English

Japanese

Home

Site Map

Next ( 1 2 3 )

Magic cubes of each order

This page shows examples of magic cubes of orders from 3 to 8. 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-3 magic cubes (the constant is 42)

There are just four order-3 magic cubes. John R. Hendricks 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
18	22	2
20	9	13
4	11	27

Plane No.2
23	3	16
7	14	21
12	25	5

Plane No.3
1	17	24
15	19	8
26	6	10

Plane No.1
16	23	3
20	9	13
6	10	26

Plane No.2
24	1	17
7	14	21
11	27	4

Plane No.3
2	18	22
15	19	8
25	5	12

Plane No.1
12	23	7
22	9	11
8	10	24

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

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

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

the source: Harvey D. Heinz's site (http://members.shaw.ca/hdhcubes/cube_3.htm)

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

Top

Order-4 magic cubes (the constant is 130)

There exist order-4 magic cubes of classes simple and pantriagonal. Associated magic cubes can exist for both classes. The number of order-4 cubes is unknown. However, 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 (horizontal-diagonal, axis symmetrical) [Yoshihiro Kurushima (?-1757)]

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

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
49	32	48	1
12	37	21	60
8	41	25	56
61	20	36	13

Plane No.2
15	34	18	63
54	27	43	6
58	23	39	10
3	46	30	51

Plane No.3
14	35	19	62
55	26	42	7
59	22	38	11
2	47	31	50

Plane No.4
52	29	45	4
9	40	24	57
5	44	28	53
64	17	33	16

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.154

The above two cubes are the first order-4 magic cubes, which were made by a Japanese mathematician, Yoshihiro Kurushima (?-1757). These cubes have been found from his manuscript.

an order-4 simple magic cube (horizontal-diagonal, associated) [Gakuho Abe]

Plane No.1
8	61	1	60
11	50	14	55
62	7	59	2
49	12	56	13

Plane No.2
45	24	44	17
34	27	39	30
23	46	18	43
28	33	29	40

Plane No.3
25	36	32	37
22	47	19	42
35	26	38	31
48	21	41	20

Plane No.4
52	9	53	16
63	6	58	3
10	51	15	54
5	64	4	57

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.107

The four planes of this cube compose the following order-8 associated magic square.

8	61	1	60	45	24	44	17
11	50	14	55	34	27	39	30
62	7	59	2	23	46	18	43
49	12	56	13	28	33	29	40
25	36	32	37	52	9	53	16
22	47	19	42	63	6	58	3
35	26	38	31	10	51	15	54
48	21	41	20	5	64	4	57

an order-4 simple magic cube (horizontal-pandiagonal, non-associated) [Gensho Abe, 1938]

Plane No.1
64	6	43	17
33	27	54	16
22	48	1	59
11	49	32	38

Plane No.2
2	60	21	47
31	37	12	50
44	18	63	5
53	15	34	28

Plane No.3
61	7	42	20
36	26	55	13
23	45	4	58
10	52	29	39

Plane No.4
3	57	24	46
30	40	9	51
41	19	62	8
56	14	35	25

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

There cannot exist an order-4 magic cube which is both horizontal-pandiagonal and associated.

(2) Order-4 pantriagonal magic cubes

an order-4 pantriagonal magic cube (non-associated, complete) [A. H. Frost, 1878]

Plane No.1
33	31	30	36
28	38	39	25
14	52	49	15
55	9	12	54

Plane No.2
24	42	43	21
45	19	18	48
59	5	8	58
2	64	61	3

Plane No.3
16	50	51	13
53	11	10	56
35	29	32	34
26	40	37	27

Plane No.4
57	7	6	60
4	62	63	1
22	44	41	23
47	17	20	46

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

an order-4 pantriagonal magic cube (non-associated, complete, 2-compact) [Arata Sakai (1908-1964), 1938]

Plane No.1
43	21	42	24
18	48	19	45
39	25	38	28
30	36	31	33

Plane No.2
6	60	7	57
63	1	62	4
10	56	11	53
51	13	50	16

Plane No.3
27	37	26	40
34	32	35	29
23	41	22	44
46	20	47	17

Plane No.4
54	12	55	9
15	49	14	52
58	8	59	5
3	61	2	64

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

an order-4 pantriagonal magic cube (non-associated, non-complete, 2-compact) [Kanji Setsuda]

