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Magic tesseracts of each order

This page shows examples of magic tesseracts of orders from 3 to 4. Every magic tesseract in this page is normal.
Definitions of terms are here. Examples of magic cubes are here.

Order-3 magic tesseracts (the constant is 123)

There are 58 order-3 magic tesseracts. All order-3 magic tesseracts are associated and simple.

an order-3 simple magic tesseract (associated) [John R. Hendricks]

plane (1,1)
65	24	34
22	35	66
36	64	23

plane (1,2)
31	71	21
72	19	32
20	33	70

plane (1,3)
27	28	68
29	69	25
67	26	30

plane (2,1)
6	43	74
44	75	4
73	5	45

plane (2,2)
80	3	40
1	41	81
42	79	2

plane (2,3)
37	77	9
78	7	38
8	39	76

plane (3,1)
52	56	15
57	13	53
14	54	55

plane (3,2)
12	49	62
50	63	10
61	11	51

plane (3,3)
59	18	46
16	47	60
48	58	17

the source: Harvey D. Heinz & John R. Hendricks, Magic Square Lexicon: Illustrated, self-published, 2000, ISBN 0-9687985-0-0, pp.91-92

Order-4 magic tesseracts (the constant is 514)

an order-4 simple magic tesseract (non-associated) [Shigematsu Urata(1889-1958), 1955]

plane (1,1)
1	255	66	192
254	4	189	67
131	125	196	62
128	130	63	193

plane (1,2)
248	10	183	73
11	245	76	182
118	140	53	203
137	119	202	56

plane (1,3)
25	231	90	168
230	28	165	91
155	101	220	38
104	154	39	217

plane (1,4)
240	18	175	81
19	237	84	174
110	148	45	211
145	111	210	48

plane (2,1)
208	50	143	113
51	205	116	142
78	180	13	243
177	79	242	16

plane (2,2)
57	199	122	136
198	60	133	123
187	69	252	6
72	186	7	249

plane (2,3)
216	42	151	105
43	213	108	150
86	172	21	235
169	87	234	24

plane (2,4)
33	223	98	160
222	36	157	99
163	93	228	30
96	162	31	225

plane (3,1)
61	195	126	132
194	64	129	127
191	65	256	2
68	190	3	253

plane (3,2)
204	54	139	117
55	201	120	138
74	184	9	247
181	75	246	12

plane (3,3)
37	219	102	156
218	40	153	103
167	89	232	26
92	166	27	229

plane (3,4)
212	46	147	109
47	209	112	146
82	176	17	239
173	83	238	20

plane (4,1)
244	14	179	77
15	241	80	178
114	144	49	207
141	115	206	52

plane (4,2)
5	251	70	188
250	8	185	71
135	121	200	58
124	134	59	197

plane (4,3)
236	22	171	85
23	233	88	170
106	152	41	215
149	107	214	44

plane (4,4)
29	227	94	164
226	32	161	95
159	97	224	34
100	158	35	221

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.176

an order-4 simple magic tesseract (non-associated, 4-compact, horizontal-pandiagonal) [Nakamura, September 2007]

plane (1,1)
1	222	103	188
120	171	18	205
154	69	256	35
239	52	137	86

plane (1,2)
175	116	201	22
218	5	192	99
56	235	82	141
65	158	39	252

plane (1,3)
248	43	146	77
129	94	231	60
111	180	9	214
26	197	128	163

plane (1,4)
90	133	64	227
47	244	73	150
193	30	167	124
184	107	210	13

plane (2,1)
170	117	208	19
223	4	185	102
49	238	87	140
72	155	34	253

plane (2,2)
8	219	98	189
113	174	23	204
159	68	249	38
234	53	144	83

plane (2,3)
95	132	57	230
42	245	80	147
200	27	162	125
177	110	215	12

plane (2,4)
241	46	151	76
136	91	226	61
106	181	16	211
31	196	121	166

plane (3,1)
255	36	153	70
138	85	240	51
104	187	2	221
17	206	119	172

plane (3,2)
81	142	55	236
40	251	66	157
202	21	176	115
191	100	217	6

plane (3,3)
10	213	112	179
127	164	25	198
145	78	247	44
232	59	130	93

plane (3,4)
168	123	194	29
209	14	183	108
63	228	89	134
74	149	48	243

plane (4,1)
88	139	50	237
33	254	71	156
207	20	169	118
186	101	224	3

plane (4,2)
250	37	160	67
143	84	233	54
97	190	7	220
24	203	114	173

plane (4,3)
161	126	199	28
216	11	178	109
58	229	96	131
79	148	41	246

plane (4,4)
15	212	105	182
122	165	32	195
152	75	242	45
225	62	135	92

the source: original.
This magic tesseract is simple but 4-compact and the 16 horizontal planes of the tesseract are pandiagonal magic squares.

