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Magic cubes of each order

   This page shows examples of magic cubes of orders from 3 to 10. Every magic cube in this page is normal.
Definitions of terms are here. Explanation of classes of magic cubes is here. Examples of magic tesseracts are here.
[Order 3] [Order 4] [Order 5] [Order 6] [Order 7] [Order 8] [Order 9] [Order 10]

Order-3 magic cubes (the constant is 42)

   There are exactly four order-3 magic cubes. John R. Hendricks (1929-2007) proved that in 1972 ([1])(*). All order-3 magic cubes are associated and simple.
(*) Before Hendricks, the following persons claimed independent of one another that the number of order-3 magic cubes is 4 ([2]):
     Eisuke Ishikawa(1933), D. N. Lehmer(1934), Gensho Abe(1939).

[1] John R. Hendricks, The Third-Order Magic Cube Complete, Journal of Recreational Mathematics 5:1(1972), 43-50.
[2] Akira Hirayama & Gakuho Abe, Researches in Magic Squares (Japanese), Osaka Kyoikutosho, 1983, 151.

Plane No.1
18222
20913
41127
Plane No.2
23316
71421
12255
Plane No.3
11724
15198
26610
Plane No.1
16233
20913
61026
Plane No.2
24117
71421
11274
Plane No.3
21822
15198
25512
Plane No.1
12237
22911
81024
Plane No.2
26115
31425
13272
Plane No.3
41820
17196
21516
Plane No.1
10248
23712
91122
Plane No.2
26115
31425
13272
Plane No.3
61719
16215
20418
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

   T. Hugel made the first order-3 magic cube in 1876. His cube is equivalent to the first of the above cubes.

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Order-4 magic cubes (the constant is 130)

   There exist order-4 magic cubes of classes simple and pantriagonal. Associated magic cubes of order 4 can exist for both classes. The number of order-4 magic cubes is unknown. Water Trump discovered in 2003 that the number of order-4 associated magic cubes is exactly 44,447,308,800.

(1) Order-4 simple magic cubes

an order-4 simple magic cube (axis symmetrical, horizontal-diagonal) [Yoshihiro Kurushima (?-1757)]
Plane No.1
162634
44232241
24434221
612364
Plane No.2
607657
17464720
45181948
859585
Plane No.3
56111053
29343532
33303136
1255549
Plane No.4
13505116
40272637
28393825
49141552
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.154

an order-4 simple magic cube (associated) [Yoshihiro Kurushima]
Plane No.1
4932481
12372160
8412556
61203613
Plane No.2
15341863
5427436
58233910
3463051
Plane No.3
14351962
5526427
59223811
2473150
Plane No.4
5229454
9402457
5442853
64173316
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.154

An order-4 magic cube was first constructed by a Japanese mathematician, Yoshihiro Kurushima (?-1757). These two magic cubes were found from his manuscript.
Either of these cubes is the first magic cube in the world.

an order-4 simple magic cube (associated, horizontal-diagonal) [Gakuho Abe]
Plane No.1
861160
11501455
627592
49125613
Plane No.2
45244417
34273930
23461843
28332940
Plane No.3
25363237
22471942
35263831
48214120
Plane No.4
5295316
636583
10511554
564457
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.107

This magic cube is associated and each of the four horizontal planes of the cube is an order-4 magic square.
The four horizontal planes compose the following order-8 associated magic square.
86116045244417
1150145534273930
62759223461843
4912561328332940
253632375295316
22471942636583
3526383110511554
48214120564457


an order-4 simple magic cube (non-associated, horizontal-pandiagonal, 3D-compact) [Gensho Abe, 1938]
Plane No.1
6464317
33275416
2248159
11493238
Plane No.2
2602147
31371250
4418635
53153428
Plane No.3
6174220
36265513
2345458
10522939
Plane No.4
3572446
3040951
4119628
56143525
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

Every horizontal plane of this cube is an order-4 pandiagonal magic square which is also complete and compact.

an order-4 simple magic cube (associated, 3D-compact) [Nakamura, November 2007]
Plane No.1
1485724
6318742
36132853
30513811
Plane No.2
6021445
6436219
25563316
39103150
Plane No.3
15345526
4932940
4632259
2061445
Plane No.4
54271435
12375229
2358472
4181764
the source: original.

