This is a real story. A long long time ago, based on the "Four fours" problem, I wondered if I could do the same using exactly five 3īs, the restrictions a bit tighter: I could use only the 4 basic math operations, exponentiation, factorial, and all parentheses I may need. Besides, I didnīt disallow me to join two "3īs" to make "33".
Using this, and only this, I succeeded in writing expressions for all integers from 0 to 100.
To narrow your work, since a great number of integers can be easily obtained, can you find expressions for 47, 50, 56, 58, 64, 70, 71, 73, 74, 76, 77, 85, 88, 94, and 95?
(In reply to
re(3): all hundred by Leming)
The following list is not a definitive proof that 145 can't be achieved, but that is the first one not found by the program. The program's design limitations are that it will not do any multiplication where either of the multiplicands is greater than 10^90 in absolute value (or the equivalent matching in division, except for the denominator not being smaller than 10^-90), and will not take factorials when the argument is larger than 17. These are to prevent overflow.
3,3/,333^ (3/3)^333 1
3,33,33/- 3-33/33 2
3,33,33/* 3*33/33 3
3,33,33/+ 3+33/33 4
3!,33,33/- 3!-33/33 5
3!,33,33/* 3!*33/33 6
33,33/,3!+ 33/33+3! 7
33,3,3*-,3/ (33-3*3)/3 8
3,3,3^-,33+ 3-3^3+33 9
33,3/,3,3/- 33/3-3/3 10
33,33+,3!/ (33+33)/3! 11
3,3/,33,3/+ 3/3+33/3 12
33,3,3++,3/ (33+3+3)/3 13
33,3,3*+,3/ (33+3*3)/3 14
33,3,3,3+*- 33-3*(3+3) 15
3,3^,33,3/- 3^3-33/3 16
3,33,3/+,3+ 3+33/3+3 17
3,33,3,3^-* 3*(33-3^3) 18
3!,3!,33+,3/+ 3!+(3!+33)/3 19
33,3,3^+,3/ (33+3^3)/3 20
33,3-,3,3*- 33-3-3*3 21
33,33,3/- 33-33/3 22
3,3!!,3,33+/+ 3+3!!/(3+33) 23
33,33/,3+! (33/33+3)! 24
33,3!,3/,3^- 33-(3!/3)^3 25
33,3!-,3,3/- 33-3!-3/3 26
33,3,3,3*-+ 33+3-3*3 27
3,3/,33+,3!- 3/3+33-3! 28
33,3,3,3/+- 33-(3+3/3) 29
33,3+,3-,3- 33+3-3-3 30
3,3/,33+,3- 3/3+33-3 31
3,33*,3-,3/ (3*33-3)/3 32
3,3-,3/,33+ (3-3)/3+33 33
