Magic Squares may be perhaps the only area of recreational mathematics to which many of us have been exposed. The classic form of a magic square is a square containing consecutive numbers starting with 1, in which the rows and columns and the diagonals all total to the same number.
I'll have to admit that I was never very much interested by magic squares, as opposed to other mathematical amusements, but a Mathematical Games column in Scientific American by Martin Gardner disclosed some new discoveries in magic squares that are of interest.
The only magic square of order 3, except for trivial translations such as reflection and rotation, is:
6 1 8 7 5 3 2 9 4
Some magic squares are very simple to construct. Magic squares of any odd order can be constructed following a pattern very similar to that of the 3 by 3 magic square:
8 1 6 17 24 1 8 15 30 39 48 1 10 19 28 47 58 69 80 1 12 23 34 45
3 5 7 23 5 7 14 16 38 47 7 9 18 27 29 57 68 79 9 11 22 33 44 46
4 9 2 4 6 13 20 22 46 6 8 17 26 35 37 67 78 8 10 21 32 43 54 56
10 12 19 21 3 5 14 16 25 34 36 45 77 7 18 20 31 42 53 55 66
11 18 25 2 9 13 15 24 33 42 44 4 6 17 19 30 41 52 63 65 76
21 23 32 41 43 3 12 16 27 29 40 51 62 64 75 5
22 31 40 49 2 11 20 26 28 39 50 61 72 74 4 15
36 38 49 60 71 73 3 14 25
37 48 59 70 81 2 13 24 35
One can also construct a magic square by making a square array of copies of a magic square, and then adding a displacement to the elements of each copy based on a plan given by another magic square:
thus, making nine copies of
8 1 6 3 5 7 4 9 2
and subtracting one and multiplying by 9 to get the offsets of
63 0 45 18 36 54 27 72 9
we form the square:
71 64 69 8 1 6 53 46 51 66 68 70 3 5 7 48 50 52 67 72 65 4 9 2 49 54 47 26 19 24 44 37 42 62 55 60 21 23 25 39 41 43 57 59 61 22 27 20 40 45 38 58 63 56 35 28 33 80 73 78 17 10 15 30 32 34 75 77 79 12 14 16 31 36 29 76 81 74 13 18 11
Starting from a 4 by 4 magic square, such as
1 12 8 13 15 6 10 3 14 7 11 2 4 9 5 16
we can trivially form magic squares of the even orders 12, 20, 28, 36 by combining this square with one of odd order.
A variation on this technique works for doubling the order of a magic square of even order. We can construct the following magic square of order 8:
1 60 8 61 17 44 24 45 63 6 58 3 47 22 42 19 14 55 11 50 30 39 27 34 52 9 53 16 36 25 37 32 33 28 40 29 49 12 56 13 31 38 26 35 15 54 10 51 46 23 43 18 62 7 59 2 20 41 21 48 4 57 5 64
from the order-4 square given above by making four copies of it, and then using the array:
1/4 2/3 3/2 4/1
instead of a real magic square, as our basis for the offsets: that is, in this case the offsets will look like:
0 48 0 48 16 32 16 32 48 0 48 0 32 16 32 16 0 48 0 48 16 32 16 32 48 0 48 0 32 16 32 16 32 16 32 16 48 0 48 0 16 32 16 32 0 48 0 48 32 16 32 16 48 0 48 0 16 32 16 32 0 48 0 48
First we could construct a magic square of any odd order, then one of four times any odd order, and now we can construct one whose order is four times any number.
Magic squares of orders 6, 10, 14, 18, and so on also exist, but the general rule for their construction is more complicated.
It has long been known that there are 880 different 4 by 4 magic squares, and it was recently discovered that the number of different 5 by 5 magic squares is 275,305,224. Other transformations in addition to reflection and rotation can relate all order-5 magic squares to a smaller number of "really" different squares, but those other transformations could affect some of the other properties of the square.
The following magic hexagon,
18 17 3
11 1 7 19
9 6 5 2 16
14 8 4 12
15 13 10
is the only magic hexagon possible of any size except the trivial one consisting of only the number 1.
In 1890, an 8 x 8 magic square was devised by one G. Pfeffermann in which not only the numbers, but the squares of the numbers as well, along each row, column, and diagonal summed to the same total. A square of this type is called bimagic.
