How far can you count using expressions formed from the digits 1 2 3 4? The answer depends on which operators you consider acceptable.
This is a listing of the best results I have. To try to please all tastes I introduce new operators and functions as far down the list as possible according to various ideas about what should or should not be permitted. If you allow integer truncation operations such as floor() and ciel() it is too easy, so we don't. There is also an obvious trick using logs and square roots which can be used to construct any integer so we don't allow logs either.
---------------------------------------------------------- Summary: allowed operations answer + - * 28 + - * ^ 33 + - * ^ dd 51 + - * ^ ! 75 + - * ^ / ! .d 76 + - * ^ / ! sqrt 84 + - * ^ / ! dd 102 + - * ^ / ! dd .d 108 + - * ^ / ! dd .d sqrt 136 + - * ^ / ! dd .d sqrt .(d) 276 + addition - subtraction * multiplication ^ exponentiation (powers) / division ! factorial dd digit concatenation .d decimal point (e.g. .2 = 1/5) sqrt square root .(d) recurring decimals (e.g. .(3) = 1/3) 0 = 1 + 4 - 2 - 3 = 12 - 3*4 1 = 3*1 + 2 - 4 2 = 2 - 1 + 4 - 3 3 = 2*1 + 4 - 3 = 12 - 3 - 4 4 = 4 + 2 + 1 - 3 5 = 4*1 + 3 - 2 = 12 - 3 - 4 6 = 1 - 2 + 3 + 4 7 = 4*(2 - 1) + 3 = 31 - 24 8 = 4 + 3 + 2 - 1 9 = 4*3 - 2 - 1 = 23 - 14 10 = 4 + 3 + 2 + 1 11 = 4*3 - 2 + 1 = 24 - 13 12 = 2*4 + 3 + 1 13 = 3*4 + 2 - 1 = 34 - 21 14 = 3*4 + 2*1 = 21 - 3 - 4 15 = 3*4 + 2 + 1 = 13 + 4 - 2 16 = 2*4*(3 - 1) 17 = (4 + 2)*3 - 1 18 = 1*3*(2 + 4) = 32 - 14 19 = (2 + 3)*4 - 1 = 12 + 3 + 4 20 = (2 + 3)*4*1 = 14 + 2*3 21 = (2 + 3)*4 + 1 = 21*(4 - 3) 22 = (4*3 - 1)*2 = 34 - 12 23 = 2*3*4 - 1 = 31 - 2*4 24 = 1*2*3*4 = 12 + 3*4 25 = 2*3*4 + 1 = 14*2 - 3 26 = 2*(3*4 + 1) = 23 + 4 - 1 27 = (4*2 + 1)*3 = 23 + 4*1 28 = 4*(2*3 + 1) = 21 + 3 + 4 29 = 2^(1+4) - 3 = 42 - 13 30 = 3*2*(4 + 1) = 13*2 + 4 31 = (2^3)*4 - 1 = 43 - 12 32 = 4*2*(3 + 1) = 12*3 - 4 33 = (2^3)*4 + 1 = 132/4 34 = 4! + 3^2 + 1 = 34*(2 - 1) 35 = 2^(1 + 4) + 3 = 34 + 2 - 1 36 = 3*4*(2 + 1) = 34 + 2*1 37 = (3^2)*4 + 1 = 13 + 24 38 = (4! - 3! + 1)*2 = 42 - 3 - 1 39 = 3!^2 + 4 - 1 = 42 - 3*1 40 = 2^3*(4 + 1) = 42 - 3 + 1 41 = 3!^2 + 4 + 1 = 43 - 2*1 42 = 4!*2 - 3!*1 = 43 - 2 + 1 43 = 4!*2 - 3! + 1 = 43*(2 - 1) 44 = 4!*2 - 3 - 1 = 43 + 2 - 1 45 = 3^2*(4 + 1) = 12*4 - 3 46 = 4!*2 - 3 + 1 = 12 + 34 47 = (2^4)*3 - 1 48 = (2^4)*3*1 = (14 + 2)*3 49 = (2^4)*3 + 1 50 = (3 + 4)^2+1 = 13*4 - 2 51 = (2^4 + 1)*3 = 12*4 + 3 52 = 4!*2 + 3 + 1 53 = 4!*2 + 3! - 1 54 = 4!*2 + 3!*1 = 13*4 + 2 55 = 4!*2 + 3! + 1 = 34 + 21 56 = (4! + 3 + 1)*2 57 = 4!