Magic rectangles

A 2-dimensional magic rectangle of order (m₁, m₂) is defined as an m₁ x m₂ rectangular array in which all rows of the array sum to the same value (called the row constant or the row sum) and all columns of the array also do (the column constant or the column sum). It is not required that diagonals be magic. An order-(m₁, m₂) magic rectangle is called normal if the rectangle consists of consecutive integers from 1 to m₁m₂, and called non-normal if not. This site is concerned only with normal magic rectangles. Similarly, n-dimensional magic rectangles are defined.

The following figures are examples of associated 2-dimensional magic rectangles:

(3,5)-rectangle
(associated)
14	10	4	5	7
1	3	8	13	15
9	11	12	6	2

(3,7)-rectangle
(associated)
10	21	9	16	5	14	2
3	4	7	11	15	18	19
20	8	17	6	13	1	12

(5,7)-rectangle
(associated)
26	19	8	31	25	13	4
20	6	34	24	14	1	27
3	7	15	18	21	29	33
9	35	22	12	2	30	16
32	23	11	5	28	17	10

Magic rectangles of (odd,odd)-order are used in order to construct magic cubes and magic tesseracts (see the pages for Nasik magic cubes and Nasik magic tesseracts). However, it is not easy to construct (odd,odd)-order magic rectangles algorithmically (discuss later). On the other hand, it is not so difficult to construct (even,even)-order magic rectangles (see the figure below). Normal magic rectangles of (odd,even)-order cannot exist.

(4,6)-rectangle
(non-associated)
1	2	3	22	23	24
19	20	21	4	5	6
18	17	16	9	8	7
12	11	10	15	14	13

Existence of magic rectangles

   For m₁, m₂ > 1, an order-(m₁, m₂) 2-dimensional normal magic rectangle exists only if one of the following conditions is hold:
(1) Both m₁ and m₂ are even and at least one of them is greater than 2.
(2) Both m₁ and m₂ are odd.
   T. Harmuth published a proof of this theorem in 1881. Thomas R. Hagedorn showed an elementary proof of the theorem with a concreat construction method in [2]. (The method is elementary but complicated.) He also proved the existence of the following magic rectangles in [1]:

(1) An order-(m₁, m₂, ..., m_n) n-dimenaional magic rectangle, where m₁, .., m_n > 1 are even integers and (m_i, m_j) is not (2,2) for 1 <= i < j <= n,
(2) An order-(m₁, m₂, m₃) 3-dimensional magic rectangle, where m₁, m₂, m₃ > 1 are odd integers and gcd(m₁, m₂) > 1, where gcd means the greatest common divisor.

   Furthermore, it can be proved the existence of the following magic rectangles by generalizing the Hagedorn's proof:
(2)' An order-(m₁, m₂, ..., m_n) n-dimenaional magic rectangle, where m₁, .., m_n > 1 are odd integers and gcd(m_i, m_j) > 1 for some i and j such that1 <= i < j <= n.

   As a result, 3-dimensional magic rectangles of orders (3,3,5), (3,3,7), (3,5,5), (3,5,9), (3,5,15), (3,7,15), etc. do exist. Hagedorn said in [1] that it was an open question whether an order-(3,5,7) 3-dimensional magic rectangle exists or not. I found an order-(3,5,7) 3-dimensional magic rectangle in April 2004, and an order-(3,5,11) and an order-(3,5,13) 3-dimensional magic rectangle in February 2005. I found these magic rectangles by searching by my personal computer (1.46GHz). It is unknown whether an order-(m₁, m₂, m₃) 3-dimensional magic rectangle can exist or not for any odd m₁, m₂, m₃ > 1.

Examples of 3-dimensional magic rectangles

**Plane No.1**
31	3	38	13	30
24	41	12	36	2
14	25	19	20	37

**Plane No.2**
29	40	4	35	7
1	18	23	28	45
39	11	42	6	17

**Plane No.3**
9	26	27	21	32
44	10	34	5	22
16	33	8	43	15

**Plane No.1**
33	4	37	11	30	33	34
35	23	32	36	13	31	12
10	51	9	31	35	14	32

**Plane No.2**
25	36	24	46	5	44	2
3	34	7	26	45	18	49
50	8	47	6	28	16	27

**Plane No.3**
20	38	17	21	43	1	42
40	21	39	16	20	29	17
18	19	22	41	15	48	19

**Plane No.1**
70	54	35	99	22	87	4
63	29	93	45	92	24	25
89	94	38	9	18	67	56
41	31	40	34	48	76	101
2	57	59	78	85	11	79

**Plane No.2**
62	10	103	32	90	23	51
91	100	8	42	1	60	69
20	26	33	53	73	80	86
37	46	105	64	98	6	15
55	83	16	74	3	96	44

**Plane No.3**
27	95	21	28	47	49	104
5	30	58	72	66	75	65
50	39	88	97	68	12	17
81	82	14	61	13	77	43
102	19	84	7	71	52	36

**Plane No.1**
51	28	120	79	145	44	124	75	5	109	133
101	152	20	50	13	130	96	110	128	106	7
119	62	67	17	151	24	126	90	78	31	148
49	141	131	164	22	158	3	1	64	150	30
95	32	77	105	84	59	66	139	140	19	97

**Plane No.2**
129	74	103	143	4	98	43	113	155	6	45
12	81	127	34	73	111	9	137	86	118	125
112	52	94	156	58	83	108	10	72	114	54
41	48	80	29	157	55	93	132	39	85	154
121	160	11	53	123	68	162	23	63	92	37

**Plane No.3**
69	147	26	27	100	107	82	61	89	134	71
136	16	102	165	163	8	144	2	35	25	117
18	135	88	76	40	142	15	149	99	104	47
159	60	38	56	70	36	153	116	146	14	65
33	57	161	91	42	122	21	87	46	138	115

**Plane No.1**
37	106	131	13	116	121	58	74	171	182	187	26	52
84	69	55	180	173	93	175	70	18	31	85	91	150
193	134	36	44	43	101	89	167	54	78	15	192	128
166	79	155	124	12	38	148	2	190	99	8	120	133
10	102	113	129	146	137	20	177	57	100	195	61	27

**Plane No.2**
88	53	162	185	39	154	60	161	73	45	24	174	56
147	149	51	17	115	7	71	66	92	191	168	86	114
33	156	77	132	109	164	98	32	87	64	119	40	163
82	110	28	5	104	130	125	189	81	179	145	47	49
140	22	172	151	123	35	136	42	157	11	34	143	108

**Plane No.3**
169	135	1	96	139	19	176	59	50	67	83	94	186
63	76	188	97	6	194	48	158	184	72	41	117	30
68	4	181	118	142	29	107	95	153	152	160	62	3
46	105	111	165	178	126	21	103	23	16	141	127	112
144	170	9	14	25	122	138	75	80	183	65	90	159

26	19	8	31	25	13	4
20	6	34	24	14	1	27
3	7	15	18	21	29	33
9	35	22	12	2	30	16
32	23	11	5	28	17	10

70	54	35	99	22	87	4
63	29	93	45	92	24	25
89	94	38	9	18	67	56
41	31	40	34	48	76	101
2	57	59	78	85	11	79

62	10	103	32	90	23	51
91	100	8	42	1	60	69
20	26	33	53	73	80	86
37	46	105	64	98	6	15
55	83	16	74	3	96	44

27	95	21	28	47	49	104
5	30	58	72	66	75	65
50	39	88	97	68	12	17
81	82	14	61	13	77	43
102	19	84	7	71	52	36

51	28	120	79	145	44	124	75	5	109	133
101	152	20	50	13	130	96	110	128	106	7
119	62	67	17	151	24	126	90	78	31	148
49	141	131	164	22	158	3	1	64	150	30
95	32	77	105	84	59	66	139	140	19	97

129	74	103	143	4	98	43	113	155	6	45
12	81	127	34	73	111	9	137	86	118	125
112	52	94	156	58	83	108	10	72	114	54
41	48	80	29	157	55	93	132	39	85	154
121	160	11	53	123	68	162	23	63	92	37

69	147	26	27	100	107	82	61	89	134	71
136	16	102	165	163	8	144	2	35	25	117
18	135	88	76	40	142	15	149	99	104	47
159	60	38	56	70	36	153	116	146	14	65
33	57	161	91	42	122	21	87	46	138	115

37	106	131	13	116	121	58	74	171	182	187	26	52
84	69	55	180	173	93	175	70	18	31	85	91	150
193	134	36	44	43	101	89	167	54	78	15	192	128
166	79	155	124	12	38	148	2	190	99	8	120	133
10	102	113	129	146	137	20	177	57	100	195	61	27

88	53	162	185	39	154	60	161	73	45	24	174	56
147	149	51	17	115	7	71	66	92	191	168	86	114
33	156	77	132	109	164	98	32	87	64	119	40	163
82	110	28	5	104	130	125	189	81	179	145	47	49
140	22	172	151	123	35	136	42	157	11	34	143	108

169	135	1	96	139	19	176	59	50	67	83	94	186
63	76	188	97	6	194	48	158	184	72	41	117	30
68	4	181	118	142	29	107	95	153	152	160	62	3
46	105	111	165	178	126	21	103	23	16	141	127	112
144	170	9	14	25	122	138	75	80	183	65	90	159

26	19	8	31	25	13	4
20	6	34	24	14	1	27
3	7	15	18	21	29	33
9	35	22	12	2	30	16
32	23	11	5	28	17	10

70	54	35	99	22	87	4
63	29	93	45	92	24	25
89	94	38	9	18	67	56
41	31	40	34	48	76	101
2	57	59	78	85	11	79

