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Since May 9, 2004 (Last updated on August 26, 2016)

Magic Cubes and Tesseracts

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What are magic cubes and magic tesseracts?

   A magic cube is defined as a cubical array such that all rows, columns, pillars, and four triagonals of the array sum to the same value (called the (magic) constant or the magic sum). Magic cubes are, as it were, three-dimensional magic squares. An order-m magic cube is called a normal magic cube if the cube consists of consecutive integers from 1 to m3, and called a non-normal magic cube if not. A magic cube in this site is normal if its normality is unspecified.
   It is not required that (2-dimensional) diagonals of a magic cube sum to the constant. A magic cube with the feature that every diagonal sums to the constant is called a diagonal magic cube. A diagonal magic cube can exist only for orders higher than 4.

   Similarly, a magic tesseract is a four-dimensional hypercube whose rows, columns, pillars, files, and eight quadragonals sum to the constant. An order-m normal magic tesseract consists of consecutive integers from 1 to m4. A magic tesseract is called a strictly magic tesseract if all diagonals and all triagonals of the tesseract are magic. The smallest known strictly magic tesseract is an order-8 strictly magic tesseract constructed by the author in 2004.

   Generally, n-dimensional magic hypercubes are defined for every integer n > 1. Marián Trenkler proved that a normal magic hypercube of dimension n and order m can exist for every integer n > 1 and every integer m > 2.

Examples

an order-3 magic cube (the magic constant is 42) : minimum magic cube

Plane No.1
10248
23712
91122
Plane No.2
26115
31425
13272
Plane No.3
61719
16215
20418

an order-4 magic cube (the magic constant is 130) [Yoshihiro Kurushima (?-1757)] : the first magic cube in the world

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

an order-5 diagonal magic cube (the magic constant is 315) [Walter Trump & Christian Boyer, 2003] : minimum diagonal magic cube

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

an order-3 magic tesseract (the magic constant is 123) : minimum magic tesseract

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

   There exist four order-3 magic cubes and 58 order-3 magic tesseracts. For order 4 or higher, the number of magic cubes or magic tesseracts is still unknown. According to Water Trump, the number of order-4 associated magic cubes is exactly 44,447,308,800.
Here are other examples of magic cubes and magic tesseracts.

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