Magic Cubes and Tesseracts

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What's new

Fixed an error of an algorithm for 2,3,4-agonal magic tesseracts of odd order (the old algorithm generated a non-normal magic tesseract for some order) (Algorithm to make strictly magic tesseracts). [June 21, 2009] New!
Added to a link to Arie Breedijk's site (Links and Bibliography). [June 21, 2009] New!
Added a magic hypercube of order 8 and dimension 5 which is associated, 3-compact, and pan-3,5-agonal (Works on magic tesseracts and hypercubes). [June 13, 2009] New!
Added a magic hypercube of order 8 and dimension 5 with a knight tour, and added a comment on magic knight tours on hypercubes (Works on magic tesseracts and hypercubes). [May 17, 2009]
Added 2 magic hypercubes of order 4 and dimensions 5 and 7 with knight tours (Works on magic tesseracts and hypercubes). [April 13, 2009]
Changed the e-mail address. [September 7, 2008]

Click here to see the history of updating this site.

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. This site is concerned only with normal magic cubes in principle.
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
10	24	8
23	7	12
9	11	22

Plane No.2
26	1	15
3	14	25
13	27	2

Plane No.3
6	17	19
16	21	5
20	4	18

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

Plane No.1
1	62	63	4
44	23	22	41
24	43	42	21
61	2	3	64

Plane No.2
60	7	6	57
17	46	47	20
45	18	19	48
8	59	58	5

Plane No.3
56	11	10	53
29	34	35	32
33	30	31	36
12	55	54	9

Plane No.4
13	50	51	16
40	27	26	37
28	39	38	25
49	14	15	52

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

Plane No.1
25	16	80	104	90
115	98	4	1	97
42	111	85	2	75
66	72	27	102	48
67	18	119	106	5

Plane No.2
91	77	71	6	70
52	64	117	69	13
30	118	21	123	23
26	39	92	44	114
116	17	14	73	95

Plane No.3
47	61	45	76	86
107	43	38	33	94
89	68	63	58	37
32	93	88	83	19
40	50	81	65	79

Plane No.4
31	53	112	109	10
12	82	34	87	100
103	3	105	8	96
113	57	9	62	74
56	120	55	49	35

Plane No.5
121	108	7	20	59
29	28	122	125	11
51	15	41	124	84
78	54	99	24	60
36	110	46	22	101

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

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

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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Mitsutoshi Nakamura (Feedback) To send an email to me, please enable JavaScript on your browser.
Changed the e-mail address. (September 7, 2008)
"Magic Cubes and Tesseracts" http://homepage2.nifty.com/googol/magcube/en/
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