Benjamin Franklin made a few magic squares with interesting and unusual properties; at least four of them have survived. Two are well-known, and are explained in Chapter III of Andrews. Another was published by Pasles. Ahmed has studied the methods by which Franklin might have constructed these squares; she gives some more examples of squares constructed by her methods.
Franklin's first square is:
52 | 61 | 4 | 13 | 20 | 29 | 36 | 45 |
14 | 3 | 62 | 51 | 46 | 35 | 30 | 19 |
53 | 60 | 5 | 12 | 21 | 28 | 37 | 44 |
11 | 6 | 59 | 54 | 43 | 38 | 27 | 22 |
55 | 58 | 7 | 10 | 23 | 26 | 39 | 42 |
9 | 8 | 57 | 56 | 41 | 40 | 25 | 24 |
50 | 63 | 2 | 15 | 18 | 31 | 34 | 47 |
16 | 1 | 64 | 49 | 48 | 33 | 32 | 17 |
It has the following magical properties:
This square is affine. Here are the affine functions which define it:
A_{1} = f_{3}+f_{4}+f_{5}+1 |
A_{2} = f_{3}+f_{5}+1 |
A_{3} = f_{3}+f_{6}+0 |
A_{4} = f_{1}+f_{2}+f_{3}+f_{6}+0 |
A_{5} = f_{2}+f_{3}+f_{6}+1 |
A_{6} = f_{1}+f_{2}+f_{6}+1 |
It is easy enough to check that the square is uniform on the subspace <e_{2}, e_{3}>, which is a half column, and consequently the sum is 130 on every parallel flat, which is to say, on every half column. Likewise for the half rows. But what about these bent rows and bent columns? They can be investigated using the applet that displays flats. It turns out that half of the bent rows and columns actually are 3-flats, but the others are not. To understand the bent rows, you can use the two linear subspaces
The affine square is uniform on both of these. Every flat that is parallel to α or to β is part of a bent row; and every bent row consists of two such flats. So the magical property of bent rows follows from the affine structure of the square. A similar analysis works for the bent columns.
As for the 2×2 blocks, they are all 2-flats. They belong to four families of parallel flats, not all of which are contiguous 2×2 blocks. Franklin hinted at "other" properties of this square, and it occurs to me that he may have known about these other 2-flats.
Is there any chance that Franklin constructed this square, and others like it, by defining an affine map with the requisite properties? Certainly he could not have used the concepts and notation of linear algebra, which did not exist in his day. But we do not know what technique he did use. One possible technique is sketched in Chapter III of Andrews. This is, to start with the numbers in their natural order, which corresponds to the identity transform as an affine map, and apply various transformations which preserve the affine property. In Andrews, figures 194 - 197, you can see a nontrivial affine map being constructed in this way.
This square was relegated to a footnote in Franklin's collected works. It is resurrected by Pasles and further treated by Ahmed.
17 | 47 | 30 | 36 | 21 | 43 | 26 | 40 |
32 | 34 | 19 | 45 | 28 | 38 | 23 | 41 |
33 | 31 | 46 | 20 | 37 | 27 | 42 | 24 |
48 | 18 | 35 | 29 | 44 | 22 | 39 | 25 |
49 | 15 | 62 | 4 | 53 | 11 | 58 | 8 |
64 | 2 | 51 | 13 | 60 | 6 | 55 | 9 |
1 | 63 | 14 | 52 | 5 | 59 | 10 | 56 |
16 | 50 | 3 | 61 | 12 | 54 | 7 | 57 |
It, too, is affine. defined by these functions:
A_{1} = f_{1}+f_{2}+f_{6}+0 |
A_{2} = f_{2}+f_{6}+1 |
A_{3} = f_{3}+f_{5}+f_{6}+0 |
A_{4} = f_{3}+f_{4}+f_{5}+f_{6}+0 |
A_{5} = f_{3}+f_{6}+0 |
A_{6} = f_{3}+f_{5}+0 |
One square of order 16, constructed by Franklin, has been reprinted in several places, including Andrews:
200 | 217 | 232 | 249 | 8 | 25 | 40 | 57 | 72 | 89 | 104 | 121 | 136 | 153 | 168 | 185 |
58 | 39 | 26 | 7 | 250 | 231 | 218 | 199 | 186 | 167 | 154 | 135 | 122 | 103 | 90 | 71 |
198 | 219 | 230 | 251 | 6 | 27 | 38 | 59 | 70 | 91 | 102 | 123 | 134 | 155 | 166 | 187 |
