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 The five 3īs (Posted on 2008-07-07)
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?

 See The Solution Submitted by pcbouhid Rating: 4.5000 (2 votes)

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 re(4): all hundred | Comment 31 of 35 |
(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

Posted by Charlie
on 2008-07-09 11:37:35

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