THE FOUR DIGITS PROBLEM The Four Digits Problem by Andy Pepperdine 2007-01-12 (re-formatted 2010-01-02) based on notes made 50 years ago. This work is released under the Creative Commons Attribution – Share-alike 2.5 license [1]. Make all integers from 0 upwards using the simplest mathematical operators, as defined in some way, and the digits 1, 2, 3 and 4. W. W. Rouse Ball gives this in his book Mathematical Recreations and Essays[2] as a more interesting exercise than the normal Four Fours where all the digits are 4. He states on page 15, that with the digits 1, 2, 3 and 4, and allowing the notation of the denary scale (including decimals), as also algebraic sums, products, and positive integral powers, all numbers from 1 up to 88 can be made. If the use of symbols for square roots and factorials, repeated a finite number of times, is also permitted, then we can get to 264; if negative indices are also permitted, to 276; and if fractional indices are permitted, to 312. NB: I interpret the last comment about “fractional indices” to mean the extraction of roots other than the square root by saying which one to extract. I cannot find any other solution to 277 and 307 otherwise. The 89 values I cannot find up to 1000 are: 313, 331, 391, 407, 419, 421, 437, 439, 443, 446, 454, 455, 457, 461, 467, 493, 557, 559, 586, 587, 614, 617, 633, 653, 659, 661, 662, 667, 772, 773, 779, 787, 788, 791, 807, 813, 815, 817, 823, 831, 833, 850, 851, 853, 857, 859, 861, 867, 871, 874, 877, 879, 881, 883, 885, 886, 887, 893, 897, 906, 907, 908, 911, 913, 914, 915, 917, 919, 921, 922, 923, 927, 929, 931, 932, 933, 935, 941, 942, 946, 947, 949, 951, 953, 956, 979, 983, 986, 989. Examples of each number from 0 upwards are given in the following table. I've tried to keep to the order in which Rouse Ball postulated the use of operators, but also, I've used factorial in preference to the square root sign, and allowed recurring decimals and integral powers in preference to all other powers, factorials and roots. Some interesting alternative methods are noted, too. Five digits Rouse Ball also gives the limits for 5 digits (1, 2, 3, 4 & 5) of 3832 without negative and fractional powers, and of 4282 with. My investigations indicate the same. Digits-1234.odt 1 THE FOUR DIGITS PROBLEM No. Method No. Method 0 14−3−2 33 341−2 1 32−4×1 34 34×2–1 2 123−4 35 342−1 3 2×31−4 36 324×1 4 4×2−3−1 37 2314 5 43−2×1 38 42–3−1 6 431−2 39 42–3×1 7 31−24 40 12×34 8 432−1 41 43–2×1 9 432×1 42 413−2 10 4321 43 43×2– 1 11 42−31 44 4×13 –2 12 4×3×2−1 45 423×1 13 34−21 46 3214 14 21–3−4 47 412×3 15 134−2 48 3×142 16 4×32−1 49 4123 17 3×142 50 13×4−2 18 32−14 51 12×43 19 23–4×1 3 52 2× −4 or 43−12 20 213−4 ·1 21 24–3×1 2 53 3 22 43−21 ·1×·4 23 31–2×4 54 4×132 24 4×123 55 4213 3 25 31–4−2 56 41 ·2 26 234−1 4 27 234×1 57 3× −1 ·2 28 32–4×1 58 31×2−4 29 42−13 59 31×3−4 30 13×24 60 4×312 31 43−12 13 61 −4 or 43 –2−1 32 3×12−4 ·2 Digits-1234.odt 2 THE FOUR DIGITS PROBLEM No. Method No. Method 31×4 88 4×23 –1 62 2 13 4! 4! −31 4 89 or 63 23 ·2 ·2 1 · 3×12 90 64 2341 · 4 13×2 65 91 23×4−1 · 4 92 23×4×1 66 31×24 93 23×41 134 67 94 2×413! 2 3 68 34×2×1 2 95 ·4 or 4!×3!