated.
I - Top
II
III
147 146
40 38 141 139 150 151 33 35 140 142 115 114 196 200 15
11
149 145 36 39 138 144 148 152 37 34 143 137 113 116 199
195 12 16
136 135 163 165 27 25 129 130 166 164 30 32 7 8 106
109 209 212
132 134 168 162 26 29 133 131 161 167 31 28 6 5 112
107 211 210
45 43 123 122 160 158 44 46 126 127 153 155 208 205 17
18 104 99
42 48 121 125 159 156 47 41 128 124 154 157 202 203 21
22 100 103
IV
V
VI -
Bottom
118 119
197 193 10 14 58 59 96 95 170 173 63 62 89 90 175
172
120 117 194 198 13 9 64 61 92 94 171 169 57 60 93
91 174 176
2 1 111 108 216 213 186 185 50 56 88 86 191 192 55
49 81 83
3 4 105 110 214 215 187 189 53 51 84 87 190 188 52
54 85 82
201 204 24 23 97 102 80 78 181 177 66 69 73 75 180
184 71 68
207 206 20 19 101 98 76 79 179 178 72 67 77 74 182
183 65 70
John Worthington, A Magic
Cube of Six, The Monist, 20, 1910, pp 303-309
W. S. Andrews, Magic Squares & Cubes, 2nd edition,
Dover Publ. 1960 (1917) page 205.
This cube is pantriagonal magic and is not associated. It has no other special characteristics.
However, it is the only pantriagonal cube I have seen for order 6 except for a rotated (disguised) version that appeared on a site by F. Poyo (which is no longer available).
I - Top
II
III
1 144
14 198 118 176 141 173 192 121 17 7 161 67 88 47 186
102
140 172 190 122 18 9 178 6 125 12 193 137 66 107 51
187 83 157
180 5 126 10 194 136 2 142 13 197 120 177 103 156 182
87 52 71
28 135 59 153 109 167 132 164 147 112 62 34 206 22 79
38 213 93
131 163 145 113 63 36 169 33 116 57 148 128 21 98 42
214 74 202
171 32 117 55 149 127 29 133 58 152 111 168 94 201 209
78 43 26
IV
V
VI -
Bottom
64 108
50 189 82 158 105 155 183 85 53 70 179 4 124 11 195
138
104 154 181 86 54 72 160 69 89 48 184 101 3 143 15
196 119 175
162 68 90 46 185 100 65 106 49 188 84 159 139 174 191
123 16 8
19 99 41 216 73 203 96 200 210 76 44 25 170 31 115
56 150 129
95 199 208 77 45 27 205 24 80 39 211 92 30 134 60
151 110 166
207 23 81 37 212 91 20 97 40 215 75 204 130 165 146
114 61 35
From Mutsumi Suzuki's Web site at http://mathforum.com/te/exchange/hosted/suzuki/MagicSquare.html
This is an associated magic cube and so is also semi-pantriagonal. 24 planar arrays have rows and columns correct. Oblique arrays; 4 with columns correct, 2 with rows correct.
I - Top
II
III
32 156
199 140 48 76 192 127 53 81 19 179 55 104 12 166 194
120
42 85 26 150 214 134 13 173 186 121 65 93 206 114 70
98 6 157
190 128 54 79 20 180 59 108 7 167 195 115 36 151 200
144 46 74
122 174 184 14 63 94 207 1 71 99 112 161 40 86 135
148 212 30
57 106 116 168 196 8 139 155 201 34 47 75 188 24 52
80 129 178
208 2 72 100 110 159 41 87 133 149 213 28 126 172 182
15 64 92
IV
V
VI -
Bottom
125 153
202 35 45 91 189 4 68 84 130 176 58 107 117 145 215
9
39 88 137 165 193 29 142 170 183 16 62 78 209 21 49
101 111 160
187 5 69 82 131 177 56 105 118 146 216 10 123 154 203
33 43 95
143 171 73 17 66 181 102 22 50 210 109 158 37 197 138
163 89 27
60 211 119 147 103 11 124 152 96 31 44 204 83 3 67
191 132 175
97 23 51 205 113 162 38 198 136 164 90 25 141 169 77
18 61 185
A. W. Johnson, Jr. Normal Magic Cubes of Order 4m+2 (Letter to the Editor), JRM 21:2, 1989, 101-103.
This is an order-6 associated magic cube. It is not normal because it is an inlay
and uses 216 of the numbers from 112 to 888. It occupies the central position of
an order 10 simple magic cube. Parallel to each face of this cube are 2 inlaid
simple magic squares as part of the shell surrounding the cube.
Although associated, it is NOT semi-pantriagonal, probably because it does not
use consecutive numbers.
I - Top
II
III
889 188
117 114 813 882 782 713 287 284 718 219 319 683 384 617 688
312
122 173 827 824 178 879 779 223 724 277 228 772 629 328 377
374 673 622
139 863 164 837 868 132 269 768 737 734 233 262 362 333 667
664 338 639
862 833 134 167 838 169 239 738 264 767 763 232 332 668 637
634 363 369
172 828 877 874 123 129 222 273 774 727 278 729 679 378 624
327 323 672
819 118 884 187 183 812 712 288 217 214 783 789 682 613 314
387 618 389
IV
V
VI -
Bottom
612 383
614 317 388 689 212 218 787 784 283 719 189 818 814 887 113
182
329 678 674 627 373 322 272 723 274 777 728 229 872 878 127
124 823 179
632 638 367 364 663 339 769 238 234 267 733 762 832 163 834
137 168 869
669 368 337 334 633 662 739 263 764 237 268 732 162 133 867
864 138 839
379 623 324 677 628 372 722 773 227 224 778 279 829 128 177
174 873 822
382 313 687 684 318 619 289 788 717 714 213 282 119 883 184
817 888 112
John R. Hendricks, Inlaid Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7, pp 148-161.
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This page last updated
October 14, 2009
Copyright © 2003 by Harvey D. Heinz