Every order-4 associated and 3D-compact magic cube belongs to the class 'simple'.

(2) Order-4 pantriagonal magic cubes

an order-4 pantriagonal magic cube (non-associated, complete) [A. H. Frost, 1878]
Plane No.1
33313036
28383925
14524915
5591254
Plane No.2
24424321
45191848
595858
264613
Plane No.3
16505113
53111056
35293234
26403727
Plane No.4
577660
462631
22444123
47172046
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

This cube seems to be the first order-4 pantriagonal magic cube.

an order-4 pantriagonal magic cube (non-associated, complete, 2D-compact) [Arata Sakai (1908-1964), 1938]
Plane No.1
43214224
18481945
39253828
30363133
Plane No.2
660757
631624
10561153
51135016
Plane No.3
27372640
34323529
23412244
46204717
Plane No.4
5412559
15491452
588595
361264
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

This magic cube is pantraigonal, complete, and 2D-compact.
Every 2D-compact magic cube is pantraigonal.

an order-4 pantriagonal magic cube (non-associated, non-complete, 2D-compact) [Kanji Setsuda]
Plane No.1
1562548
6494017
5522944
60133621
Plane No.2
63103918
2552647
59143522
6513043
Plane No.3
4532845
61123720
8493241
57163324
Plane No.4
62113819
3542746
58153423
7503142
the source: Kanji Setsuda's site (http://kanjisetsuda.la.coocan.jp/pages/EnglishP1.html)

This magic cube is pantraigonal and 2D-compact but not complete. (A 2D-compact magic cube cannot be associated.)
Kanji Setsuda has beed made a lot of studies of order-4 2D-compact pantriagonal magic cubes.

an order-4 pantriagonal magic cube (non-associated, complete, horizontal-diagonal) [Shigematsu Urata (1889-1958), 1941]
Plane No.1
1406227
5930833
24494314
46111756
Plane No.2
6029734
2396128
45121855
23504413
Plane No.3
22514116
4891954
3386425
5732635
Plane No.4
4710205
21524215
5831536
4376326
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

This magic cube is pantriagonal and complete, and each of the four horizontal planes of the cube is an order-4 magic square.
An order-4 pantriagonal magic cube cannot be 2D-compact or 3D-compact if all of its horizontal planes are magic.

an order-4 pantriagonal magic cube (non-associated, non-complete, 'knight-tour') [Guenter Stertenbrink, 2003]
Plane No.1
20411455
3930574
10512445
6183526
Plane No.2
15541744
6013831
21481150
3427645
Plane No.3
42195613
2940358
5294623
7622536
Plane No.4
53164318
2593237
47224912
2833663
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

This magic cube is pantriagonal and the 64 consecutive integers of the cube trace out a magic knight tour.

an order-4 pantriagonal magic cube (plane-symmetrical, 2D-compact) [Nakamura, May 2004]
Plane No.1
1323364
5641249
13204552
6037285
Plane No.2
6235303
11224354
50471815
7263958
Plane No.3
4293661
53442112
16174849
5740258
Plane No.4
6334312
10234255
51461914
6273859
Plane No.1
1323364
5641249
13204552
6037285
Plane No.2
6235303
11224354
50471815
7263958
Plane No.3
19145146
3859627
3126334
42551023
Plane No.4
48491617
2585740
3661429
21125344
the source: original

Both of these magic cubes are pantriagonal, plane symmetrical, and 2D-compact.
These pantriagonal magic cubes are essentially diffrent from each other though they share the 1st and the 2nd plane.

an order-4 pantriagonal magic cube (associated) [Nakamura, May 2004]
Plane No.1
1551460
40294318
30441739
5925613
Plane No.2
31382041
5335816
4571554
42323719
Plane No.3
46283323
1150861
4976212
24452734
Plane No.4
529636
26482135
47223625
5511064
the source: original. Here is this cube of CSV format.

