Introduction
1. Some time ago, I was given a copy of `The Wonders of Magic Squares' by
Jim Moran , published by Vintage Books. It made much of the difficulty of
making 6 by 6 magic squares and contained a letter from Benjamin Franklin
to Peter Collinson in which he claimed to be able to make magic squares as
fast as he could write down the numbers. This claim was said to be "pushing
reasonable exaggeration beyond the limit" in a letter from Herb the falconer
(Kahn) to Jim Moran.
2. I found a method of making 2n by 2n magic squares from n by n magic
squares which extends to a method of making order 2n magic cubes from those
of order n. With this method and some practice, magic squares can be made in
about the time it takes to write down the numbers twice.
3. The method can be thought of as generalisation of the well known method
of making mn by mn magic squares from m by m and n by n squares. The well
known method uses n^2 order m squares where the i'th square is a magic
square made of the numbers 1 + (i-1)*m^2,... i*m^2 and it uses the order
n square to place them in the order mn square. My method makes order 2n
squares from 4 order n squares and uses a `construction square' to place
their elements in the larger square.
4. The following describes the method and shows the making of a 6 by 6
magic square and a 6 by 6 by 6 magic cube. I note that in Magic Squares
and Cubes by W.S Andrews p.189, there is a reference to an article which
states that the first magic cube of this order was found in 1889 by
W. Firth, Scholar of Emanuel, Cambridge.
A 6 by 6 magic square
5. To illustrate the method, I will construct a 6 by 6 magic square. The
first stage is to construct a square which I will call a
`construction square'. These are easily construced as I'll show later.
1 3 2 | 3 2 4 It has the following two defining properties:
2 4 4 | 1 2 2 i) Each row, column and diagonal sum to the same number -
4 3 3 | 2 2 1 in this case 15 (5*n).
------|------ ii) It is made from the numbers 1,2,3 and 4 and when split
2 1 1 | 4 4 3 into 4 3 by 3 squares no two squares have the same number
3 3 1 | 4 1 3 in the same position.
3 1 4 | 1 4 2
6. If we let C denote the above construction square and M_3 the 3 by 3 magic
square, then the 6 by 6 magic square is given by
M_6[i,j] = M_3[i mod 3,j mod 3] + 9*(C[i,j]-1)
7. In English, to fill a cell in a 2n by 2n magic square look at the
corresponding cell in the construction cube. The number in that cell tells
you which of the 4 n by n magic squares you are to use and the position of
that cell in one of the n by n subsquares tells you which cell of the n by n
magic square to use.
8. Property i) of C ensures that the rows columns and diagonals of M 6 add
up to the same number and property ii) ensures that we use each of the numbers
1, 2... (2n)^2 once and once only.
9. The above construction gives the following magic square. The first number
in each expression and the first number after each open bracket come from
the 3 by 3 magic square the construction square respectively.
8+9(1-1) 1+9(3-1) 6+9(2-1) | 8+9(3-1) 1+9(2-1) 6+9(4-1)
3+9(2-1) 5+9(4-1) 7+9(4-1) | 3+9(1-1) 5+9(2-1) 7+9(2-1)
4+9(4-1) 9+9(3-1) 2+9(3-1) | 4+9(2-1) 9+9(2-1) 2+9(1-1)
-----------------------------|-----------------------------
8+9(2-1) 1+9(1-1) 6+9(1-1) | 8+9(4-1) 1+9(4-1) 6+9(3-1)
3+9(3-1) 5+9(3-1) 7+9(1-1) | 3+9(4-1) 5+9(1-1) 7+9(3-1)
4+9(3-1) 9+9(1-1) 2+9(4-1) | 4+9(1-1) 9+9(4-1) 2+9(2-1)
Constructing Construction Squares
10. The order 6 construction square was easily made. I filled in the corners
of the 4 order 3 subsquares using the following 4 by 4 construction square
and filled the rest in easily by hand.
1 2 3 4
4 3 2 1
2 1 4 3
3 4 1 2
11. Given a 2n by 2n construction square A B , we can make a
C D
2(n + 2) by 2(n + 2) square as follows:
1 | 1 . . ._1_| 2 | 3 | 4 . . ._4_| 4
--|-----------|---|---|-----------|--
1 | | 4 | 2 | | 3
. | | . | . | | .