33,3*,3+,3/ (33*3+3)/3 34
33,3+,3,3/- 33+3-3/3 35
3,33+,3,3// (3+33)/(3/3) 36
333,3,3*/ 333/(3*3) 37
3,3^,33,3/+ 3^3+33/3 38
33,3,3*,3-+ 33+3*3-3 39
3,3/,3!+,33+ 3/3+3!+33 40
3!,3/,3^,33+ (3!/3)^3+33 41
33,3+,3+,3+ 33+3+3+3 42
3,3!,3/+!,3/,3+ (3+3!/3)!/3+3 43
33,33,3/+ 33+33/3 44
33,3+,3,3*+ 33+3+3*3 45
3!,3/,3!^,3,3!*- (3!/3)^3!-3*3! 46
33,3/,3!,3!*+ 33/3+3!*3! 47
3,3,3^*,33- 3*3^3-33 48
3,3/,3!+,3!,3/^ (3/3+3!)^(3!/3) 49
3!,3/,3^!,3!!/,3!- (3!/3)^3!/3!!-3! 50
3,3,3+*,33+ 3*(3+3)+33 51
3,3*,3!*,3!,3/- 3*3*3!-3!/3 52
3,3!,3**,3,3/- 3*3!*3-3/3 53
33,3!,3*,3++ 33+3!*3+3 54
333,3-,3!/ (333-3)/3! 55
3,333+,3!/ (3+333)/3! 56
3,3^,33+,3- 3^3+33-3 57
3,3/,3+,3^,3!- (3/3+3)^3-3! 58
3!!,3!,3!+/,3,3/- 3!!/(3!+3!)-3/3 59
33,33+,3!- 33+33-3! 60
3!,3^,33-,3/ (3!^3-33)/3 61
3,3!!,3,3^/,3!-* 3*(3!!/3^3-3!) 62
33,33,3-+ 33+33-3 63
3,3,3*+,3/,3^ ((3+3*3)/3)^3 64
3!,33*,3-,3/ (3!*33-3)/3 65
33,3*,33- 33*3-33 66
33,3!*,3+,3/ (33*3!+3)/3 67
33,3!*,3!+,3/ (33*3!+3!)/3 68
33,33+,3+ 33+33+3 69
3,3/,3+,3^,3!+ (3/3+3)^3+3! 70
3,3+,3^,3-,3/ ((3+3)^3-3)/3 71
33,3!,33++ 33+3!+33 72
3,3+,3^,3+,3/ ((3+3)^3+3)/3 73
3,3!,3^,3++,3/ (3+3!^3+3)/3 74
3!,3!,3**,33- 3!*3!*3-33 75
3,3!,3^,3+,3/+ 3+(3!^3+3)/3 76
3,3+!,3,3*/,3- (3+3)!/(3*3)-3 77
33,3!-,3*,3- (33-3!)*3-3 78
3,3^,3*,3!,3/- 3^3*3-3!/3 79
3,3,3^*,3,3/- 3*3^3-3/3 80
33,3,3+-,3* (33-(3+3))*3 81
3,3/,3,3,3^*+ 3/3+3*3^3 82
3!,3^,33+,3/ (3!^3+33)/3 83
33,3-,3*,3!- (33-3)*3-3! 84
3,3!,3!!,3/+,3/+ 3+(3!+3!!/3)/3 85
3!!,3/,3/,3+,3+ (3!!/3)/3+3+3 86
3,33,3-*,3- 3*(33-3)-3 87
3!!,3!+,3!/,33- (3!!+3!)/3!-33 88
3!!,3,3^/,3+,3* (3!!/3^3+3)*3 89
3,33*,3,3*- 3*33-3*3 90
33,3!!,3/+,3/ (33+3!!/3)/3 91
3!,3!!,3,3*/,3!++ 3!+3!!/(3*3)+3! 92
3,33,3-*,3+ 3*(33-3)+3 93
3!,3!!+,3!/,3,3^- (3!+3!!)/3!-3^3 94
3!,3^,3!!,3!+,3!/- 3!^3-(3!!+3!)/3! 95
3,33,3,3/-* 3*(33-3/3) 96
33,3*,3!,3/- 33*3-3!/3 97
3,33*,3,3/- 3*33-3/3 98
33,3*,3/,3* (33*3/3)*3 99
33,3-,3/,3^ ((33-3)/3)^3 0