Plane No.1
1	56	25	48
64	9	40	17
5	52	29	44
60	13	36	21

Plane No.2
63	10	39	18
2	55	26	47
59	14	35	22
6	51	30	43

Plane No.3
4	53	28	45
61	12	37	20
8	49	32	41
57	16	33	24

Plane No.4
62	11	38	19
3	54	27	46
58	15	34	23
7	50	31	42

the source: Kanji Setsuda's site (http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html)

Kanji Setsuda has beed made a lot of studies of order-4 2-compact pantriagonal magic cubes.
A (normal) magic cube cannot be both horizontal-diagonal and 2-compact. (This fact also holds for any dimension higher than 3.)
Furthermore, a (normal) magic cube cannot be both associated and 2-compact. (This fact also holds for any odd dimension.)

an order-4 pantriagonal magic cube (non-associated, non-complete, 'knight-tour') [Guenter Stertenbrink, 2003]

Plane No.1
20	41	14	55
39	30	57	4
10	51	24	45
61	8	35	26

Plane No.2
15	54	17	44
60	1	38	31
21	48	11	50
34	27	64	5

Plane No.3
42	19	56	13
29	40	3	58
52	9	46	23
7	62	25	36

Plane No.4
53	16	43	18
2	59	32	37
47	22	49	12
28	33	6	63

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

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

an order-4 pantriagonal magic cube (associated) [Nakamura, May 2004]

Plane No.1
1	55	14	60
40	29	43	18
30	44	17	39
59	2	56	13

Plane No.2
31	38	20	41
53	3	58	16
4	57	15	54
42	32	37	19

Plane No.3
46	28	33	23
11	50	8	61
49	7	62	12
24	45	27	34

Plane No.4
52	9	63	6
26	48	21	35
47	22	36	25
5	51	10	64

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 37,824 order-4 associated pantriagonal magic cubes.
An order-4 associated panmagic hypercube can exist only if the dimension is odd. This cube is an example for dimension 3 (a panmagic hypercube of dimension 3 means a pantriagonal magic cube). Here are examples for dimensions 5 and 7.
There is a study of order-4 associated pantriagonal magic cubes by Kanji Setsuda.

an order-4 pantriagonal magic cube (horizontal-diagonal, non-associated, complete) [Shigematsu Urata (1889-1958), 1941]

Plane No.1
1	40	62	27
59	30	8	33
24	49	43	14
46	11	17	56

Plane No.2
60	29	7	34
2	39	61	28
45	12	18	55
23	50	44	13

Plane No.3
22	51	41	16
48	9	19	54
3	38	64	25
57	32	6	35

Plane No.4
47	10	20	5
21	52	42	15
58	31	5	36
4	37	63	26

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

Every order-4 pantriagonal magic cube also horizontal-diagonal is complete, so an order-4 (normal) pantriagonal magic cube cannot be both horizontal-diagonal and associated.
Furthermore, an order-4 pantriagonal magic cube also horizontal-pandiagonal cannot exist.

Top

Order-5 magic cubes (the constant is 315)

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

an order-5 simple magic cube (associated) [Hermann Schubert, 1898]

Plane No.1
121	27	83	14	70
10	61	117	48	79
44	100	1	57	113
53	109	40	91	22
87	18	74	105	31

Plane No.2
2	58	114	45	96
36	92	23	54	110
75	101	32	88	19
84	15	66	122	28
118	49	80	6	62

Plane No.3
33	89	20	71	102
67	123	29	85	11
76	7	63	119	50
115	41	97	3	59
24	55	106	37	93

Plane No.4
64	120	46	77	8
98	4	60	111	42
107	38	94	25	51
16	72	103	34	90
30	81	12	68	124

Plane No.5
95	21	52	108	39
104	35	86	17	73
13	69	125	26	82
47	78	9	65	116
56	112	43	99	5

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

an order-5 simple magic cube (horizontal-diagonal, associated) [Theodor Hugel, 1876]