. | A | . | . | B | .
. | | . | . | | .
1 | | 4 | 2 | | 3
--|-----------|---|---|-----------|--
4 | 4 . . ._4_| 3 | 2 | 1 . . ._1_| 1
--|-----------|---|---|-----------|--
2 | 2 . . ._2_| 1 | 4 | 3 . . ._3_| 3
__|___________|___|___|___________|__
4 | | 1 | 3 | | 2
. | | . | . | | .
. | C | . | . | D | .
. | | . | . | | .
4 | | 1 | 3 | | 2
--|-----------|---|---|-----------|--
3 | 3 . . ._3_| 4 | 1 | 2 . . ._2_|
The corners of the four subsquares come from a 4 by 4 construction square,
the centers come from the the 2n by 2n construction and the remaining gaps
have been filled in easily in sets of 8 (as indicated by the _i_ numbers)
so as to make sure that the two properties hold.
12.
1 2 | 1 2 | 3 4 | 3 4 This 8 by 8 square demonstrates the construction
4 3 | 4 3 | 2 1 | 2 1 of 4n by 4n construction squares directly from the
----|-----|-----|---- 4 by 4 square.
1 2 | 1 2 | 3 4 | 3 4
4 3 | 4 3 | 2 1 | 2 1 We can now make any 2n by 2n construction square
----|-----|-----|---- starting from either a 4 by 4 or 6 by 6 squares
2 1 | 2 1 | 4 3 | 4 3 and using the above expansion method.
3 4 | 3 4 | 1 2 | 1 2
----|-----|-----|----
2 1 | 2 1 | 4 3 | 4 3
3 4 | 3 4 | 1 2 | 1 2
Extension To Magic Cubes
13. This idea of making magic squares from smaller ones and construction
squares extends to magic cubes and as with squares, we can make any even
order construction cube.
14. In some of the following cubes, I have had to use more than one line
so I should make it clear that the layers are to be read from left to right.
Reading them this way means that neighbouring layers are also neighbours in
the cube.
15. The following are 4 layers of an order 4 construction cube - analagous
to the 4 by 4 construction square.
1 8 | 8 1 6 3 | 3 6 4 5 | 5 4 7 2 | 2 7
8 1 | 1 8 3 6 | 6 3 5 4 | 4 5 2 7 | 7 2
----|---- ----|---- ----|---- ----|----
7 2 | 2 7 4 5 | 5 4 6 3 | 3 6 1 8 | 8 1
2 7 | 7 2 5 4 | 4 5 3 6 | 6 3 8 1 | 1 8
16. These construction cubes have the additive properties required of the
magic cube, they are made from the numbers 1,2,...,8 and no 2 of the 8
subcubes have the same number in the same postion.
17. I made the following order 6 construction cube by filling in the corners
of the 8 order 3 cubes using the above order 4 cube and then filling in the
rest by trial and error. This was much harder than the 6 by 6 construction
square, but not too hard to be done by hand.
1 6 8 | 8 3 1 1 8 8 | 7 1 2 6 3 3 | 3 6 6
6 8 5 | 4 3 1 8 1 4 | 6 6 2 3 2 6 | 1 8 7
8 2 1 | 1 7 8 6 1 4 | 1 8 7 3 6 6 | 6 3 3
------|------ ------|------ ------|------
7 4 2 | 2 5 7 4 6 5 | 6 3 3 4 5 5 | 5 4 4
3 6 4 | 5 1 8 1 7 5 | 3 4 7 6 3 3 | 8 5 2
2 1 7 | 7 8 2 7 4 1 | 4 5 6 5 8 4 | 4 1 5
4 1 5 | 5 8 4 8 2 1 | 2 7 7 7 7 2 | 2 2 7
1 5 3 | 7 4 7 2 5 8 | 4 2 6 7 6 1 | 5 4 4
5 4 4 | 4 5 5 3 7 5 | 8 2 2 2 7 7 | 7 2 2
------|------ ------|------ ------|------
6 7 3 | 3 2 6 5 4 4 | 3 5 6 1 1 8 | 8 8 1
8 7 6 | 2 2 2 7 3 1 | 5 8 3 2 1 8 | 4 7 5
3 3 6 | 6 6 3 2 6 8 | 5 3 3 8 5 1 | 1 4 8
18. We can get a different construction cube by swapping over the following two parts of the 2'nd and 5'th layers
8 1 2 7
8 1 4 6 6 2 2 5 8 4 2 6
1 8 7 2
6 3 4 5
1 7 5 3 4 7 7 3 1 5 8 3
4 5 6 3
or just parts of them to give
8 1 2 7
8 1 4 6 6 2 2 5 8 4 2 6
1 8 7 2
4 5 6 3
1 7 5 3 4 7 7 3 1 5 8 3
6 3 4 5
19.
