Here's an amusing passage from Benjamin Franklin's autobiography: "Being one day in the country at the house of our common friend, the late learned Mr. Logan, he showed me a folio French book filled with magic squares, wrote, if I forget not, by one M. Frenicle [Bernard Frenicle de Bessy], in which, he said, the author had discovered great ingenuity and dexterity in the management of numbers; and, though several other foreigners had distinguished themselves in the same way, he did not recollect that any one Englishman had done anything of the kind remarkable. I said it was perhaps a mark of the good sense of our English mathematicians that they would not spend their time in things that were merely 'difficiles nugae', incapable of any useful application." Logan disagreed, pointing out that many of the math questions publically posed and answered in England were equally trifling and useless. After some further discussion about how things of this sort might perhaps be useful for sharpening the mind, Franklin says "I then confessed to him that in my younger days, having once some leisure which I still think I might have employed more usefully, I had amused myself in making these kind of magic squares..." Franklin then described an 8x8 magic square he had devised in his youth, and the special properties it possessed. Here is the square: 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 As explained by Franklin, each row and column of the square have the common sum 260. Also, he noted that half of each row or column sums to half of 260. In addition, each of the "bent rows" (as Franklin called them) have the sum 260. The "bent rows" are patterns of 8 numbers with any of the shapes and orientations shown below # - - - - - - - # - - - - - - # - # - - - - - - - # - - - - # - - - # - - - - - - - # - - # - - - - - # - - - - - - - # # - - - - - - # - - - - - - - - - - - - - - # - - - - - - - - - - - - - - # - - - - - - - - - - - - - - # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - # - - - - - - - - - - - - - - # - - - - - - - - - - - - - - # - - - - - - - - - - - - - - # - - - - - - # # - - - - - - - # - - - - - # - - # - - - - - - - # - - - # - - - - # - - - - - - - # - # - - - - - - # - - - - - - - # It isn't clear from his verbal description whether Franklin was claiming just the five parallel patterns of each of these types that fall strictly within the square, or if he was claiming all eight, counting those that "wrap around". In any case, his square does possess this property. For example, if we shift the first "bent row" to the left, wrapping the ends around, we have the patterns - - - - - - - # - - - - - - # - - - - - - # - - # - - - - - - - - - - - - - - # - - - - - - # - - # - - - - - - # - - - - - - - - - - - - - - # - - # - - - - - - # - - - - - - # - - - - - - - - - # - - - - - - # - - - - - - # - - - - - - - - # - - - - - - # - - - - - - - - - - - - - - # # - - - - - - - - - - - - - - # - - - - - - # - - - - - - - - # - - - - - - # - - - - - - # - - In addition, Franklin noted that the "shortened bent rows" plus the "corners" also sum to 260. An example of this pattern is shown below: # - # - - # - # - # - - - - # - # - - - - - - # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - As with the previous patterns, this template can be rotated in any of the four directions, and shifted parallel into any of the eight positions (with wrap-around), and the sum of the highlighted numbers is always 260. Finally, Franklin noted that the following two sets of eight numbers also sum to 260 - # - - - - # - # - - - - - - # # - - - - - - # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - # # - - - - - - - - - - - - - - # # - - - - - - - - - - - - - - - - - - - # - - - - - - # - - - - - - - - - # - - - - # - # - - - - - - # He doesn't explicitly mention it, but these patterns can also be translated (with wrap-around), and since they are symmetrical between horizontal and vertical, they can be translated in either direction. After showing off this 8x8 square to his friend, Franklin continued: "Mr Logan then showed me an old arithmetical book in quarto, wrote, I think, by one [Michel] Stifelius, which contained a square of 16x16 that he said he should imagine must have been a work of great labor; but if I forget not, it had only the common properties of making the same sum, viz 2056, in every row, horizontal, vertical, and diagonal. Not willing to be outdone by Mr. Stifelius, even in the size of my square, I went home and made that evening the following magical square of 16, which, besides having all the [special] properties of [his earlier 8x8 square], had this added: that a four-square hole being cut in a piece of paper of such a size as to take in and show through it just 16 of the little squares, when laid on the greater square, the sum of the 16 numbers so appearing through the hole, wherever it was placed on the greater square, should likewise make 2056. This I sent to our friend the next morning, who, after some days, sent it back in a letter with these words: 'I return to thee thy astonishing or most stupendous piece of the magical square, in which' - but the compliment is too extravagant, and therefore, for his sake as well as my own, I ought not to repeat it. Nor is it necessary; for I make no question but you will readily allow this square of 16 to be the most magically magical of any magic square ever made by any magician." Both the 8x8 and the 16x16 square are reproduced in Van Doren's book, taken from Franklin's autobiography, but Van Doren notes that "Barbeu Dubourg, when translating Franklin's works twenty years later, found two mistakes in the [16x16] square". This is interesting because if we compare the square as reproduced in Van Doren with the supposedly correct version given in Christopher Henrich's nice discussion of Franklin's squares in the article "Magic Squares and Linear Algebra" (Americal Mathematical Monthly, vol 98, no 6, 1991), we find THREE discrepancies. However, one of them turns out to be an error in Henrich's version, proving that even in the computer age it's still possible to mis-transcribe a large square. Here is what I believe to be Franklin's 16x16 square in it's true and correct form: 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 I assume the two errors found by Dubourg were at the locations in square brackets, [211] and [185], where Franklin had written 241 and 181 respectively. The entry in parentheses, (231), seems to be correct in Franklin's version, but Henrich's article gives that entry as 31. (Henrich cites W. S. Andrews's book "Magic Squares and Cubes" as his source, but I've checked the latter and it gives the correct values.) It's worth noting that neither of Franklin's squares satisfies the main diagonal sums. Thus, although they possess numerous interesting "affine" properties as described in Henrich's article, they don't strictly qualify as "magic squares" according to the common definition of the term that includes the diagonal sums.

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