The square
58 12 51 1 47 29 38 24 6 56 15 61 19 33 26 44 36 18 41 27 53 7 64 14 32 46 21 39 9 59 4 50 11 57 2 52 30 48 23 37 55 5 62 16 34 20 43 25 17 35 28 42 8 54 13 63 45 31 40 22 60 10 49 3
is one example of such a square.
There are 80 possible 8 by 8 bimagic squares. A link previously provided for a listing of all of them is now broken, but it should be possible through a web search to find sites about magic squares with more information than this page.
In 1905, a 128 by 128 magic square was devised by Gaston Tarry where the numbers, their squares, and their cubes were all magic; this is called a trimagic square. This was cut in half to 64 by 64 by General Cazalas in 1933, and halved again by William Benson to 32 by 32.
Very recently, although a 16 by 16 trimagic square with the numbers from 1 to 256 is not yet known, a 12 by 12 trimagic square was discovered by Walter Trump in June 2002:
1 22 33 41 62 66 79 83 104 112 123 144 9 119 45 115 107 93 52 38 30 100 26 136 75 141 35 48 57 14 131 88 97 110 4 70 74 8 106 49 12 43 102 133 96 39 137 71 140 101 124 42 60 37 108 85 103 21 44 5 122 76 142 86 67 126 19 78 59 3 69 23 55 27 95 135 130 89 56 15 10 50 118 90 132 117 68 91 11 99 46 134 54 77 28 13 73 64 2 121 109 32 113 36 24 143 81 72 58 98 84 116 138 16 129 7 29 61 47 87 80 34 105 6 92 127 18 53 139 40 111 65 51 63 31 20 25 128 17 120 125 114 82 94
It has the property that if it is reflected left to right, and the value in each square is replaced by 145 minus that value, that it remains the same square.
In 2001, a 512 by 512 square with powers up to the fourth power being magic, and a 1024 by 1024 square with powers up to the fifth power being magic, were announced by C. Boyer and A. Viricel.
The traditional definition of a magic cube has been one in which the rows in both directions, the columns, and the diagonals between opposite corners going through the center of the cube all add to the same total.
An example of such a cube of order 3 is:
19 15 8 5 25 12 18 2 22 17 1 24 21 14 7 4 27 11 6 26 10 16 3 23 20 13 9
However, there are other diagonals in a cube.
A cube of order 6 in which the diagonals on the faces of the cube are also magic was devised by John Worthington some time before 1917:
111 1 2 213 216 108 194 117 120 9 13 198 197 119 118 14 10 193 20 206 207 98 101 19 24 204 201 102 97 23 105 4 3 215 214 110 50 185 186 86 88 56 92 61 64 169 171 94 96 59 58 173 170 95 179 79 76 67 72 178 181 78 80 69 66 177 53 189 187 87 84 51 55 192 191 83 81 49 93 60 57 176 174 91 89 62 63 172 175 90 182 74 77 70 65 183 180 75 73 68 71 184 52 188 190 82 85 54 163 135 136 25 27 165 36 145 149 144 138 39 40 146 147 139 141 38 121 48 42 156 159 125 123 43 45 158 160 122 168 134 132 29 26 162 166 130 129 32 30 164 37 152 148 137 143 34 33 151 150 142 140 35 128 41 47 157 154 124 126 46 44 155 153 127 161 131 133 28 31 167 106 8 7 212 209 109 199 116 113 16 12 195 196 114 115 11 15 200 21 203 202 103 100 22 17 205 208 99 104 18 112 5 6 210 211 107
Like other cubes and squares presented here, I have oriented it to place the lowest numbers in the earliest positions - in this case, only by reversing the order of the layers from the version originally constructed.
Order 6 is a difficult order in which to construct magic squares as well as cubes. From any Graeco-Latin square, it is possible to construct trivially a square that is a magic square except for its diagonals (although they possibly can add to the right total as well, so some magic squares do correspond directly to such squares). There can be no Graeco-Latin square of order 6, and for orders 10, 14, 18, 22, 26, and so on, such squares can only be produced by a method which was very recently discovered.
For orders 7 and 8, there are magic cubes that are perfect, in which not only the space diagonals, but the diagonals of every layer in each of the three directions, add to the correct total. The first order 8 perfect magic cube was discovered in 1875, but the first order 7 one was discovered only in 1962. There are such cubes for higher orders as well.