By Paul Hoffman
An advent to the delights and demanding situations of recent arithmetic.
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Additional info for Archimedes' Revenge: The Joys and Perils of Mathematics
Example text
136 35 38 17 16 39 18S5 40 19 6 41/2O 7/8 42 21 22 43/744 45 34 33 32 15 14 13 4 3 12 1/2\11/28 9 10 23 24 25 4647 48 31 30 29 27 26 49 ULAM'S LITTLE DOODLE To his surprise, the prime numbers in his doodle, which I've marked in gray, tended to fall on diagonal lines. 41 Prime Prostitution 421 420 419 415 417 416 415 414 413 412 411 410 409 408 407 486 405 464 403 4U2 422 347 3" 34 344 343 342 341 1 3H338 337 33 335 334 333 332 331 33= 401 423 1 233 Mii 29 275 277 278 275 274 273 m 2 270 269 23 267 2C 3Za4o 424 349 32Z223 22* 221 E2N219 218 217 216 215 214 213 212 211 210 2S 32S399 425 3S 283 7_ 113 I72 171 174 I IN 16 INb 16 S 164 163 162 2 264 327 398 426 311 34 23 174 131 1N 12 128 127 12 427 352 35 226 175 132 97 N 95 94 03 428 353 2 227 176 133 9S 71 70 69 68 25 4IU123 122 161 23 3 326 391 92 91 9 121 10 267 26 325 396 67 C6 89 120 153C 261 324 395 429 354 28723 177 134 n n 53 52 Si So 13516 73 54 43 42 49 43' 36S3 22 m17 431 X 20 236 179 136 101 74 55 44 41 41 42 357 2Z 23 IN 137 IU- 75 56 45 46 47 433 318 291 232 1IS 138 103 76 57 SS 59 0 434 435 436 437 359 292 233 18 139 104 77 78 79 80 360 293 234 18340 115 106 107 108 109 361 294 235 1U 141 14Z143 144 145 14C 362 295 236 185 18 18f7l U1 13 6 191 363 296 237 238 239 24 241 242 3U3 4 a 1# IN 15CZ Z 4 87 I 1 2029 IS 63 S 117 156 263 258 42 85 11U155 ZU 257 61 84 IIS 154 261 256 81 82 83 114 153 20 255 310 111 12 113 152 199 254 10-414 9 156 1116 253 1*91w494 195 196 197 252 244 * 247 248 249 250 51 23394 122 393 321 392 33Z 391 31 390 318 389 31; 388 31C 3S7 31s 386 314 385 364 297 29 2" 360391 M 38 , 304 315 J 307 1 3663 310 311 312 311 384 36S 3M6367 368 369 376 372 373 374 375 7 377 378 3' 380 381 382 383 ULAM'S BIGDOODLE Inspired by this serendipitous discovery, Ulam and two co-workers, Mark2Wells and Myron Stein, investigated square spirals that started with whole numbers other than 1.
If, on the basis of a frequency count, the cryptoanalyst can decipher the nine most common letters e, t, a, o, n, i, r, s, and h, respectively, he has generally broken 70 percent of the cipher. The most modem of code-cracking techniques is based on the ageold method of frequency analysis. Frequency analysis also applies to the positions of letters within words and to combinations of letters. For example, the first letter of more than half of all English words comes from the five-letter group t, a, o, s, and w.
Alberti's system makes use of the following table. Above the table are uppercase letters known as key letters, which identify the cipher alphabets in the table. To the left of the table are plaintext letters, also in upper case. IU x I y y z y z d d My Y y z 7aM b. o ni p u b e r d o ,t I x. f y I I IC v wx v y MyC, I b 8 d , h d f x d Y y / f Ip F g h i i k k I I m1 I h a h b . f9 f g i k 9 op "Pl P P q r4 qr r U n n It P 4 I u uI\ p p k I I m I m n m n 11 r CI r g j p jk k I mn ne d g gh hi I C i 7 7 I uv I W !