How to make perfect magic squares & cubes - [More than] perfect magic 9x9x9 cube

See an ultra magic 9x9 square on page '[ultra] panmagic 9x9 square (1)'. We use this square and the row
grid of this square to produce the fifth level of the perfect magic 9x9x9 cube and we use the (diagonal) shif-
ted versions of this square and the (diagonal) shifted versions of the row grid of this square to produce the
other levels of the more than perfect magic 9x9x9 cube; see below.

*(1) Shifted version 9x9*									*row grid of (9)*									*9x9x9 magic cube, first level*
41	27	60	43	20	62	39	22	55	3	0	4	8	5	6	1	7	2	284	27	384	691	425	548	120	589	217
75	31	10	77	36	15	79	29	17	1	7	2	3	0	4	8	5	6	156	598	172	320	36	339	727	434	503
52	2	71	48	4	64	50	9	69	8	5	6	1	7	2	3	0	4	700	407	557	129	571	226	293	9	393
59	45	24	61	38	26	57	40	19	3	0	4	8	5	6	1	7	2	302	45	348	709	443	512	138	607	181
12	76	28	14	81	33	16	74	35	1	7	2	3	0	4	8	5	6	93	643	190	257	81	357	664	479	521
70	47	8	66	49	1	68	54	6	8	5	6	1	7	2	3	0	4	718	452	494	147	616	163	311	54	330
23	63	42	25	56	44	21	58	37	3	0	4	8	5	6	1	7	2	266	63	366	673	461	530	102	625	199
30	13	73	32	18	78	34	11	80	1	7	2	3	0	4	8	5	6	111	580	235	275	18	402	682	416	566
7	65	53	3	67	46	5	72	51	8	5	6	1	7	2	3	0	4	655	470	539	84	634	208	248	72	375


*(2) Shifted version 9x9*									*row grid of (8)*									*9x9x9 magic cube second level*
51	7	65	53	3	67	46	5	72	7	2	3	0	4	8	5	6	1	618	169	308	53	327	715	451	491	153
55	41	27	60	43	20	62	39	22	5	6	1	7	2	3	0	4	8	460	527	108	627	205	263	62	363	670
17	75	31	10	77	36	15	79	29	0	4	8	5	6	1	7	2	3	17	399	679	415	563	117	582	241	272
69	52	2	71	48	4	64	50	9	7	2	3	0	4	8	5	6	1	636	214	245	71	372	652	469	536	90
19	59	45	24	61	38	26	57	40	5	6	1	7	2	3	0	4	8	424	545	126	591	223	281	26	381	688
35	12	76	28	14	81	33	16	74	0	4	8	5	6	1	7	2	3	35	336	724	433	500	162	600	178	317
6	70	47	8	66	49	1	68	54	7	2	3	0	4	8	5	6	1	573	232	290	8	390	697	406	554	135
37	23	63	42	25	56	44	21	58	5	6	1	7	2	3	0	4	8	442	509	144	609	187	299	44	345	706
80	30	13	73	32	18	78	34	11	0	4	8	5	6	1	7	2	3	80	354	661	478	518	99	645	196	254


*(3) Shifted version 9x9*									*row grid of (7)*									*9x9x9 magic cube, third level*
11	80	30	13	73	32	18	78	34	6	1	7	2	3	0	4	8	5	497	161	597	175	316	32	342	726	439
72	51	7	65	53	3	67	46	5	4	8	5	6	1	7	2	3	0	396	699	412	551	134	570	229	289	5
22	55	41	27	60	43	20	62	39	2	3	0	4	8	5	6	1	7	184	298	41	351	708	448	506	143	606
29	17	75	31	10	77	36	15	79	6	1	7	2	3	0	4	8	5	515	98	642	193	253	77	360	663	484
9	69	52	2	71	48	4	64	50	4	8	5	6	1	7	2	3	0	333	717	457	488	152	615	166	307	50
40	19	59	45	24	61	38	26	57	2	3	0	4	8	5	6	1	7	202	262	59	369	672	466	524	107	624
74	35	12	76	28	14	81	33	16	6	1	7	2	3	0	4	8	5	560	116	579	238	271	14	405	681	421
54	6	70	47	8	66	49	1	68	4	8	5	6	1	7	2	3	0	378	654	475	533	89	633	211	244	68
58	37	23	63	42	25	56	44	21	2	3	0	4	8	5	6	1	7	220	280	23	387	690	430	542	125	588