60 | 37 | 28 | 5 | 252 | 229 | 220 | 197 | 188 | 165 | 156 | 133 | 124 | 101 | 92 | 69 |
201 | 216 | 233 | 248 | 9 | 24 | 41 | 56 | 73 | 88 | 105 | 120 | 137 | 152 | 169 | 184 |
55 | 42 | 23 | 10 | 247 | 234 | 215 | 202 | 183 | 170 | 151 | 138 | 119 | 106 | 87 | 74 |
203 | 214 | 235 | 246 | 11 | 22 | 43 | 54 | 75 | 86 | 107 | 118 | 139 | 150 | 171 | 182 |
53 | 44 | 21 | 12 | 245 | 236 | 213 | 204 | 181 | 172 | 149 | 140 | 117 | 108 | 85 | 76 |
205 | 212 | 237 | 244 | 13 | 20 | 45 | 52 | 77 | 84 | 109 | 116 | 141 | 148 | 173 | 180 |
51 | 46 | 19 | 14 | 243 | 238 | 211 | 206 | 179 | 174 | 147 | 142 | 115 | 110 | 83 | 78 |
207 | 210 | 239 | 242 | 15 | 18 | 47 | 50 | 79 | 82 | 111 | 114 | 143 | 146 | 175 | 178 |
49 | 48 | 17 | 16 | 241 | 240 | 209 | 208 | 177 | 176 | 145 | 144 | 113 | 112 | 81 | 80 |
196 | 221 | 228 | 253 | 4 | 29 | 36 | 61 | 68 | 93 | 100 | 125 | 132 | 157 | 164 | 189 |
62 | 35 | 30 | 3 | 254 | 227 | 222 | 195 | 190 | 163 | 158 | 131 | 126 | 99 | 94 | 67 |
194 | 223 | 226 | 255 | 2 | 31 | 34 | 63 | 66 | 95 | 98 | 127 | 130 | 159 | 162 | 191 |
64 | 33 | 32 | 1 | 256 | 225 | 224 | 193 | 192 | 161 | 160 | 129 | 128 | 97 | 96 | 65 |
Like his squares of order 8, this is magic for rows, columns, bent rows, and bent columns, the magical sum being 2056. It is also magical for all contiguous 4×4 blocks; this is a corollary of the fact that every contiguous 2×2 block adds up to 514. Franklin affirmed this "to be the most magically magical of any magic square ever made by any magician." But only up to that time, which was about 1752. This square is affine, and here are its defining functions.
A_{1} = f_{4}+f_{5}+f_{6}+1 |
A_{2} = f_{4}+f_{6}+1 |
A_{3} = f_{4}+f_{7}+0 |
A_{4} = f_{4}+f_{8}+0 |
A_{5} = f_{1}+f_{2}+f_{4}+f_{8}+0 |
A_{6} = f_{2}+f_{4}+f_{8}+1 |
A_{7} = f_{1}+f_{2}+f_{3}+f_{4}+f_{8}+1 |
A_{8} = f_{1}+f_{2}+f_{8}+1 |
In a 1765 letter to the electrical investigator John Canton, Franklin included another square of order 16, which was first published by Pasles and can best be seen at his web site. It is:
16 | 255 | 2 | 241 | 14 | 253 | 4 | 243 | 12 | 251 | 6 | 245 | 10 | 249 | 8 | 247 |
1 | 242 | 15 | 256 | 3 | 244 | 13 | 254 | 5 | 246 | 11 | 252 | 7 | 248 | 9 | 250 |
240 | 31 | 226 | 17 | 238 | 29 | 228 | 19 | 236 | 27 | 230 | 21 | 234 | 25 | 232 | 23 |
225 | 18 | 239 | 32 | 227 | 20 | 237 | 30 | 229 | 22 | 235 | 28 | 231 | 24 | 233 | 26 |
223 | 48 | 209 | 34 | 221 | 46 | 211 | 36 | 219 | 44 | 213 | 38 | 217 | 42 | 215 | 40 |
210 | 33 | 224 | 47 | 212 | 35 | 222 | 45 | 214 | 37 | 220 | 43 | 216 | 39 | 218 | 41 |
63 | 208 | 49 | 194 | 61 | 206 | 51 | 196 | 59 | 204 | 53 | 198 | 57 | 202 | 55 | 200 |
50 | 193 | 64 | 207 | 52 | 195 | 62 | 205 | 54 | 197 | 60 | 203 | 56 | 199 | 58 | 201 |
80 | 191 | 66 | 177 | 78 | 189 | 68 | 179 | 76 | 187 | 70 | 181 | 74 | 185 | 72 | 183 |
65 | 178 | 79 | 192 | 67 | 180 | 77 | 190 | 69 | 182 | 75 | 188 | 71 | 184 | 73 | 186 |
176 | 95 | 162 | 81 | 174 | 93 | 164 | 83 | 172 | 91 | 166 | 85 | 170 | 89 | 168 | 87 |
161 | 82 | 175 | 96 | 163 | 84 | 173 | 94 | 165 | 86 | 171 | 92 | 167 | 88 | 169 | 90 |
159 | 112 | 145 | 98 | 157 | 110 | 147 | 100 | 155 | 108 | 149 | 102 | 153 | 106 | 151 | 104 |
146 | 97 | 160 | 111 | 148 | 99 | 158 | 109 | 150 | 101 | 156 | 107 | 152 | 103 | 154 | 105 |
127 | 144 | 113 | 130 | 125 | 142 | 115 | 132 | 123 | 140 | 117 | 134 | 121 | 138 | 119 | 136 |
114 | 129 | 128 | 143 | 116 | 131 | 126 | 141 | 118 | 133 | 124 | 139 | 120 | 135 | 122 | 137 |
Sure enough, it too is affine; the defining functions are:
A_{1} = f_{2}+f_{3}+f_{8}+0 |
A_{2} = f_{1}+f_{2}+f_{3}+f_{8}+0 |
A_{3} = f_{3}+f_{8}+0 |
A_{4} = f_{2}+f_{8}+0 |
A_{5} = f_{4}+f_{7}+1 |