10 26 6 23 3 16 9 13 20 This order 3 magic cube and the above order
24 1 17 7 14 21 11 27 4 6 construction cube gives the following
8 15 19 12 25 5 22 2 18 order 6 magic cube
10 161 195 199 80 6 23 192 205 185 3 43
159 190 125 105 55 17 196 14 102 142 149 48
197 42 19 8 177 208 147 25 86 12 214 167
172 107 33 37 134 168 104 138 124 158 57 70
78 136 98 132 1 206 7 176 129 61 95 183
35 15 181 170 204 46 174 106 5 93 133 140
144 67 74 63 148 155 91 26 114 118 215 87
65 54 139 11 216 166 24 109 71 186 82 178
76 137 153 157 56 72 116 96 100 89 123 127
90 121 128 117 94 101 145 188 60 64 53 141
146 81 58 200 135 31 213 163 152 51 28 44
130 191 99 103 2 126 62 69 154 143 150 73
212 30 16 50 165 178 171 175 47 36 40 182
34 122 210 88 41 156 173 162 4 119 108 85
66 187 113 210 52 32 49 164 180 184 29 45
131 84 97 77 111 151 9 13 209 198 202 20
169 68 21 115 203 75 38 27 193 92 189 112
39 160 194 120 79 59 211 110 18 22 83 207
Conctructing Construction Cubes
20. When I made a 2(n+2) by 2(n+2) construction square, I filled in the
corners and centers of the 4 subsquares using a 4 by 4 and a 2n by 2n
construction square. The remaining cells were split into sets of eight
which were filled using the `construction rectangle'
1 4 2 3
4 1 3 2
21. We can make an order 2(n+2) construction cube with the same strategy,
but when the corners and centers of 8 subcubes have been filled with the
order 4 and order 2n cubes, the remaining cells are split into sets of 32
which are filled with a 4 by 4 by 2 `construction cuboid' one example of
which is:
8 6 | 3 1 1 3 | 6 8
7 5 | 4 2 2 4 | 5 7
----|---- ----|----
2 4 | 5 7 7 5 | 4 2
1 4 | 6 8 8 6 | 3 1
Back To Odd Order Squares
22. I said earlier that this method is a generalisation of the well known
method of making mn by mn magic squares from those of order m and those of
order n. To demonstrate this, the following are two construction squares
for making 9 by 9 magic squares from 3 by 3 magic squares. It is clear that
using the first one is equivalent to using the well known method.
8 8 8 | 1 1 1 | 6 6 6 8 2 8 | 1 9 1 | 6 4 6
8 8 8 | 1 1 1 | 6 6 6 2 8 2 | 9 1 9 | 4 6 4
8 8 8 | 1 1 1 | 6 6 6 8 2 8 | 1 9 1 | 6 4 6
------|-------|------ ------|-------|------
3 3 3 | 5 5 5 | 7 7 7 3 7 3 | 5 5 5 | 7 3 7
3 3 3 | 5 5 5 | 7 7 7 7 3 7 | 5 5 5 | 3 7 3
3 3 3 | 5 5 5 | 7 7 7 3 7 3 | 5 5 5 | 7 3 7
------|-------|------ ------|-------|------
4 4 4 | 9 9 9 | 2 2 2 4 6 4 | 9 1 9 | 2 8 2
4 4 4 | 9 9 9 | 2 2 2 6 4 6 | 1 9 1 | 8 2 8
4 4 4 | 9 9 9 | 2 2 2 4 6 4 | 9 1 9 | 2 8 2