This is an order-4 magic cube which is both pantriagonal and associated. It was once believed that such a magic cube could not exist. According to Water Trump, there are exactly 37,824 order-4 associated pantriagonal magic cubes.
An order-4 associated pantriagonal magic cube cannot be 2D-compact or 3D-compact.
There is a study of order-4 associated pantriagonal magic cubes by Kanji Setsuda.

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Order-5 magic cubes (the constant is 315)

   There exist order-5 magic cubes of classes simple, pantriagonal, and diagonal. An associated magic cube of order 5 can exist for the classes simple and pantriagonal. It is unknown whether an associated diagonal magic cube of order 5 can exist or not.

an order-5 simple magic cube (associated) [Hermann Schubert, 1898]
Plane No.1
12127831470
10611174879
44100157113
53109409122
87187410531
Plane No.2
2581144596
36922354110
75101328819
84156612228
1184980662
Plane No.3
33892071102
67123298511
7676311950
1154197359
24551063793
Plane No.4
6412046778
9846011142
10738942551
16721033490
30811268124
Plane No.5
95215210839
10435861773
13691252682
4778965116
5611243995
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

an order-5 simple magic cube (associated, horizontal-diagonal) [Theodor Hugel, 1876]
Plane No.1
9312162435
5294598111
50881015719
10667244078
14308311672
Plane No.2
12298511871
9512361234
51744100113
49901035617
10866223980
Plane No.3
11068213779
11278412073
9412563132
5364299115
47891055816
Plane No.4
46871046018
10970233677
13268211975
9212465331
5584197114
Plane No.5
54104396112
48861025920
10769253876
15288111774
9112264533
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

This cube is associated and every horizontal plane of the cube is an order-5 pandiagonal magic square.
This cube seems to be the first order-5 magic cube in the world.

an order-5 simple magic cube (non-associated, s-magic, bordered) [Walter Trump, 2003]
Plane No.1
332211342105
2891061999
8582411925
989061129
9732862377
Plane No.2
95801111118
9155656935
16667053110
5685467121
108461511531
Plane No.3
100174834116
11475526212
8350637643
8647451118
10109789226
Plane No.4
38102312547
8159725845
3073566096
8757617139
7924123188
Plane No.5
49944010329
273720107124
10144122741
117361201428
21104138493
the source: Walter Trump's site (http://www.trump.de/magic-squares/magic-cubes/cubes-1.html)

This cube is an s-magic cube. Furthermore, the cube contains within it an order-3 magic cube, which consists of consecutive integers from 50 to 76, so it is also a bordered (namely, consecutively concentric) magic cube.
Trump says that an order-5 bordered diagonal magic cube cannot exist.

an order-5 pantriagonal magic cube (associated) [Yoshio Moriyama, 1967]
Plane No.1
21405493107
1121455998
7811765064
69831221130
35748810216
Plane No.2
37519510923
34256100114
1198476180
85124132766
71901041832
Plane No.3
53921062539
4458971115
10496377116
12115296882
87101203473
Plane No.4
94108223655
6099113241
4665791187
12267084123
10317317589
Plane No.5
11024385291
9611544357
6276120948
28678112514
19337286105
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, pp.163-164

Is this cube the first order-5 pantriagonal magic cube?

an order-5 diagonal magic cube (non-associated) [Walter Trump & Christian Boyer, 2003]
Plane No.1
25168010490
115984197
4211185275
66722710248
67181191065
Plane No.2
917771670
52641176913
301182112323
26399244114
11617147395
Plane No.3
4761457686
10743383394
8968635837
3293888319
4050816579
Plane No.4
315311210910
12823487100
1033105896
1135796274
56120554935
Plane No.5
12110872059
292812212511
51154112484
7854992460
361104622101
the source: Walter Trump's site (http://www.trump.de/magic-squares/magic-cubes/cubes-1.html)

This cube by Walter Trump & Christian Boyer is the first order-5 diagonal magic cube in the world. It was once believed that such a magic cube could not exist. They found this cube by computer. They also found other order-5 diagonal magic cubes, which are listed to Aale de Winkel's site.

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This page was last updated on August 8, 2016.
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