A_{6} = f_{4}+f_{5}+f_{7}+1 |
A_{7} = f_{4}+f_{6}+f_{7}+1 |
A_{8} = f_{2}+f_{4}+f_{8}+1 |
And this one is even more magical than the previous square; in particular, it is pandiagonal.
We can see that the magic properties of Franklin's squares can be derived from their being (a) affine, and (b) uniform on certain linear subspaces of various dimensions. On how many subspaces is an affine square uniform? Here is a "calculator" which gives the answer: the number of uniform subspaces of dimension m, and the number of m-flats parallel to those spaces, in a vector space F_{p}^{d}.
P: | This selector gives you the order of the prime field. | |
D: | This selector chooses the dimension of the vector space. | |
M: | This selector chooses the dimension m of the linear subspaces and flats being counted. |
The number of cells is 000.
There are 000 uniform linear subspaces of dimension 0, and 000 uniform 0-flats.
The calculator starts up with p = 2, d = 4, and m = 2, which pertain to such squares as the Dürer square. Try p = 2, d = 8, and m = 4 to see how much magic there is in one of Franklin's squares. Franklin's "most magically magical" statement was in fact modest.The theory behind this calculator is described in some notes, along with the theory of counting affine squares of any prime-power order.
Benson and Jacoby constructed a trimagic square of order 32. This is the smallest trimagic square of prime-power order that I know of. I investigated it using Mathematica™. It is an affine square. The rows and columns are uniform of degree 3, but the diagonals are not. However, they are not very non-uniform. In degree two, there are two pairs of digit coordinates that are not linearly independent on the diagonal subspace: A_{3}A_{10} and A_{4}A_{9}. On the main diagonal, most of the products of two digit coordinates take the value 1 in 1/4 of the cells; but A_{3}A_{10} has the value 1 nowhere, and A_{4}A_{9} has the value 1 at 1/2 of the cells. Note that 3 + 10 = 4 + 9; these two products together contribute as much to the sum of squares as if they were both uniform. Similarly, on the opposite diagonal, A_{4}A_{9} contributes nothing whereas A_{3}A_{10} contributes a double amount.
Christian Boyer has constructed a web site with much information on multimagic squares. He provides downloadable copies of several multimagic squares of impressively high degree. I have investigated them to see if they were affine. Here are some findings.
Trimagic, order 64 The first such square was made by General Eutrope Cazalas, in 1933. It is affine, and is uniform of degree 3 in the rows, columns, and diagonals.
Trimagic, order 128 The first such square was made by Gaston Tarry in 1905. It is not affine, but the discrepancy is less than it might be. My method of checking for the affine property was to find the affine square which would agree with Tarry's square at 15 points, namely those corresponding to {0 , e_{1},..., e_{14}}. I compared the digit coordinates of this square with those of Tarry's. There were 14×16384 pairs of bits to check. The two squares differed in 3×16384 cases. Six of the digit coordinates agreed completely. As for the others, two were discrepant at 5120 places, four at 6144 places, and two at 7168 places. Most surely, Tarry's square has some hidden structure; it is not as simple as an affine square, but it is not very much more complicated.
Tetramagic, order 243 A square of this description was constructed by Pan Fengchu in 2004; this appears to be the smallest tetramagic square currently known. Alas, it is not affine. I compared it with the affine square determined by 11 points, and found discrepancies at somewhat more than 50% of the data.