*2 + 3^2 = (2 + 4)/.1 - 3 58 = 4^3 - (1 + 2)! = 31*2 - 4 59 = 3!^2 + 4! - 1 = 21*3 - 4 60 = 3!^2 + 4!*1 = (13 + 2)*4 61 = 4^3 - 2 - 1 62 = 4^3 - 2*1 63 = 4^3 - 2 + 1 64 = 4^3*(2 - 1) = 41 + 23 65 = 4^3 + 2 - 1 66 = 4^3 + 2*1 = (34 - 1)*2 67 = 4^3 + 2 + 1 = 21*3 + 4 68 = 2^(3!) + 4*1 = 34*2*1 69 = 4!*3 - 2 - 1 = 34*2 + 1 70 = 4!*3 - 1*2 = (34 + 1)*2 71 = 4!*3 - 2 + 1 = 24*3 - 1 72 = (2 + 1)!*3*4 = 24*3*1 73 = 4!*3 + 2 - 1 = 31 + 42 74 = 4!*3 + 1*2 75 = 4!*3 + 2 + 1 = (24 + 1)*3 76 = sqrt(3!!/2+1)*4 = 41 * 2 - 3! = 2^3/.1 + 4 77 = (4! + 1)*3 + 2 78 = 3^4 - 2 - 1 = 13*(2 + 4) 79 = 3^4 - 2*1 = 41*2 - 3 80 = 3^4 - 2 + 1 81 = 3^4*(2 - 1) = 21*4 - 3 82 = 3^4 + 2 - 1 83 = 3^4 + 2*1 84 = 3^4 + 2 + 1 = 14*2*3 85 = 43*2 - 1 86 = 43*2*1 87 = 4! + 2^(3!) - 1 = 43*2 + 1 88 = 2^(3!) + 4!*1 = (23 - 1)*4 89 = 4! + 2^(3!) + 1 90 = 3!*(2^4 - 1) = (12 + 3)*4 91 = 3!!/4/2 + 1 = 23*4 - 1 92 = (4! - 1)*(3! - 2) = 23*4*1 93 = 23*4 + 1 94 = (3! - 1)! - 4! - 2 95 = (2 + 3)! - 4! - 1 96 = (3 + 2 - 1)!*4 = (21 + 3)*4 97 = (2 + 3)! - 4! + 1 98 = (3 + 1)!*4 + 2 99 = (3! + 4)^2 - 1 100 = (3! + 4)^2*1 101 = (3! + 4)^2 + 1 102 = (2^4 + 1)*3! = 34*(2 + 1) 103 = (.1)^(-4/2) + 3 104 = 13*2*4 105 = 41 + 2^(3!) 106 = (2 + 3)! - 14 107 = (.1)^(-2) + 4 + 3 108 = 132 - 4! 109 = (.1)^(-2) + 3^sqrt(4) 110 = (31 + 4!)*2 111 = (1 + 4)! - 3^2 112 = (2^3)*14 113 = (1/.2)! - 3 - 4 114 = (1 + 4)! - 2*3 115 = (2 + 3)! - 1 - 4 116 = (2 + 3)!*1 - 4 117 = (3 + 2)! - 4 + 1 118 = 124 - 3! 119 = 123 - 4 120 = (4 + 1)!*(3 - 2) 121 = 124-3 122 = 31*4 - 2 123 = (1+4)^3 - 2 124 = (2 + 3)! + 4*1 125 = 42*3 - 1 126 = 42*3*1 127 = 123 + 4 128 = 132 - 4 129 = 2^(3 + 4) + 1 130 = 124 + 3! 131 = (4! - 2)*3! - 1 132 = 134 - 2 133 = (4! - 2)*3! + 1 134 = (2 + 3)! + 14 135 = (4 - 1)^3 / .2 136 = 132 + 4 137 = 3/.1/.(2) + sqrt(4) 138 = 4!*3! - (2 + 1)! 139 = 142 - 3 140 = (3!^2 - 1)*4 141 = 143 - 2 142 = (3! - 1)! + 4! - 2 143 = 4!*3*2 - 1 144 = 12*3*4 145 = 4!*3*2 + 1 146 = (3! - 1)! + 4! + 2 147 = 123 + 4! 148 = 142 + 3! 149 = .2^-3 + 4!*1 150 = 4!*3! + (2 + 1)! 151 = 31/.2 - 4 152 = 32 + (1 + 4)! 153 = (3/.2 + sqrt(4))/.(1) 154 = 3/.2/.1 + 4 155 = (4! + 2)*3! - 1 156 = 132 + 4! 157 = (4! + 2)*3! + 1 158 = 3!!*.(1)*2 - sqrt(4) 159 = 3!!/4 - 21 160 = (3^4 - 1)*2 161 = (2 + 3)! + 41 162 = 2*1*(3^4) 163 = (3^4)*2 + 1 164 = 41*(3! - 2) 165 = 13^2 - 4 166 = (3! + 1)*4! - 2 167 = 13^2 - sqrt(4) 168 = 42*(3 + 1) 169 = (3*4 + 1)^2 170 = (3! + 1)*4! + 2 171 = 34/.2 + 1 = 13^2 + sqrt(4) 172 = .1^-2 + 4!*3 173 = 13^2 + 4 174 = 3!!/4 - (2 + 1)! 175 = (34 + 1)/.2 176 = .2*(.1^(-3)) - 4! 