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A 4x4x4 magic cube consists of an array of cells that each
hold a number. These numbers are such that each row, column, pillar, and each of
the four triagonals sum to the constant.
Imagine a structure such that each cell was actually itself a small cube. If we
place a number on each of the six surfaces of each cubelet, it is possible to
have 6 magic cubes, one of which is represented by each face of the cubelets.
Herein is described a physical model of such a cube.
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This model is constructed
using sixty-four 3/4" wooden blocks and connected by 1/8" hardwood dowels
showing rows, columns, pillars and triagonals. Numbers used are 1 to 384,
which is 43 times 6. This model was completed on July 28, 2002
The six faces of each cubelet are each painted a
different color. The 64 cubelet faces of each color form a pantriagonal
magic cube with two special features.
It is compact because all 2x2 square arrays within the cube, in all 3
orientations, sum to the constant. This includes wraparound.
It is complete because every pantriagonal contains m/2 complement
pairs spaced m/2 apart.
Each of the six cubes is also pantriagonal, because all two and three
segment lines parallel to each of the 4 triagonals sums to the correct
value!
Unfortunately, the magic constants for these 6
squares are not the same, but vary from 760 to 780. They are:
White = 760, Blue = 764, Red = 768, Pink = 772, Green = 776, Yellow = 780.
Some examples of the 2x2 square arrays are: white, 271 + 133 + 115 + 241 =
760; green, 305 + 95 + 77 + 299 = 776; green (wrap-around), 101 + 281 + 95
+ 299 = 776. Compare these with the large copy of picture 1.
Combinations in each cube that equal the magic
constant are: rows, columns, pillars = 3 x 42, pantriagonals =
4m2, so total lines = 112. 2x2 squares that sum
correctly: 43 x 3 orientations = 192. total combinations for
each cube = 304.
The six faces of each of the 64 cubelets sum to 1155. This is because the
numbers appearing on opposite faces of each cubelet are members of a
complement pair (3 times 385 = 1155).
(Click on a
picture for an enlarged view.) |
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The Six cubes listed
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White S = 760
Top Top – 1
1
355
217
187
367
37
151
205
373
31
157
199
19
337
235
169
55
301
271
133
313
91
97
259
331
73
115
241
61
295
277
127
Bottom +
1 Bottom
109
247
325
79
283
121
67
289
265
139
49
307
103
253
319
85
163
193
379
25
229
175
13
343
223
181
7
349
145
211
361
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Blue S = 764
Top
Top - 1 Bottom + 1 Bottom
2
356
218
188
368
38
152
206
110
248
326
80
284
122
68
290
374
32
158
200
20
338
236
170
266
140
50
308
104
254
320
86
56
302
272
134
314
92
98
260
164
194
380
26
230
176
14
344
332
74
116
242
62
296
278
128
224
182
8
350
146
212
362
44
Red S = 768
Top
Top - 1 Bottom + 1 Bottom
3
357
219
189
369
39
153
207
111
249
327
81
285
123
69
291
375
33
159
201
21
339
237
171
267
141
51
309
105
255
321
87
57
303
273
135
315
93
99
261
165
195
381
27
231
177
15
345
333 75 117 243 63
297 279 129
225
183 9 351
147
213 363 45
Pink S = 772
Top
Top - 1 Bottom + 1 Bottom
94 316
262 100 304 58 136 274 178 232 346 16 196 166 28 382
298
64 130 280 76
334 244 118
214
148 46 364
184
226 352 10
40 370 208 154
358
4 190 220 124
286 292 70
250
112 82 328
340 22 172 238
34
376 202 160
256
106 88 322
142
268 310 52
Green S = 776
Top
Top - 1 Bottom + 1 Bottom
95
317
263
101
305
59
137
275
179
233
347
17
197
167
29
383
299
65
131
281
77
335
245
119
215
149
47
365
185
227
353
11
41
371
209
155
359
5
191
221
125
287
293
71
251
113
83
329
341 23 173 239
35
377 203 161
257
107 89 323
143
269 311 53
Yellow S = 780
Top
Top - 1 Bottom + 1 Bottom
240
174
24
342
162
204
378
36
324
90
108
258
54
312
270
144
156
210
372
42
222
192
6
360
72
294
288
126
330
84
114
252
282
132
66
300
120
246
336
78
366
48
150
216
12
354
228
186
102
264 318 96
276
138 60 306
18
348 234 180
384
30 168 198
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