3,33*,3,3/+ 3*33+3/3 100
33,3*,3!,3/+ 33*3+3!/3 101
3,3,3/,33+* 3*(3/3+33) 102
3!,3!!+,3!/,3!,3*- (3!+3!!)/3!-3!*3 103
3!!,3!,3^+,3,3*/ (3!!+3!^3)/(3*3) 104
333,3/,3!- 333/3-3! 105
3,3!,3!**,3!,3/- 3*3!*3!-3!/3 106
3!,3!*,3*,3,3/- 3!*3!*3-3/3 107
333,3/,3- 333/3-3 108
333,3!-,3/ (333-3!)/3 109
333,3-,3/ (333-3)/3 110
333,3!,3-/ 333/(3!-3) 111
3,333+,3/ (3+333)/3 112
3!,333+,3/ (3!+333)/3 113
3,333,3/+ 3+333/3 114
3,33,3!!--,3!/ (3-(33-3!!))/3! 115
3,3!^,33-,3!/ (3^3!-33)/3! 116
333,3/,3!+ 333/3+3! 117
3,3!^,3-,3!/,3- (3^3!-3)/3!-3 118
3,3!^,3+,3!/,3- (3^3!+3)/3!-3 119
3!,33,33/-! (3!-33/33)! 120
33,3/,3!,3/^ (33/3)^(3!/3) 121
3,3,3+^,3+,3!/ (3^(3+3)+3)/3! 122
3,33,3/,3!-!+ 3+(33/3-3!)! 123
3,3!^,3-,3!/,3+ (3^3!-3)/3!+3 124
33,3/,3!-,3^ (33/3-3!)^3 125
3,33,3,3*+* 3*(33+3*3) 126
3,3!^,33+,3!/ (3^3!+33)/3! 127
3,3!,3/,3+,3^+ 3+(3!/3+3)^3 128
3!!,333-,3/ (3!!-333)/3 129
3,3*,3!,3!!+,3!/+ 3*3+(3!+3!!)/3! 130
3,3!,3!!*+,33/ (3+3!*3!!)/33 131
33,33,3*+ 33+33*3 132
3!!,3!+,3!/,3!+,3!+ (3!!+3!)/3!+3!+3! 133
3!!,3!/,3!!+,3!/,3!- (3!!/3!+3!!)/3!-3! 134
3,3!,3!,33++* 3*(3!+3!+33) 135
3!,3^,3!!,3,3*/- 3!^3-3!!/(3*3) 136
3,3!*,3!,3!!-,3!/- 3*3!-(3!-3!!)/3! 137
3,3^,3-,3!*,3!- (3^3-3)*3!-3! 138
3!,3!!+,3!/,3,3!*+ (3!+3!!)/3!+3*3! 139
3,3/,3!+!,3!,3!*/ (3/3+3!)!/(3!*3!) 140
33,3!,3!*,3*+ 33+3!*3!*3 141
3!,3!!,3,3^/,3-* 3!*(3!!/3^3-3) 142
3!!,3!!,3!/+,3!/,3+ (3!!+3!!/3!)/3!+3 143
3!,33,3,3*-* 3!*(33-3*3) 144
3,3^,3!!,3!-,3!/+ 3^3+(3!!-3!)/3! 146
3,3!,3,3,3^-*- 3-3!*(3-3^3) 147
3!!,3!+,3!/,3,3^+ (3!!+3!)/3!+3^3 148
3!!,3!/,3-,33+ 3!!/3!-3+33 150
33,3!,3!!-,3!/- 33-(3!-3!!)/3! 152
3,33,3!,3*+* 3*(33+3!*3) 153
3!,3!!+,3!/,33+ (3!+3!!)/3!+33 154
3!!,3!,3^,3!-+,3!/ (3!!+3!^3-3!)/3! 155
3,3^,3!*,3-,3- 3^3*3!-3-3 156
3!,3,3^,3!!//,3- 3!/(3^3/3!!)-3 157
3!!,3!!+,3/,3!-,3/ ((3!!+3!!)/3-3!)/3 158
3!,33,3!-*,3- 3!*(33-3!)-3 159
3,3+,3,3^,3!!// (3+3)/(3^3/3!!) 160
3,3^,3!*,3,3/- 3^3*3!-3/3 161
33,3-,3-,3!* (33-3-3)*3! 162
