Pandiagonal magic & symmetric 7x7x7 cube (shift method)

Use the same method to construct the panmagic 5x5 square (shift) to construct a symmetric & panmagic 7x7x7 cube.

Choose as first row of the first grid in the first level: 3-4-5-6-0-1-2. Construct row 2 up to 7 of the first grid of the first level by shifting the first row each time 4 places to the left. Construct the first grid of level 2 up to 7 by shifting the columns of the first grid of the first level each time 2 places to the left.

Choose as first row of the second grid in the first level: 6-0-1-2-3-4-5. Construct row 2 up to 7 of the second grid of the first level by shifting the first row each time 4 places to the right. Construct the second grid of level 2 up to 7 by shifting the columns of the second grid of the first level each time 2 places to the left.

The third grid is the same as the second grid, but the levels are put in reversed order (7 up to 1 instead of 1 up to 7).

Take 1x number from first grid + 7x number from second grid + 49x number from third grid to get a symmetric & panmagic 7x7x7 cube.

*1x number +1 [level 1] + 7x number [Level 1] + 49x number [Level 1]*
3	4	5	6	0	1	2	6	0	1	2	3	4	5	4	5	6	0	1	2	3
0	1	2	3	4	5	6	2	3	4	5	6	0	1	0	1	2	3	4	5	6
4	5	6	0	1	2	3	5	6	0	1	2	3	4	3	4	5	6	0	1	2
1	2	3	4	5	6	0	1	2	3	4	5	6	0	6	0	1	2	3	4	5
5	6	0	1	2	3	4	4	5	6	0	1	2	3	2	3	4	5	6	0	1
2	3	4	5	6	0	1	0	1	2	3	4	5	6	5	6	0	1	2	3	4
6	0	1	2	3	4	5	3	4	5	6	0	1	2	1	2	3	4	5	6	0

*1x number +1 [level 2] + 7x number [Level 2] + 49x number [Level 2]*
5	6	0	1	2	3	4	1	2	3	4	5	6	0	2	3	4	5	6	0	1
2	3	4	5	6	0	1	4	5	6	0	1	2	3	5	6	0	1	2	3	4
6	0	1	2	3	4	5	0	1	2	3	4	5	6	1	2	3	4	5	6	0
3	4	5	6	0	1	2	3	4	5	6	0	1	2	4	5	6	0	1	2	3
0	1	2	3	4	5	6	6	0	1	2	3	4	5	0	1	2	3	4	5	6
4	5	6	0	1	2	3	2	3	4	5	6	0	1	3	4	5	6	0	1	2
1	2	3	4	5	6	0	5	6	0	1	2	3	4	6	0	1	2	3	4	5

*1x number +1 [level 3] + 7x number [Level 3] + 49x number [Level 3]*
0	1	2	3	4	5	6	3	4	5	6	0	1	2	0	1	2	3	4	5	6
4	5	6	0	1	2	3	6	0	1	2	3	4	5	3	4	5	6	0	1	2
1	2	3	4	5	6	0	2	3	4	5	6	0	1	6	0	1	2	3	4	5
5	6	0	1	2	3	4	5	6	0	1	2	3	4	2	3	4	5	6	0	1
2	3	4	5	6	0	1	1	2	3	4	5	6	0	5	6	0	1	2	3	4
6	0	1	2	3	4	5	4	5	6	0	1	2	3	1	2	3	4	5	6	0
3	4	5	6	0	1	2	0	1	2	3	4	5	6	4	5	6	0	1	2	3

*1x number +1 [level 4] + 7x number [Level 4] + 49x number [Level 4]*
2	3	4	5	6	0	1	5	6	0	1	2	3	4	5	6	0	1	2	3	4
6	0	1	2	3	4	5	1	2	3	4	5	6	0	1	2	3	4	5	6	0
3	4	5	6	0	1	2	4	5	6	0	1	2	3	4	5	6	0	1	2	3
0	1	2	3	4	5	6	0	1	2	3	4	5	6	0	1	2	3	4	5	6
4	5	6	0	1	2	3	3	4	5	6	0	1	2	3	4	5	6	0	1	2
1	2	3	4	5	6	0	6	0	1	2	3	4	5	6	0	1	2	3	4	5
5	6	0	1	2	3	4	2	3	4	5	6	0	1	2	3	4	5	6	0	1

*1x number +1 [level 5] + 7x number [Level 5] + 49x number [Level 5]*
4	5	6	0	1	2	3	0	1	2	3	4	5	6	3	4	5	6	0	1	2
1	2	3	4	5	6	0	3	4	5	6	0	1	2	6	0	1	2	3	4	5
5	6	0	1	2	3	4	6	0	1	2	3	4	5	2	3	4	5	6	0	1
2	3	4	5	6	0	1	2	3	4	5	6	0	1	5	6	0	1	2	3	4
6	0	1	2	3	4	5	5	6	0	1	2	3	4	1	2	3	4	5	6	0
3	4	5	6	0	1	2	1	2	3	4	5	6	0	4	5	6	0	1	2	3
0	1	2	3	4	5	6	4	5	6	0	1	2	3	0	1	2	3	4	5	6