*(4) Shifted version 9x9*									*row grid of (6)*									*9x9x9 magic cube, fourth level*
21	58	37	23	63	42	25	56	44	8	5	6	1	7	2	3	0	4	669	463	523	104	630	204	268	56	368
34	11	80	30	13	73	32	18	78	3	0	4	8	5	6	1	7	2	277	11	404	678	418	559	113	585	240
5	72	51	7	65	53	3	67	46	1	7	2	3	0	4	8	5	6	86	639	213	250	65	377	651	472	532
39	22	55	41	27	60	43	20	62	8	5	6	1	7	2	3	0	4	687	427	541	122	594	222	286	20	386
79	29	17	75	31	10	77	36	15	3	0	4	8	5	6	1	7	2	322	29	341	723	436	496	158	603	177
50	9	69	52	2	71	48	4	64	1	7	2	3	0	4	8	5	6	131	576	231	295	2	395	696	409	550
57	40	19	59	45	24	61	38	26	8	5	6	1	7	2	3	0	4	705	445	505	140	612	186	304	38	350
16	74	35	12	76	28	14	81	33	3	0	4	8	5	6	1	7	2	259	74	359	660	481	514	95	648	195
68	54	6	70	47	8	66	49	1	1	7	2	3	0	4	8	5	6	149	621	168	313	47	332	714	454	487


*(5) Ultra magic 9x9*									*row grid of (5)*									*9x9x9 magic cube, fifth level*
1	68	54	6	70	47	8	66	49	0	4	8	5	6	1	7	2	3	1	392	702	411	556	128	575	228	292
44	21	58	37	23	63	42	25	56	7	2	3	0	4	8	5	6	1	611	183	301	37	347	711	447	511	137
78	34	11	80	30	13	73	32	18	5	6	1	7	2	3	0	4	8	483	520	92	647	192	256	73	356	666
46	5	72	51	7	65	53	3	67	0	4	8	5	6	1	7	2	3	46	329	720	456	493	146	620	165	310
62	39	22	55	41	27	60	43	20	7	2	3	0	4	8	5	6	1	629	201	265	55	365	675	465	529	101
15	79	29	17	75	31	10	77	36	5	6	1	7	2	3	0	4	8	420	565	110	584	237	274	10	401	684
64	50	9	69	52	2	71	48	4	0	4	8	5	6	1	7	2	3	64	374	657	474	538	83	638	210	247
26	57	40	19	59	45	24	61	38	7	2	3	0	4	8	5	6	1	593	219	283	19	383	693	429	547	119
33	16	74	35	12	76	28	14	81	5	6	1	7	2	3	0	4	8	438	502	155	602	174	319	28	338	729


*(6) Shifted version 9x9*									*row grid of (4)*									*9x9x9 magic cube, sixth level*
81	33	16	74	35	12	76	28	14	2	3	0	4	8	5	6	1	7	243	276	16	398	683	417	562	109	581
49	1	68	54	6	70	47	8	66	6	1	7	2	3	0	4	8	5	535	82	635	216	249	70	371	656	471
56	44	21	58	37	23	63	42	25	4	8	5	6	1	7	2	3	0	380	692	426	544	118	590	225	285	25
18	78	34	11	80	30	13	73	32	2	3	0	4	8	5	6	1	7	180	321	34	335	728	435	499	154	599
67	46	5	72	51	7	65	53	3	6	1	7	2	3	0	4	8	5	553	127	572	234	294	7	389	701	408
20	62	39	22	55	41	27	60	43	4	8	5	6	1	7	2	3	0	344	710	444	508	136	608	189	303	43
36	15	79	29	17	75	31	10	77	2	3	0	4	8	5	6	1	7	198	258	79	353	665	480	517	91	644
4	64	50	9	69	52	2	71	48	6	1	7	2	3	0	4	8	5	490	145	617	171	312	52	326	719	453
38	26	57	40	19	59	45	24	61	4	8	5	6	1	7	2	3	0	362	674	462	526	100	626	207	267	61


*(7) Shifted version 9x9*									*row grid of (3)*									*9x9x9 magic cube, seventh level*
61	38	26	57	40	19	59	45	24	1	7	2	3	0	4	8	5	6	142	605	188	300	40	343	707	450	510
14	81	33	16	74	35	12	76	28	8	5	6	1	7	2	3	0	4	662	486	519	97	641	197	255	76	352
66	49	1	68	54	6	70	47	8	3	0	4	8	5	6	1	7	2	309	49	325	716	459	492	151	614	170
25	56	44	21	58	37	23	63	42	1	7	2	3	0	4	8	5	6	106	623	206	264	58	361	671	468	528
32	18	78	34	11	80	30	13	73	8	5	6	1	7	2	3	0	4	680	423	564	115	578	242	273	13	397
3	67	46	5	72	51	7	65	53	3	0	4	8	5	6	1	7	2	246	67	370	653	477	537	88	632	215
43	20	62	39	22	55	41	27	60	1	7	2	3	0	4	8	5	6	124	587	224	282	22	379	689	432	546
77	36	15	79	29	17	75	31	10	8	5	6	1	7	2	3	0	4	725	441	501	160	596	179	318	31	334
48	4	64	50	9	69	52	2	71	3	0	4	8	5	6	1	7	2	291	4	388	698	414	555	133	569	233


