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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)
652434
223566
366423
plane (1,2)
317121
721932
203370
plane (1,3)
272868
296925
672630

plane (2,1)
64374
44754
73545
plane (2,2)
80340
14181
42792
plane (2,3)
37779
78738
83976

plane (3,1)
525615
571353
145455
plane (3,2)
124962
506310
611151
plane (3,3)
591846
164760
485817
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)
125566192
254418967
13112519662
12813063193
plane (1,2)
2481018373
1124576182
11814053203
13711920256
plane (1,3)
2523190168
2302816591
15510122038
10415439217
plane (1,4)
2401817581
1923784174
11014845211
14511121048

plane (2,1)
20850143113
51205116142
7818013243
1777924216
plane (2,2)
57199122136
19860133123
187692526
721867249
plane (2,3)
21642151105
43213108150
8617221235
1698723424
plane (2,4)
3322398160
2223615799
1639322830
9616231225

plane (3,1)
61195126132
19464129127
191652562
681903253
plane (3,2)
20454139117
55201120138
741849247
1817524612
plane (3,3)
37219102156
21840153103
1678923226
9216627229
plane (3,4)
21246147109
47209112146
8217617239
1738323820

plane (4,1)
2441417977
1524180178
11414449207
14111520652
plane (4,2)
525170188
250818571
13512120058
12413459197
plane (4,3)
2362217185
2323388170
10615241215
14910721444
plane (4,4)
2922794164
2263216195
1599722434
10015835221
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)
1222103188
12017118205
1546925635
2395213786
plane (1,2)
17511620122
218519299
5623582141
6515839252
plane (1,3)
2484314677
1299423160
1111809214
26197128163
plane (1,4)
9013364227
4724473150
19330167124
18410721013

plane (2,1)
17011720819
2234185102
4923887140
7215534253
plane (2,2)
821998189
11317423204
1596824938
2345314483
plane (2,3)
9513257230
4224580147
20027162125
17711021512
plane (2,4)
2414615176
1369122661
10618116211
31196121166

plane (3,1)
2553615370
1388524051
1041872221
17206119172
plane (3,2)
8114255236
4025166157
20221176115
1911002176
plane (3,3)
10213112179
12716425198
1457824744
2325913093
plane (3,4)
16812319429
20914183108
6322889134
7414948243

plane (4,1)
8813950237
3325471156
20720169118
1861012243
plane (4,2)
2503716067
1438423354
971907220
24203114173
plane (4,3)
16112619928
21611178109
5822996131
7914841246
plane (4,4)
15212105182
12216532195
1527524245
2256213592
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)
194175244
2141887339
23513312026
6499146205
plane (1,2)
8825110165
1913321278
13032237115
10519855156
plane (1,3)
172724689
6722148178
12622817143
14958203104
plane (1,4)
2531628316
4272181219
23121140230
19615911049

plane (2,1)
11823625135
176324293
14562207100
7521340186
plane (2,2)
23912911630
986167252
56107154197
2101927735
plane (2,3)
19125144226
245170918
20415110257
4668177223
plane (2,4)
13824229123
8425514161
10919451160
1834122070

plane (3,1)
1873721674
9720663148
962432173
13428233119
plane (3,2)
3480189211
20015510653
2491668712
31113132238
plane (3,3)
2221806547
60103150201
590171248
22714112818
plane (3,4)
7121744182
15750195112
1641525481
12223221139

plane (4,1)
2081479861
27117136234
3876185215
241174954
plane (4,2)
15354199108
11424029131
7920936190
1681125085
plane (4,3)
10120259152
14220225127
1794522466
922476169
plane (4,4)
52111158193
23113712422
2181846943
1382163256
the source: original. Here is this tesseract of CSV format.

an order-4 pantriagonal magic tesseract (associated) [Nakamura, September 2007]
plane (1,1)
148243222
3249238195
242223445
2391942952
plane (1,2)
14416112683
1451929978
12782141164
9879148189
plane (1,3)
2492161138
2322012259
1039252213
2358229204
plane (1,4)
12089134171
10572155182
13517011792
15418310869

plane (2,1)
8097190147
81128163142
19114677100
16214384125
plane (2,2)
1932405130
224241463
5031196237
472221244
plane (2,3)
18415370107
16913691118
71106181156
90119172133
plane (2,4)
5724203230
409214251
2022316021
2152503712

plane (3,1)
245220742
2361972655
643248217
2754233200
plane (3,2)
12485138167
10176151186
13916612188
15018710473
plane (3,3)
1336255210
2061226207
2542111633
2272061764
plane (3,4)
13217311495
15718011166
11594129176
11067160177

plane (4,1)
18814974103
16514087122
75102185152
86123168137
plane (4,2)
5328199234
445218247
1982355625
219246418
plane (4,3)
68109178159
93116175130
17915865112
17413196113
plane (4,4)
2052286318
2122533415
6219208225
3514209256
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)
23911630129
15319910854
3418921180
8810165251
plane (1,2)
56154197107
7936190209
2498712166
13023711532
plane (1,3)
2107735192
1682508511
31132238113
10555156198
plane (1,4)
916725286
11429131240
20010653155
1912127833