Plane No.1
93	121	62	4	35
52	9	45	98	111
50	88	101	57	19
106	67	24	40	78
14	30	83	116	72

Plane No.2
12	29	85	118	71
95	123	61	2	34
51	7	44	100	113
49	90	103	56	17
108	66	22	39	80

Plane No.3
110	68	21	37	79
11	27	84	120	73
94	125	63	1	32
53	6	42	99	115
47	89	105	58	16

Plane No.4
46	87	104	60	18
109	70	23	36	77
13	26	82	119	75
92	124	65	3	31
55	8	41	97	114

Plane No.5
54	10	43	96	112
48	86	102	59	20
107	69	25	38	76
15	28	81	117	74
91	122	64	5	33

the source: Harvey D. Heinz's site (http://members.shaw.ca/hdhcubes/cube_5.htm)

an order-5 simple magic cube (s-magic, bordered, non-associated) [Walter Trump, 2003]

Plane No.1
33	22	113	42	105
2	89	106	19	99
85	82	4	119	25
98	90	6	112	9
97	32	86	23	77

Plane No.2
95	80	111	11	18
91	55	65	69	35
16	66	70	53	110
5	68	54	67	121
108	46	15	115	31

Plane No.3
100	17	48	34	116
114	75	52	62	12
83	50	63	76	43
8	64	74	51	118
10	109	78	92	26

Plane No.4
38	102	3	125	47
81	59	72	58	45
30	73	56	60	96
87	57	61	71	39
79	24	123	1	88

Plane No.5
49	94	40	103	29
27	37	20	107	124
101	44	122	7	41
117	36	120	14	28
21	104	13	84	93

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
21	40	54	93	107
112	1	45	59	98
78	117	6	50	64
69	83	122	11	30
35	74	88	102	16

Plane No.2
37	51	95	109	23
3	42	56	100	114
119	8	47	61	80
85	124	13	27	66
71	90	104	18	32

Plane No.3
53	92	106	25	39
44	58	97	111	5
10	49	63	77	116
121	15	29	68	82
87	101	20	34	73

Plane No.4
94	108	22	36	55
60	99	113	2	41
46	65	79	118	7
12	26	70	84	123
103	17	31	75	89

Plane No.5
110	24	38	52	91
96	115	4	43	57
62	76	120	9	48
28	67	81	125	14
19	33	72	86	105

the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, pp.163-164

an order-5 diagonal magic cube (non-associated) [Walter Trump & Christian Boyer, 2003]

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

the source: Walter Trump's site (http://www.trump.de/magic-squares/magic-cubes/cubes-1.html)
The term perfect means diagonal in his site.

This cube by Walter Trump & Christian Boyer is the first order-5 diagonal magic cube in the world. They found the cube by computer. They also found other order-5 diagonal magic cubes, which are listed to Aale de Winkel's site.

Top

Home

Site Map

Next ( 1 2 3 )

Mitsutoshi Nakamura (Feedback) To send an email to me, please enable JavaScript on your browser.
This page was last updated on March 2, 2008.
"Magic Cubes and Tesseracts" http://homepage2.nifty.com/googol/magcube/en/
Copyright © 2004-2008, Mitsutoshi Nakamura. All rights reserved.

	blog SEO tool

8	61	1	60	45	24	44	17
11	50	14	55	34	27	39	30
62	7	59	2	23	46	18	43
49	12	56	13	28	33	29	40
25	36	32	37	52	9	53	16
22	47	19	42	63	6	58	3
35	26	38	31	10	51	15	54
48	21	41	20	5	64	4	57

8	61	1	60	45	24	44	17
11	50	14	55	34	27	39	30
62	7	59	2	23	46	18	43
49	12	56	13	28	33	29	40
25	36	32	37	52	9	53	16
22	47	19	42	63	6	58	3
35	26	38	31	10	51	15	54
48	21	41	20	5	64	4	57

8	61	1	60	45	24	44	17
11	50	14	55	34	27	39	30
62	7	59	2	23	46	18	43
49	12	56	13	28	33	29	40
25	36	32	37	52	9	53	16
22	47	19	42	63	6	58	3
35	26	38	31	10	51	15	54
48	21	41	20	5	64	4	57