177 = 3!!/4 - 2 - 1 178 = 3!!*1/4 - 2 179 = 3!!/4 - 2 + 1 180 = 3!!/4*(2-1) 181 = 3!!/4 + 2 - 1 182 = 3!!*1/4 + 2 183 = 3!!/4 + 2 + 1 184 = 2^(3!) + (1 + 4)! 185 = 3!!/4 + 1/.2 186 = 3!!/4 + (2 + 1)! 187 = (4! - 3)/.(1) - 2 188 = 3!!*.4 - (.1^(-2)) 189 = 213 - 4! 190 = 14^2 - 3! 191 = (4!^2)/3 - 1 192 = (4!^2)/3*1 193 = 13^2 + 4! 194 = 4/.2/.1 - 3! 195 = sqrt(sqrt(sqrt(3!^4!))) - 21 196 = .1^(-3)*.2 - 4 197 = 4/.2/.1 - 3 198 = .1^(-3)*.2 - sqrt(4) 199 = 14^2 + 3 200 = (4! + 1)*(2^3) 201 = 3!!/4 + 21 202 = 14^2 + 3! 203 = 4/.2/.1 + 3 204 = 34*(2 + 1)! 205 = 41*(2 + 3) 206 = 4/.2/.1 + 3! 207 = 231 - 4! 208 = 214 - 3! 209 = 213 - 4 210 = 42*(3! - 1) 211 = 214 - 3 212 = (3! + 1)!/4! + 2 213 = 4!/.(1) - 3!/2 214 = (4! - 2)/.1 - 3! 215 = (4 + 2)^3 - 1 216 = 3*4!*(2 + 1) 217 = 213 + 4 218 = 3!!*sqrt(.(1)) - 4! + 2 219 = 4!/.(1) + 3!/2 220 = 214 + 3! 221 = sqrt(sqrt(sqrt(3!^4!))) + 1/.2 222 = sqrt(sqrt(sqrt((2 + 1)!^4!))) + 3! 223 = (4!-2)/.1 + 3 224 = 4!/.(1) + 2^3 225 = (4! + 1)*(3^2) 226 = 23/.1 - 4 227 = 231 - 4 228 = 4!/.1 - 3*4 229 = 231 - sqrt(4) 230 = (4! - 3 + 2)/.1 231 = 4!/.1 - 3^2 232 = (3!! - 4!)/(2 + 1) 233 = 234 - 1 234 = 234*1 235 = 234 + 1 236 = 2*(3! - 1)! - 4 237 = 213 + 4! 238 = 241 - 3 239 = (4 + 2)!/3 - 1 240 = (4 + 1)! + (2 + 3)! 241 = (4 + 2)!/3 + 1 242 = 243 - 1 243 = 243*1 244 = 243 + 1 245 = 4!/.1 + 2 + 3 246 = 41*2*3 247 = 241 + 3! 248 = 31*2*4 249 = 4!/.1 + 3^2 250 = (4! + 3 - 2)/.1 251 = (.2^-3)*sqrt(4) + 1 252 = 3*4*21 253 = (.1^-2)/.4 + 3 254 = 4^(3 + 1) - 2 255 = 231 + 4! 256 = 2^(1 + 3 + 4) 257 = 4^(3! - 2) + 1 258 = 4^(3 + 1) + 2 259 = 2^(4!*sqrt(.(1))) + 3 260 = (2^(3!) + 1)*4 261 = 3!!/4 + (.(1))^-2 262 = 3!!*sqrt(.(1)) + 4! - 2 263 = 4!/.1 + 23 264 = 4!*(13 - 2) 265 = 3!!*sqrt(.(1)) + sqrt(.2^-4) 266 = (4! + 2)/.1 + 3! 267 = 3!!*.4 - 21 268 = 134*2 269 = (2 + 3)!/.(4) - 1 270 = (21 + 4!)*3! 271 = (2 + 3)!/.(4) + 1 272 = 4!/.1 + 32 273 = (1/.2)!/.(4) + 3 274 = 3!/.2/.(1) + 4 275 = (31 + 4!)/.2 276 = (4! - 1)*3!*2 277 = 278 = 3!!*.4 - sqrt(.1^-2) 279 = 3!!*.4 - sqrt(.(1))^-2 280 = 3!!/4 + .1^-2 281 = 282 = (4!*2 - 1)*3! 283 = 3!!*.4 - 1/.2 284 = 3!!*sqrt(.(1)) + sqrt(sqrt(2^4!)) 285 = 4!*12 - 3 286 = (4!*3! - 1)*2 287 = 4!*3!*2 - 1 288 = 4!*3!*2*1 289 = (4! - 3! - 1)^2 290 = (4!*3! + 1)*2 291 = 4!*12 + 3 292 = 32/sqrt(.(1)) + 4 293 = 3!!*.4 + 1/.2 294 = (4!*2 + 1)*3! 295 = 296 = 3!/.1/.2 - 4 297 = 321 - 4! 298 = (4! + 3!)/.1 - 2 299 = (2 + 3)!/.4 - 1 300 = (4! + 1)*2*3!
This page was last updated 22 May 1996.
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