plane (2,1)
13822912324
52158193111
7144182217
2538316162
plane (2,2)
10951160194
2186943184
1642548115
23140230121
plane (2,3)
1832207041
1316325682
12221139232
19611049159
plane (2,4)
8414161255
23112422137
15719511250
4218121972

plane (3,1)
19144226125
10159152202
2226547180
172246897
plane (3,2)
20410257151
1792246645
517124890
12617143228
plane (3,3)
4617722368
926169247
22712818141
14920310458
plane (3,4)
245918170
14222512720
60150201103
6748178221

plane (4,1)
11825135236
2089861147
1872167437
117524494
plane (4,2)
14520710062
3818521576
962173243
23512026133
plane (4,3)
7540186213
241954174
13423311928
6414620599
plane (4,4)
176242933
27136234117
9763148206
2147339188
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

   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)
116825095
12322213237
16057103194
2306729188
plane (1,2)
21912636133
161890255
7022718928
6415319998
plane (1,3)
11220115150
2217923776
2418810175
13946116213
plane (1,4)
1821977236
20810555146
43142212117
8124817015

plane (2,1)
1741185244
21611347138
51150204109
7324017823
plane (2,2)
12020914342
1417124584
2338018183
14754108205
plane (2,3)
19510260157
1853266231
942511654
40129223122
plane (2,4)
2519222671
9919815661
13633127218
254915164

plane (3,1)
2478216169
14144118211
10620714556
2018123578
plane (3,2)
45140214115
872421769
1802175238
20211149152
plane (3,3)
1546397200
2286927190
716225689
12522013435
plane (3,4)
6822918730
58159193104
22112438131
167296249

plane (4,1)
922531636
34135217128
19710062155
1912672225
plane (4,2)
13039121224
252933166
3118623265
10119615859
plane (4,3)
53148206107
7923418417
1721383246
21011941144
plane (4,4)
2397424177
14952110203
11421513748
1217324386
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)
010155
71382
93612
144111
plane (1,2)
13728
100515
414111
39126
plane (1,3)
61293
111144
155010
82713
plane (1,4)
111414
12639
28137
515100

plane (2,1)
100515
13728
39126
414111
plane (2,2)
71382
010155
144111
93612
plane (2,3)
12639
111414
515100
28137
plane (2,4)
111144
61293
82713
155010

plane (3,1)
155010
82713
61293
111144
plane (3,2)
28137
515100
111414
12639
plane (3,3)
93612
144111
010155
71382
plane (3,4)
414111
39126
13728
100515

plane (4,1)
515100
28137
12639
111414
plane (4,2)
82713
155010
111144
61293
plane (4,3)
39126
414111
100515
13728
plane (4,4)
144111
93612
71382
010155
lower elements
plane (1,1)
07914
101334
15861
521211
plane (1,2)
101334
07914
521211
15861
plane (1,3)
15861
521211
07914
101334
plane (1,4)
521211
15861
101334
07914

plane (2,1)
131043
70149
251112
81516
plane (2,2)
70149
131043
81516
251112
plane (2,3)
251112
81516
131043
70149
plane (2,4)
81516
251112
70149
131043

plane (3,1)
61158
121152
91407
341013
plane (3,2)
121152
61158
341013
91407
plane (3,3)
91407
341013
61158
121152
plane (3,4)
341013
91407
121152
61158

plane (4,1)
111225
16815
431310
14970
plane (4,2)
16815
111225
14970
431310
plane (4,3)
431310
14970
111225
16815
plane (4,4)
14970
431310
16815
111225

an order-4 pan-3,4-agonal magic tesseract (complete, non-associated, 2-compact) [C. Planck]
plane (1,1)
1128193192
2401454881
4980241144
2241613297
plane (1,2)
2541316267
19110211174
20617914115
3594227158
plane (1,3)
4125196189
2371484584
5277244141
22116429100
plane (1,4)
2551306366
18111210175
20717815114
3495226159

plane (2,1)
2481375673
25104217168
2001858121
4188233152
plane (2,2)
11118203182
2301553891
5970251134
21417122107
plane (2,3)
2451405376
28101220165
1971885124
4485236149
plane (2,4)
10119202183
2311543990
5871250135
21517023106

plane (3,1)
13116205180
2281573693
6168253132
21217320109
plane (3,2)
2421435079
3198223162
1941912127
4782239146
plane (3,3)
16113208177
2251603396
6465256129
20917617112
plane (3,4)
2431425178
3099222163
1951903126
4683238147

plane (4,1)
2521336069
21108213172
20418112117
3792229156
plane (4,2)
7122199186
2341514287
5574247138
21816726103
plane (4,3)
2491365772
24105216169
2011849120
4089232153
plane (4,4)
6123198187
2351504386
5475246139
21916627102
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 Nasik (namely, pan-2,3,4-agonal) magic tesseract.

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 August 8, 2016.
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