−1−1 69 3×24−1 ·1 70 2×314 96 4×213 71 24×3−1 97 14!–23 72 4×21 –3 98 3!–1!–4!2 73 3241 3×4 99 or 123–4! 21 ·1˙ 2˙ 74 4 ·3 3×4 100 or 4−3×·1−2 75 3×214 ·12 76 2×41–3 4 2 101 3 or ·2−3– 4! 31−·2 ·1 77 ·4 102 34×21 78 13×24 103 3!!×·2−41 79 41×2−3 104 13×4×2 32×1 42 80 or 34−21 105 ·4 ·3·1 81 21×4−3 3×4 106 −2 41 ·1˙ 82 ·2·3 4! 107 −13 2×4 ·2 83 3 ·1 108 4×123 84 14×3×2 4!–1 109 –3! 85 41×23 ·2 2×43 86 43×2×1 110 87 21×43 ·1 Digits-1234.odt 3 THE FOUR DIGITS PROBLEM No. Method No. Method 111 14!−32 31–4 135 ·2 3 112 4× −2 ·1 136 1342 3 1 ·4 3 113 !−4−3 137 ·1˙ or 4 ·2 ·2×·1˙ ·2 4 114 3× −2 4 ·1 2 4 138 ·3 or 3!–1! ·2˙ 115 23×41 ·1˙ 116 4×31 –2 139 142−3 117 3×41 –2 42×1 140 3×4 ·3 118 −2 ·1 141 143−2 119 123−4 142 3!×4!–2×1 24 2 120 143 3×4 −1 ·3−·1 144 12×4×3 121 124−3 145 1423 122 31×4−2 3 146 −4 123 3×42–1 ·1×·2 124 4×32 –1 147 21×34 125 41×32 3 −·4 126 42×3×1 148 ·1 3!× 4!1−2 127 1234 ·2 128 132−4 149 3!!×·241 3×2 129 3×412 150 ·1×·4 41–2 130 or 1243! 31 ·3 151 −4 ·2 3 131 −4 3 ·2×·1˙ ·4 152 ·1 132 134−2 ·2 3 −·4 4! 31−·4 133 ·1˙ or 13 153 ·2 ·2 ·2 3 4 154 4 ·2 ·1×·2 134 ·1 or 32!14 ·3 Digits-1234.odt 4 THE FOUR DIGITS PROBLEM No. Method No. Method 31×2 4 155 177 −3 ·4 ·2×·1˙ 156 ! 4! 132 4 −2 178 ˙ 314 ·13 157 2 31 179 4! ·2˙ ·2 158 −4 or 3!!×·214 ·1˙ 3 3×42 180 31 ·1 159 4 ·2 3×2! 181 1 160 32×41 4 4 4! 161 2×3 −1 or 32!41 182 2 ·13˙ 162 1×2×34 4 4 183 3 163 2×3 1 ·2×·1˙ 4 164 2×3 1 184 23×4!−1 34–1 4−·3 165 185 ·2 ·1×·2 ·2˙ 166 4 186 31×24 1˙ 3 · 4!–3 187 −2 167 3!!×·24!−1 ·1˙ 168 42×31 4! 188 3!!×·1˙ 34 ·2˙ 169 −1 ·2 3×14 189 34×1 ·2˙ 170 ·2 41–3 190 34 ·2 171 1 ·2 191 4!×23−1 4 3 172 3!!×·2˙ 192 4 ×21 ·1˙ 193 142−3 173 1324 43·1˙ 3 3!! 194 174 4! or –21! .2˙ ·1×·2 4 4 3 4!–2 314 195 ·1˙ or −3 175 .1˙ ·2 ·2 ·2 – 4! ·2 176 3 196 −4 ·1 ·13 Digits-1234.odt 5 THE FOUR DIGITS PROBLEM No. Method No. Method 4 24 197 −3 216 3 ·1×·2 ·1˙ 413 217 2134 198 ˙ ·2 218 3!4–12 2 199 14 3 24 219 3 4×32 ·1˙ 200 ·1 413 220 4! ·2 201 3 ·1˙ 2˙ 4! 221 32 41 ·1˙ 202 −3 ·2 24 222 3! 4 ·1˙ 203 3 ·1×·2 4!–2 223 3 ·2 ·1 204 3 4 ·1 224 42!×·31˙ 205 41×23 225 3×412 23 23 206 – 4! 226 −4 ·1 ·1 4 3 4! 2 227 231−4 207 ·2 or −3 ·1˙ 4 23 ·1˙ 228 3!× −2 or –4 ·1 ·1 41 4! 3 208 3 or −2 ·2 ·1˙ 229 231 –4 4 209 213−4 3 4!–1 230 ·2 or 14×3 ·3−·2 210 ·1 ·2 23 211 214−3 231 4! ·1˙ 4!–3 212 2 3! 4! 3 ·1 232 4× – 2 or −2 .1 ·1 24 213 −3 ·1˙ 233 234−1 43 234 234×1 214 −1 ·2 235 2341 43×1 236 2×3!–1!−4 215 ·2 24 237 −3 ·1 Digits-1234.odt 6 THE FOUR DIGITS PROBLEM No. Method No. Method 238 241−3 264 4!×13−2 1 −2 ·2 ·1 3! 239 3 −4 265 ·4 4×3×2 240 3! ·1 266 −4 ·2×·1˙ 241 41 3 −2 1 4! 3! 267 ·2 or 242 243−1 3 4! ·1 ·2˙ 243 243×1 268 2×134 244 2431 23! 269 −1 41 ˙ 245 3 2 ·4 234 246 41×3×2 270 1 ·1 247 ·2 3 4 23! 271 1 248 31×4×2 ·4˙ 4! 2 4! 249 3 272 32 ·1 ·1 3 4! 250 2×41 273 3 ·2−·1˙ 251 3!×42−1 3! 252 21×4×3 274 4 ·2×·1˙ 253 3!×421 4!31 31 275 254 4 −2 ·2 255 2314! 23×4 276 41!×2·3 or 256 432−1 ·1˙ −·2 4!2 277 34 ·1˙ 257 −3 ·1 − 3!!×·4−·1 2 or 258 4312 278 −4! 4 ·1 –3!!−2 259 3! −1 ×·2 3·4˙ 13×4 279 260 ·1˙ 2 ·2 21×4 4!32 280 261 ·3 ·1˙ ·2−3−·1˙ 262 2 4!×·1˙ 3! or 3!!×·1˙4!−2 281 ·4˙ 4! 263 23 282 4!×12–3! ·1 Digits-1234.odt 7 THE FOUR DIGITS PROBLEM No.