an order-4 diagonal magic tesseract (associated) [Nakamura, December 2004]

plane (1,1)
1	94	175	244
214	188	73	39
235	133	120	26
64	99	146	205

plane (1,2)
88	251	10	165
191	33	212	78
130	32	237	115
105	198	55	156

plane (1,3)
172	7	246	89
67	221	48	178
126	228	17	143
149	58	203	104

plane (1,4)
253	162	83	16
42	72	181	219
23	121	140	230
196	159	110	49

plane (2,1)
118	236	25	135
176	3	242	93
145	62	207	100
75	213	40	186

plane (2,2)
239	129	116	30
9	86	167	252
56	107	154	197
210	192	77	35

plane (2,3)
19	125	144	226
245	170	91	8
204	151	102	57
46	68	177	223

plane (2,4)
138	24	229	123
84	255	14	161
109	194	51	160
183	41	220	70

plane (3,1)
187	37	216	74
97	206	63	148
96	243	2	173
134	28	233	119

plane (3,2)
34	80	189	211
200	155	106	53
249	166	87	12
31	113	132	238

plane (3,3)
222	180	65	47
60	103	150	201
5	90	171	248
227	141	128	18

plane (3,4)
71	217	44	182
157	50	195	112
164	15	254	81
122	232	21	139

plane (4,1)
208	147	98	61
27	117	136	234
38	76	185	215
241	174	95	4

plane (4,2)
153	54	199	108
114	240	29	131
79	209	36	190
168	11	250	85

plane (4,3)
101	202	59	152
142	20	225	127
179	45	224	66
92	247	6	169

plane (4,4)
52	111	158	193
231	137	124	22
218	184	69	43
13	82	163	256

the source: original. Here is this tesseract of CSV format.

an order-4 pantriagonal magic tesseract (associated) [Nakamura, September 2007]

plane (1,1)
1	48	243	222
32	49	238	195
242	223	4	45
239	194	29	52

plane (1,2)
144	161	126	83
145	192	99	78
127	82	141	164
98	79	148	189

plane (1,3)
249	216	11	38
232	201	22	59
10	39	252	213
23	58	229	204

plane (1,4)
120	89	134	171
105	72	155	182
135	170	117	92
154	183	108	69

plane (2,1)
80	97	190	147
81	128	163	142
191	146	77	100
162	143	84	125

plane (2,2)
193	240	51	30
224	241	46	3
50	31	196	237
47	2	221	244

plane (2,3)
184	153	70	107
169	136	91	118
71	106	181	156
90	119	172	133

plane (2,4)
57	24	203	230
40	9	214	251
202	231	60	21
215	250	37	12

plane (3,1)
245	220	7	42
236	197	26	55
6	43	248	217
27	54	233	200

plane (3,2)
124	85	138	167
101	76	151	186
139	166	121	88
150	187	104	73

plane (3,3)
13	36	255	210
20	61	226	207
254	211	16	33
227	206	17	64

plane (3,4)
132	173	114	95
157	180	111	66
115	94	129	176
110	67	160	177

plane (4,1)
188	149	74	103
165	140	87	122
75	102	185	152
86	123	168	137

plane (4,2)
53	28	199	234
44	5	218	247
198	235	56	25
219	246	41	8

plane (4,3)
68	109	178	159
93	116	175	130
179	158	65	112
174	131	96	113

plane (4,4)
205	228	63	18
212	253	34	15
62	19	208	225
35	14	209	256

the source: original. Here is this tesseract of CSV format.

an order-4 panmagic tesseract (complete, non-associated, 4-compact) [John R. Hendricks]