3,3^,3!*,3,3/+ 3^3*3!+3/3 163
3!,3/,3,3^,3!*+ 3!/3+3^3*3! 164
33,3!*,33- 33*3!-33 165
3!!,3!,3!,3^+-,3/ (3!!-(3!+3!^3))/3 166
3,3*!,3!!/,3-,3/ ((3*3)!/3!!-3)/3 167
3,3,3!,3,3^*++ 3+3+3!*3^3 168
3,3!,3^-,3!!+,3/ (3-3!^3+3!!)/3 169
3,3*!,3!!/,3!+,3/ ((3*3)!/3!!+3!)/3 170
3!,33*,3,3^- 3!*33-3^3 171
33,3-,3!*,3!- (33-3)*3!-3! 174
3!!,3/,3!,3/,3!^- 3!!/3-(3!/3)^3! 176
3!,33,3-*,3- 3!*(33-3)-3 177
3,3!!,3,3^/+,3!* (3+3!!/3^3)*3! 178
3,3+,33,3-* (3+3)*(33-3) 180
3,3+,3^,33- (3+3)^3-33 183
3!!,3!/,3!,3/,3!^+ 3!!/3!+(3!/3)^3! 184
3,33,3!,3^-- 3-(33-3!^3) 186
33,3!*,3,3*- 33*3!-3*3 189
33,3,3+*,3!- 33*(3+3)-3! 192
3!!,3!!,3!/-,3/,3!- (3!!-3!!/3!)/3-3! 194
3,3+,33*,3- (3+3)*33-3 195
33,3!*,3!,3/- 33*3!-3!/3 196
3!,33*,3,3/- 3!*33-3/3 197
3,33,33+* 3*(33+33) 198
3,3/,3!,33*+ 3/3+3!*33 199
3!,3/,33,3!*+ 3!/3+33*3! 200
33,3,3+*,3+ 33*(3+3)+3 201
3!!,3!-,3/,3!,3!*- (3!!-3!)/3-3!*3! 202
3,3!^,3!!,3!/-,3/ (3^3!-3!!/3!)/3 203
3!,3,3/,33+* 3!*(3/3+33) 204
3!,3^,33,3/- 3!^3-33/3 205
3!!,3-,3/,33- (3!!-3)/3-33 206
3,3+!,3/,33- (3+3)!/3-33 207
3!!,3+,3/,33- (3!!+3)/3-33 208
3!!,3!+,3/,33- (3!!+3!)/3-33 209
3,3!^,3/,33- 3^3!/3-33 210
3!,3^,3-,3!,3/- 3!^3-3-3!/3 211
3!,3^,3-,3,3/- 3!^3-3-3/3 212
3!,3,33+*,3- 3!*(3+33)-3 213
3,3/,3!,3^,3-+ 3/3+3!^3-3 214
3,3+,3^,3,3/- (3+3)^3-3/3 215
3,3+,33,3+* (3+3)*(33+3) 216
3,3+,3^,3,3/+ (3+3)^3+3/3 217
3,3+,3^,3!,3/+ (3+3)^3+3!/3 218
33,3+,3!*,3+ (33+3)*3!+3 219
3!,3^,3+,3,3/+ 3!^3+3+3/3 220
3!,3^,3+,3!,3/+ 3!^3+3+3!/3 221
3!,33,3+*,3!+ 3!*(33+3)+3! 222
3!!,33-,3/,3!- (3!!-33)/3-3! 223
3!,3^,3!,3/,3^+ 3!^3+(3!/3)^3 224
3!,33*,3,3^+ 3!*33+3^3 225
3!!,33-,3/,3- (3!!-33)/3-3 226
33,3/,3!,3^+ 33/3+3!^3 227
3!!,3-,33-,3/ (3!!-3-33)/3 228
3,3+!,33-,3/ ((3+3)!-33)/3 229
3!!,33,3--,3/ (3!!-(33-3))/3 230
33,33,3!*+ 33+33*3! 231
3,3!^,33-,3/ (3^3!-33)/3 232
3!!,3-,3/,3-,3- (3!!-3)/3-3-3 233
33,3!+,3,3+* (33+3!)*(3+3) 234
3!,3!!,33-,3/+ 3!+(3!!-33)/3 235
3,3+!,3-,3/,3- ((3+3)!-3)/3-3 236