*(8) Shifted version 9x9*									*row grid of (2)*									*9x9x9 magic cube, eighth level*
71	48	4	64	50	9	69	52	2	5	6	1	7	2	3	0	4	8	476	534	85	631	212	252	69	376	650
24	61	38	26	57	40	19	59	45	0	4	8	5	6	1	7	2	3	24	385	686	431	543	121	586	221	288
28	14	81	33	16	74	35	12	76	7	2	3	0	4	8	5	6	1	595	176	324	33	340	722	440	498	157
8	66	49	1	68	54	6	70	47	5	6	1	7	2	3	0	4	8	413	552	130	568	230	297	6	394	695
42	25	56	44	21	58	37	23	63	0	4	8	5	6	1	7	2	3	42	349	704	449	507	139	604	185	306
73	32	18	78	34	11	80	30	13	7	2	3	0	4	8	5	6	1	640	194	261	78	358	659	485	516	94
53	3	67	46	5	72	51	7	65	5	6	1	7	2	3	0	4	8	458	489	148	613	167	315	51	331	713
60	43	20	62	39	22	55	41	27	0	4	8	5	6	1	7	2	3	60	367	668	467	525	103	622	203	270
10	77	36	15	79	29	17	75	31	7	2	3	0	4	8	5	6	1	577	239	279	15	403	677	422	561	112


*(9) Shifted version 9x9*									*row grid of (1)*									*9x9x9 magic cube, ninth level*
31	10	77	36	15	79	29	17	75	4	8	5	6	1	7	2	3	0	355	658	482	522	96	646	191	260	75
2	71	48	4	64	50	9	69	52	2	3	0	4	8	5	6	1	7	164	314	48	328	712	455	495	150	619
45	24	61	38	26	57	40	19	59	6	1	7	2	3	0	4	8	5	531	105	628	200	269	57	364	667	464
76	28	14	81	33	16	74	35	12	4	8	5	6	1	7	2	3	0	400	676	419	567	114	583	236	278	12
47	8	66	49	1	68	54	6	70	2	3	0	4	8	5	6	1	7	209	251	66	373	649	473	540	87	637
63	42	25	56	44	21	58	37	23	6	1	7	2	3	0	4	8	5	549	123	592	218	287	21	382	685	428
13	73	32	18	78	34	11	80	30	4	8	5	6	1	7	2	3	0	337	721	437	504	159	601	173	323	30
65	53	3	67	46	5	72	51	7	2	3	0	4	8	5	6	1	7	227	296	3	391	694	410	558	132	574
27	60	43	20	62	39	22	55	41	6	1	7	2	3	0	4	8	5	513	141	610	182	305	39	346	703	446

N.B.: 1x digit from 9x9 magic square + 81x digit from row grid = digit in magic cube

This magic 9x9x9 cube is symmetric, pandiagonal magic (in each level and through the levels, i.e. 27+308+
175+630+128+562+450+650+355=3285), 3x3(x3) compact, but only 2/4 triagonal magic (i.e. 27+108+351+
594+675+189+432+270+513=3159; 27+460+606+20+465+608+22+467+610=3285; 27+661+533+612+274
+389+468+157+164=3285; 27+80+68+38+10+7+58+33+48=369). In comparison with above mentioned
magic cube, Frost's magic 9x9x9 cube is not 3x3 compact, but Frost's magic 9x9x9 cube is fully triagonal
magic; see http://www.magic-squares.net/c-t-htm/c_update-6.htm.

Use an ultra panmagic 15x15 square to produce a 15x15x15 cube, that is symmetric, panmagic in the levels,
pandiagonal magic through the levels, 2/4 pantriagonal magic through the levels and double (3x3x3 & b5x5x5)
compact. Use the above mentioned method to produce cubes with order is odd multiple of 3.

The secret of Frost's 9x9x9 is not difficult. Basis of the 9x9x9 cube is the following magic 9x9 square, which is
symmetric and panmagic but not compact:

60	76	52	37	29	5	26	18	66
54	39	33	4	25	10	65	59	80
32	8	27	12	69	58	79	46	38
19	11	68	62	81	48	42	31	7
67	61	73	47	41	35	9	21	15
75	51	40	34	1	20	14	71	63
44	36	3	24	13	70	55	74	50
2	23	17	72	57	78	49	43	28
16	64	56	77	53	45	30	6	22

In the levels 1 up to 9 you find [the horizontal shifted versions of] the 9x9 square 2-4-6-8-1-3-5-7-9 (= square
with in the top left corner the digit 54, 19, 75, 2, 60, 32, 67, 44 respectively 16) and the (reversed) column
(instead of row) grids 9 up to 1.

You can use Frost's method to produce magic cubes of order is odd multiple of 3; see for example perfect (nasik)
magic 15x15x15 cube.

N.B.: It is not possible to get a compact perfect 9x9x9 cube, which is fully triagonal magic.

The following downloads for analysis and construction in EXCEL are available: