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A 2-dimensional magic rectangle of order (m_{1}, m_{2}) is defined as an m_{1} x m_{2} 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-(m_{1}, m_{2}) magic rectangle is called normal if the rectangle consists of consecutive integers from 1 to m_{1}m_{2}, 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:

14 | 10 | 4 | 5 | 7 |

1 | 3 | 8 | 13 | 15 |

9 | 11 | 12 | 6 | 2 |

10 | 21 | 9 | 16 | 5 | 14 | 2 |

3 | 4 | 7 | 11 | 15 | 18 | 19 |

20 | 8 | 17 | 6 | 13 | 1 | 12 |

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.

1 | 2 | 3 | 22 | 23 | 24 |

19 | 20 | 21 | 4 | 5 | 6 |

18 | 17 | 16 | 9 | 8 | 7 |

12 | 11 | 10 | 15 | 14 | 13 |

For m_{1}, m_{2} > 1, an order-(m_{1}, m_{2}) 2-dimensional normal magic rectangle exists only if one of the following conditions is hold:

**(1)** Both m_{1} and m_{2} are **even** and at least one of them is greater than 2.

**(2)** Both m_{1} and m_{2} are **odd**.

T. Harmuth published a proof of this theorem in 1881. Thomas R. Hagedorn showed an elementary proof of the theorem with a concrete 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-(m_{1}, m_{2}, ..., m_{n}) n-dimensional magic rectangle, where m_{1}, .., m_{n} > 1 are **even** integers and (m_{i}, m_{j}) is **not** (2,2) for 1 <= i < j <= n,

**(2)** An order-(m_{1}, m_{2}, m_{3}) 3-dimensional magic rectangle, where m_{1}, m_{2}, m_{3} > 1 are **odd** integers and **gcd**(m_{1}, m_{2}) > 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-(m_{1}, m_{2}, ..., m_{n}) n-dimensional magic rectangle, where m_{1}, .., 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-(m_{1}, m_{2}, m_{3}) 3-dimensional magic rectangle can exist or not for any odd m_{1}, m_{2}, m_{3} > 1.

31 | 3 | 38 | 13 | 30 |

24 | 41 | 12 | 36 | 2 |

14 | 25 | 19 | 20 | 37 |

29 | 40 | 4 | 35 | 7 |

1 | 18 | 23 | 28 | 45 |

39 | 11 | 42 | 6 | 17 |

9 | 26 | 27 | 21 | 32 |

44 | 10 | 34 | 5 | 22 |

16 | 33 | 8 | 43 | 15 |

33 | 4 | 37 | 11 | 30 | 33 | 34 |

35 | 23 | 32 | 36 | 13 | 31 | 12 |

10 | 51 | 9 | 31 | 35 | 14 | 32 |

25 | 36 | 24 | 46 | 5 | 44 | 2 |

3 | 34 | 7 | 26 | 45 | 18 | 49 |

50 | 8 | 47 | 6 | 28 | 16 | 27 |

20 | 38 | 17 | 21 | 43 | 1 | 42 |

40 | 21 | 39 | 16 | 20 | 29 | 17 |

18 | 19 | 22 | 41 | 15 | 48 | 19 |

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 |

[1] Thomas R. Hagedorn,

[2] Thomas R. Hagedorn,

[3] Marián Trenkler,

[4] Harvey D. Heinz & John R. Hendricks,

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This page was last updated on August 8, 2016.

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