plane (1,1)
239	116	30	129
153	199	108	54
34	189	211	80
88	10	165	251

plane (1,2)
56	154	197	107
79	36	190	209
249	87	12	166
130	237	115	32

plane (1,3)
210	77	35	192
168	250	85	11
31	132	238	113
105	55	156	198

plane (1,4)
9	167	252	86
114	29	131	240
200	106	53	155
191	212	78	33

plane (2,1)
138	229	123	24
52	158	193	111
71	44	182	217
253	83	16	162

plane (2,2)
109	51	160	194
218	69	43	184
164	254	81	15
23	140	230	121

plane (2,3)
183	220	70	41
13	163	256	82
122	21	139	232
196	110	49	159

plane (2,4)
84	14	161	255
231	124	22	137
157	195	112	50
42	181	219	72

plane (3,1)
19	144	226	125
101	59	152	202
222	65	47	180
172	246	89	7

plane (3,2)
204	102	57	151
179	224	66	45
5	171	248	90
126	17	143	228

plane (3,3)
46	177	223	68
92	6	169	247
227	128	18	141
149	203	104	58

plane (3,4)
245	91	8	170
142	225	127	20
60	150	201	103
67	48	178	221

plane (4,1)
118	25	135	236
208	98	61	147
187	216	74	37
1	175	244	94

plane (4,2)
145	207	100	62
38	185	215	76
96	2	173	243
235	120	26	133

plane (4,3)
75	40	186	213
241	95	4	174
134	233	119	28
64	146	205	99

plane (4,4)
176	242	93	3
27	136	234	117
97	63	148	206
214	73	39	188

the source: Harvey D. Heinz's site (http://members.shaw.ca/hdhcubes/cube_perfect.htm)

Every order-4 panmagic tesseract is complete and 4-compact, so an order-4 normal panmagic tesseract cannot be associated. (This fact does not hold for order higher than 4.)
On the other hand, an order-4 associated normal panmagic cube does exist.

an order-4 panmagic tesseract (complete, non-associated, 4-compact, horizontal-diagonal, sudoku-type) [Nakamura, September 2007]

plane (1,1)
1	168	250	95
123	222	132	37
160	57	103	194
230	67	29	188

plane (1,2)
219	126	36	133
161	8	90	255
70	227	189	28
64	153	199	98

plane (1,3)
112	201	151	50
22	179	237	76
241	88	10	175
139	46	116	213

plane (1,4)
182	19	77	236
208	105	55	146
43	142	212	117
81	248	170	15

plane (2,1)
174	11	85	244
216	113	47	138
51	150	204	109
73	240	178	23

plane (2,2)
120	209	143	42
14	171	245	84
233	80	18	183
147	54	108	205

plane (2,3)
195	102	60	157
185	32	66	231
94	251	165	4
40	129	223	122

plane (2,4)
25	192	226	71
99	198	156	61
136	33	127	218
254	91	5	164

plane (3,1)
247	82	16	169
141	44	118	211
106	207	145	56
20	181	235	78

plane (3,2)
45	140	214	115
87	242	176	9
180	21	75	238
202	111	49	152

plane (3,3)
154	63	97	200
228	69	27	190
7	162	256	89
125	220	134	35

plane (3,4)
68	229	187	30
58	159	193	104
221	124	38	131
167	2	96	249

plane (4,1)
92	253	163	6
34	135	217	128
197	100	62	155
191	26	72	225

plane (4,2)
130	39	121	224
252	93	3	166
31	186	232	65
101	196	158	59

plane (4,3)
53	148	206	107
79	234	184	17
172	13	83	246
210	119	41	144

plane (4,4)
239	74	24	177
149	52	110	203
114	215	137	48
12	173	243	86

the source: original.
This order-4 magic tesseract is panmagic (therefore complete and 4-compact) and the 16 horizontal planes of the tesseract are magic squares.
By analyzing this tesseract by hexadecimal digits, we can get two tesseracts shown below. By connecting the 16 planes of each of the two tesseract, we can get two order-16 diagonal Latin squares which are sudoku type, that is, Latin squares which have no 4x4 block containing duplicate numbers.
I made this magic tesseract by using a matrix over the finite field F₄.