62	10	103	32	90	23	51
91	100	8	42	1	60	69
20	26	33	53	73	80	86
37	46	105	64	98	6	15
55	83	16	74	3	96	44

27	95	21	28	47	49	104
5	30	58	72	66	75	65
50	39	88	97	68	12	17
81	82	14	61	13	77	43
102	19	84	7	71	52	36

51	28	120	79	145	44	124	75	5	109	133
101	152	20	50	13	130	96	110	128	106	7
119	62	67	17	151	24	126	90	78	31	148
49	141	131	164	22	158	3	1	64	150	30
95	32	77	105	84	59	66	139	140	19	97

129	74	103	143	4	98	43	113	155	6	45
12	81	127	34	73	111	9	137	86	118	125
112	52	94	156	58	83	108	10	72	114	54
41	48	80	29	157	55	93	132	39	85	154
121	160	11	53	123	68	162	23	63	92	37

69	147	26	27	100	107	82	61	89	134	71
136	16	102	165	163	8	144	2	35	25	117
18	135	88	76	40	142	15	149	99	104	47
159	60	38	56	70	36	153	116	146	14	65
33	57	161	91	42	122	21	87	46	138	115

37	106	131	13	116	121	58	74	171	182	187	26	52
84	69	55	180	173	93	175	70	18	31	85	91	150
193	134	36	44	43	101	89	167	54	78	15	192	128
166	79	155	124	12	38	148	2	190	99	8	120	133
10	102	113	129	146	137	20	177	57	100	195	61	27

88	53	162	185	39	154	60	161	73	45	24	174	56
147	149	51	17	115	7	71	66	92	191	168	86	114
33	156	77	132	109	164	98	32	87	64	119	40	163
82	110	28	5	104	130	125	189	81	179	145	47	49
140	22	172	151	123	35	136	42	157	11	34	143	108

169	135	1	96	139	19	176	59	50	67	83	94	186
63	76	188	97	6	194	48	158	184	72	41	117	30
68	4	181	118	142	29	107	95	153	152	160	62	3
46	105	111	165	178	126	21	103	23	16	141	127	112
144	170	9	14	25	122	138	75	80	183	65	90	159

Magic rectangles

What are magic rectangles?

Existence of magic rectangles

Examples of 3-dimensional magic rectangles

26	19	8	31	25	13	4
20	6	34	24	14	1	27
3	7	15	18	21	29	33
9	35	22	12	2	30	16
32	23	11	5	28	17	10

70	54	35	99	22	87	4
63	29	93	45	92	24	25
89	94	38	9	18	67	56
41	31	40	34	48	76	101
2	57	59	78	85	11	79

62	10	103	32	90	23	51
91	100	8	42	1	60	69
20	26	33	53	73	80	86
37	46	105	64	98	6	15
55	83	16	74	3	96	44

27	95	21	28	47	49	104
5	30	58	72	66	75	65
50	39	88	97	68	12	17
81	82	14	61	13	77	43
102	19	84	7	71	52	36

51	28	120	79	145	44	124	75	5	109	133
101	152	20	50	13	130	96	110	128	106	7
119	62	67	17	151	24	126	90	78	31	148
49	141	131	164	22	158	3	1	64	150	30
95	32	77	105	84	59	66	139	140	19	97

129	74	103	143	4	98	43	113	155	6	45
12	81	127	34	73	111	9	137	86	118	125
112	52	94	156	58	83	108	10	72	114	54
41	48	80	29	157	55	93	132	39	85	154
121	160	11	53	123	68	162	23	63	92	37

69	147	26	27	100	107	82	61	89	134	71
136	16	102	165	163	8	144	2	35	25	117
18	135	88	76	40	142	15	149	99	104	47
159	60	38	56	70	36	153	116	146	14	65
33	57	161	91	42	122	21	87	46	138	115

37	106	131	13	116	121	58	74	171	182	187	26	52
84	69	55	180	173	93	175	70	18	31	85	91	150
193	134	36	44	43	101	89	167	54	78	15	192	128
166	79	155	124	12	38	148	2	190	99	8	120	133
10	102	113	129	146	137	20	177	57	100	195	61	27

88	53	162	185	39	154	60	161	73	45	24	174	56
147	149	51	17	115	7	71	66	92	191	168	86	114
33	156	77	132	109	164	98	32	87	64	119	40	163
82	110	28	5	104	130	125	189	81	179	145	47	49
140	22	172	151	123	35	136	42	157	11	34	143	108

169	135	1	96	139	19	176	59	50	67	83	94	186
63	76	188	97	6	194	48	158	184	72	41	117	30
68	4	181	118	142	29	107	95	153	152	160	62	3
46	105	111	165	178	126	21	103	23	16	141	127	112
144	170	9	14	25	122	138	75	80	183	65	90	159