3,3!,3!,33+*+ 3+3!*(3!+33) 237
3,3+!,3,3+-,3/ ((3+3)!-(3+3))/3 238
3,3*,3-!,3-,3/ ((3*3-3)!-3)/3 239
33,3,3^-!,3/ (33-3^3)!/3 240
3,3+!,3/,3,3/+ (3+3)!/3+3/3 241
3,3,3+^,3-,3/ (3^(3+3)-3)/3 242
33,3!-,3,3** (33-3!)*3*3 243
3,3*,3^,3+,3/ ((3*3)^3+3)/3 244
3!!,33+,3/,3!- (3!!+33)/3-3! 245
33,3-,3!,3^+ 33-3+3!^3 246
3,3,3,3!^+,3/+ 3+(3+3^3!)/3 247
33,3!!+,3/,3- (33+3!!)/3-3 248
3,3+,3^,33+ (3+3)^3+33 249
3!!,3,33--,3/ (3!!-(3-33))/3 250
33,3,3+!+,3/ (33+(3+3)!)/3 251
3!,3^,33,3++ 3!^3+33+3 252
33,3!!,3!++,3/ (33+3!!+3!)/3 253
3,3!^,33+,3/ (3^3!+33)/3 254
3!,3^,33+,3!+ 3!^3+33+3! 255
3!,3/,3!,3/,3^^ (3!/3)^(3!/3)^3 256
33,3!!+,3/,3!+ (33+3!!)/3+3! 257
3,3+!,3/,3,3!*+ (3+3)!/3+3*3! 258
3!!,3+,3/,3,3!*+ (3!!+3)/3+3*3! 259
3,3!*,3!!,3!+,3/+ 3*3!+(3!!+3!)/3 260
3!!,3!/,33-,3* (3!!/3!-33)*3 261
33,3!,3/,3^* 33*(3!/3)^3 264
3,3^,3!,3!!-,3/- 3^3-(3!-3!!)/3 265
3,3^,3!!,3-,3/+ 3^3+(3!!-3)/3 266
3,3+!,3/,3,3^+ (3+3)!/3+3^3 267
3,3^,3,3!!+,3/+ 3^3+(3+3!!)/3 268
3!!,3!+,3/,3,3^+ (3!!+3!)/3+3^3 269
3,3,33,3-** 3*3*(33-3) 270
33,3!!,3!-,3/+ 33+(3!!-3!)/3 271
33,3!!,3-,3/+ 33+(3!!-3)/3 272
33,3,3+!,3/+ 33+(3+3)!/3 273
33,3,3!!+,3/+ 33+(3+3!!)/3 274
33,3!!,3!+,3/+ 33+(3!!+3!)/3 275
3,3!^,3/,33+ 3^3!/3+33 276
3!!,3+,3/,3!,3!*+ (3!!+3)/3+3!*3! 277
3!,3!*,3!,3!!+,3/+ 3!*3!+(3!+3!!)/3 278
333,3*,3!!- 333*3-3!! 279
3!,3/,3!^,3!,3^+ (3!/3)^3!+3!^3 280
3!!,3!/,3!!+,3+,3/ (3!!/3!+3!!+3)/3 281
3!!,3,3,3!/-/,3!- 3!!/(3-3/3!)-3! 282
3,3!^,3!!,3!/+,3/ (3^3!+3!!/3!)/3 283
3!,3!+,3^,3!/,3- (3!+3!)^3/3!-3 285
3!,3!!,3!/,3!!+,3/+ 3!+(3!!/3!+3!!)/3 286
3!,3!+,3^,3!-,3!/ ((3!+3!)^3-3!)/3! 287
3,33,3*,3-* 3*(33*3-3) 288
3!,3!,3!+,3^+,3!/ (3!+(3!+3!)^3)/3! 289
3,33*,3*,3!- 3*33*3-3! 291
33,3*,3*,3- 33*3*3-3 294
3!!,3/,3/,3!,3^+ (3!!/3)/3+3!^3 296
33,3/,3,3^* (33/3)*3^3 297
333,33- 333-33 &nbs
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Posted by Charlie
on 2008-07-09 11:37:35 |
re(4): all hundred