upper elements

plane (1,1)
0	10	15	5
7	13	8	2
9	3	6	12
14	4	1	11

plane (1,2)
13	7	2	8
10	0	5	15
4	14	11	1
3	9	12	6

plane (1,3)
6	12	9	3
1	11	14	4
15	5	0	10
8	2	7	13

plane (1,4)
11	1	4	14
12	6	3	9
2	8	13	7
5	15	10	0

plane (2,1)
10	0	5	15
13	7	2	8
3	9	12	6
4	14	11	1

plane (2,2)
7	13	8	2
0	10	15	5
14	4	1	11
9	3	6	12

plane (2,3)
12	6	3	9
11	1	4	14
5	15	10	0
2	8	13	7

plane (2,4)
1	11	14	4
6	12	9	3
8	2	7	13
15	5	0	10

plane (3,1)
15	5	0	10
8	2	7	13
6	12	9	3
1	11	14	4

plane (3,2)
2	8	13	7
5	15	10	0
11	1	4	14
12	6	3	9

plane (3,3)
9	3	6	12
14	4	1	11
0	10	15	5
7	13	8	2

plane (3,4)
4	14	11	1
3	9	12	6
13	7	2	8
10	0	5	15

plane (4,1)
5	15	10	0
2	8	13	7
12	6	3	9
11	1	4	14

plane (4,2)
8	2	7	13
15	5	0	10
1	11	14	4
6	12	9	3

plane (4,3)
3	9	12	6
4	14	11	1
10	0	5	15
13	7	2	8

plane (4,4)
14	4	1	11
9	3	6	12
7	13	8	2
0	10	15	5

lower elements

plane (1,1)
0	7	9	14
10	13	3	4
15	8	6	1
5	2	12	11

plane (1,2)
10	13	3	4
0	7	9	14
5	2	12	11
15	8	6	1

plane (1,3)
15	8	6	1
5	2	12	11
0	7	9	14
10	13	3	4

plane (1,4)
5	2	12	11
15	8	6	1
10	13	3	4
0	7	9	14

plane (2,1)
13	10	4	3
7	0	14	9
2	5	11	12
8	15	1	6

plane (2,2)
7	0	14	9
13	10	4	3
8	15	1	6
2	5	11	12

plane (2,3)
2	5	11	12
8	15	1	6
13	10	4	3
7	0	14	9

plane (2,4)
8	15	1	6
2	5	11	12
7	0	14	9
13	10	4	3

plane (3,1)
6	1	15	8
12	11	5	2
9	14	0	7
3	4	10	13

plane (3,2)
12	11	5	2
6	1	15	8
3	4	10	13
9	14	0	7

plane (3,3)
9	14	0	7
3	4	10	13
6	1	15	8
12	11	5	2

plane (3,4)
3	4	10	13
9	14	0	7
12	11	5	2
6	1	15	8

plane (4,1)
11	12	2	5
1	6	8	15
4	3	13	10
14	9	7	0

plane (4,2)
1	6	8	15
11	12	2	5
14	9	7	0
4	3	13	10

plane (4,3)
4	3	13	10
14	9	7	0
11	12	2	5
1	6	8	15

plane (4,4)
14	9	7	0
4	3	13	10
1	6	8	15
11	12	2	5

an order-4 pan-3,4-agonal magic tesseract (complete, non-associated, 2-compact) [C. Planck]

plane (1,1)
1	128	193	192
240	145	48	81
49	80	241	144
224	161	32	97

plane (1,2)
254	131	62	67
19	110	211	174
206	179	14	115
35	94	227	158

plane (1,3)
4	125	196	189
237	148	45	84
52	77	244	141
221	164	29	100

plane (1,4)
255	130	63	66
18	111	210	175
207	178	15	114
34	95	226	159

plane (2,1)
248	137	56	73
25	104	217	168
200	185	8	121
41	88	233	152

plane (2,2)
11	118	203	182
230	155	38	91
59	70	251	134
214	171	22	107

plane (2,3)
245	140	53	76
28	101	220	165
197	188	5	124
44	85	236	149

plane (2,4)
10	119	202	183
231	154	39	90
58	71	250	135
215	170	23	106

plane (3,1)
13	116	205	180
228	157	36	93
61	68	253	132
212	173	20	109

plane (3,2)
242	143	50	79
31	98	223	162
194	191	2	127
47	82	239	146

plane (3,3)
16	113	208	177
225	160	33	96
64	65	256	129
209	176	17	112

plane (3,4)
243	142	51	78
30	99	222	163
195	190	3	126
46	83	238	147

plane (4,1)
252	133	60	69
21	108	213	172
204	181	12	117
37	92	229	156

plane (4,2)
7	122	199	186
234	151	42	87
55	74	247	138
218	167	26	103

plane (4,3)
249	136	57	72
24	105	216	169
201	184	9	120
40	89	232	153

plane (4,4)
6	123	198	187
235	150	43	86
54	75	246	139
219	166	27	102

the source: W. S. Andrews, Magic Squares and Cubes, Dover, 1960, p.374.
This edition is a republication of the 2nd edition published by Open Court in 1917.
C. Planck made this tesseract by using a matrix.
Every 2-compact magic tesseract is pan-3,4-agonal.

Other magic tesseracts

John R. Hendricks made excellent magic tesseracts as follows:
an order-6 inlaid magic tesseract (with an inlaid order-3 magic tesseract),
an order-16 perfect (namely, pan-2,3,4-agonal) magic tesseract.
For more information, see Hendricks's site (http://members.shaw.ca/johnhendricksmath/).

I made an order-8 associated strictly magic tesseract. Look at the page about works on magic tesseracts and hypercubes.

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This page was last updated on October 1, 2007.
"Magic Cubes and Tesseracts" http://homepage2.nifty.com/googol/magcube/en/
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