*1x number +1 [level 6] + 7x number [Level 6] + 49x number [Level 6]*
6	0	1	2	3	4	5	2	3	4	5	6	0	1	1	2	3	4	5	6	0
3	4	5	6	0	1	2	5	6	0	1	2	3	4	4	5	6	0	1	2	3
0	1	2	3	4	5	6	1	2	3	4	5	6	0	0	1	2	3	4	5	6
4	5	6	0	1	2	3	4	5	6	0	1	2	3	3	4	5	6	0	1	2
1	2	3	4	5	6	0	0	1	2	3	4	5	6	6	0	1	2	3	4	5
5	6	0	1	2	3	4	3	4	5	6	0	1	2	2	3	4	5	6	0	1
2	3	4	5	6	0	1	6	0	1	2	3	4	5	5	6	0	1	2	3	4

*1x number +1 [level 7] + 7x number [Level 7] + 49x number [Level 7]*
1	2	3	4	5	6	0	4	5	6	0	1	2	3	6	0	1	2	3	4	5
5	6	0	1	2	3	4	0	1	2	3	4	5	6	2	3	4	5	6	0	1
2	3	4	5	6	0	1	3	4	5	6	0	1	2	5	6	0	1	2	3	4
6	0	1	2	3	4	5	6	0	1	2	3	4	5	1	2	3	4	5	6	0
3	4	5	6	0	1	2	2	3	4	5	6	0	1	4	5	6	0	1	2	3
0	1	2	3	4	5	6	5	6	0	1	2	3	4	0	1	2	3	4	5	6
4	5	6	0	1	2	3	1	2	3	4	5	6	0	3	4	5	6	0	1	2

*Sym. & panm. 7x7x7 [Level 1]*
242	250	307	21	71	128	185
15	72	129	186	243	251	308
187	244	252	302	16	73	130
303	17	74	131	188	245	246
132	189	239	247	304	18	75
248	305	19	76	133	183	240
77	127	184	241	249	306	20


*Sym. & panm. 7x7x7 [Level 2]*
111	168	218	275	332	46	54
276	333	47	55	112	162	219
56	106	163	220	277	334	48
221	278	335	49	50	107	164
43	51	108	165	222	279	336
166	223	280	330	44	52	109
331	45	53	110	167	224	274


*Sym. & panm. 7x7x7 [Level 3]*
22	79	136	193	201	258	315
194	202	259	309	23	80	137
310	24	81	138	195	203	253
139	196	197	254	311	25	82
255	312	26	83	140	190	198
84	134	191	199	256	313	27
200	257	314	28	78	135	192


*Sym. & panm. 7x7x7 [Level 4]*
283	340	5	62	119	169	226
63	113	170	227	284	341	6
228	285	342	7	57	114	171
1	58	115	172	229	286	343
173	230	287	337	2	59	116
338	3	60	117	174	231	281
118	175	225	282	339	4	61


*Sym. & panm. 7x7x7 [Level 5]*
152	209	266	316	30	87	144
317	31	88	145	153	210	260
146	154	204	261	318	32	89
262	319	33	90	147	148	205
91	141	149	206	263	320	34
207	264	321	35	85	142	150
29	86	143	151	208	265	322


*Sym. & panm. 7x7x7 [Level 6]*
70	120	177	234	291	299	13
235	292	300	14	64	121	178
8	65	122	179	236	293	301
180	237	294	295	9	66	123
296	10	67	124	181	238	288
125	182	232	289	297	11	68
290	298	12	69	126	176	233


*Sym. & panm. 7x7x7 [Level 7]*
324	38	95	103	160	217	267
104	161	211	268	325	39	96
269	326	40	97	105	155	212
98	99	156	213	270	327	41
214	271	328	42	92	100	157
36	93	101	158	215	272	329
159	216	273	323	37	94	102

Extra magic features:

The 7x7x7 cube is symmetric;
The 7x7x7 cube is panmagic in each level;
Thee 7x7x7 cube is pandiagonal magic through the levels

History
See on website http://www.multimagie.com/English/Perfectcubes.htm#FirstPerfect that Rev. A.H. Frost constructed the first symmetric panmagic 7x7x7 cube in 1866!!! Probably he used the same construction method (but with alternate shifts).

Use the shift method to construct magic cubes of odd order and from 9x9x9 and up the result is Nasik. See on this website the shift method for:

5x5x5, 7x7x7, 9x9x9, 11x11x11, 13x13x13, 15x15x15, 17x17x17, 19x19x19, 21x21x21,

23x23x23, 25x25x25, 27x27x27, 29x29x29, 31x31x31

7x7x7, panmagic.xls

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7x7x7, Analysis Frost's 7x7x7 magic cube

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7x7x7 magic cube