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(4): all the asked-for numbers (spoilers) by Charlie)

3!!,3!!+,3!,3*/,3!-          (3!!+3!!)/(3!*3)-3!            74 
3!!,3!!+,3!/,3/,3!-          ((3!!+3!!)/3!)/3-3!            74 
3!!,3!!+,3,3!*/,3!-          (3!!+3!!)/(3*3!)-3!            74 
3!!,3!!+,3/,3!/,3!-          ((3!!+3!!)/3)/3!-3!            74 
3!!,3!!,3/-,3!/,3!-          (3!!-3!!/3)/3!-3!              74 
3!!,3!,3!+,3-/,3!-           3!!/(3!+3!-3)-3!               74 
3!!,3!,3!,3-+/,3!-           3!!/(3!+3!-3)-3!               74 
3!!,3!,3+/,3!,3-!-           3!!/(3!+3)-(3!-3)!             74 
3!!,3!,3+/,3,3+-             3!!/(3!+3)-(3+3)               74 
3!!,3!,3+/,3-,3-             3!!/(3!+3)-3-3                 74 
3!!,3!,3,3!--/,3!-           3!!/(3!-(3-3!))-3!             74 
3!!,3!,3-!,3+/,3!-           3!!/((3!-3)!+3)-3!             74 
3!!,3!,3-,3!+/,3!-           3!!/(3!-3+3!)-3!               74 
3!!,3!,3-,3*/,3!-            3!!/((3!-3)*3)-3!              74 
3!!,3!,3-/,3/,3!-            (3!!/(3!-3))/3-3!              74 
3!!,3*,3,3^/,3!-             3!!*3/3^3-3!                   74 
3!!,3,3!+/,3!,3-!-           3!!/(3+3!)-(3!-3)!             74 
3!!,3,3!+/,3,3+-             3!!/(3+3!)-(3+3)               74 
3!!,3,3!+/,3-,3-             3!!/(3+3!)-3-3                 74 
3!!,3,3!,3-!+/,3!-           3!!/(3+(3!-3)!)-3!             74 
3!!,3,3!,3-*/,3!-            3!!/(3*(3!-3))-3!              74 
3!!,3,3!,3/^/,3!-            3!!/3^(3!/3)-3!                74 
3!!,3,3!^,3!!-/,3!-          3!!/(3^3!-3!!)-3!              74 
3!!,3,3*/,3!,3-!-            3!!/(3*3)-(3!-3)!              74 
3!!,3,3*/,3,3+-              3!!/(3*3)-(3+3)                74 
3!!,3,3*/,3-,3-              3!!/(3*3)-3-3                  74 
3!!,3,3+,3+/,3!-             3!!/(3+3+3)-3!                 74 
3!!,3,3,3++/,3!-             3!!/(3+3+3)-3!                 74 
3!!,3,3,3^/*,3!-             3!!*3/3^3-3!                   74 
3!!,3,3^,3//,3!-             3!!/(3^3/3)-3!                 74 
3!!,3,3^/,3*,3!-             (3!!/3^3)*3-3!                 74 
3!!,3/,3!,3*-,3/             (3!!/3-3!*3)/3                 74 
3!!,3/,3!,3-/,3!-            (3!!/3)/(3!-3)-3!              74 
3!!,3/,3,3!*-,3/             (3!!/3-3*3!)/3                 74 
3!!,3/,3/,3!,3-!-            (3!!/3)/3-(3!-3)!              74 
3!!,3/,3/,3,3+-              (3!!/3)/3-(3+3)                74 
3!!,3/,3/,3-,3-              (3!!/3)/3-3-3                  74 
3!,3!*,3!*,3!+,3/            (3!*3!*3!+3!)/3                74 
3!,3!+,3!*,3!,3/+            (3!+3!)*3!+3!/3                74 
3!,3!,3!**,3!+,3/            (3!*3!*3!+3!)/3                74 
3!,3!,3!*,3!*+,3/            (3!+3!*3!*3!)/3                74 
3!,3!,3!+*,3!,3/+            3!*(3!+3!)+3!/3                74 
3!,3!,3!,3!**+,3/            (3!+3!*3!*3!)/3                74 
3!,3!,3!,3-^+,3/             (3!+3!^(3!-3))/3               74 
3!,3!,3-!,3^+,3/             (3!+(3!-3)!^3)/3               74 
3!,3!,3-^,3!+,3/             (3!^(3!-3)+3!)/3               74 
3!,3!,3^+,3!,3-/             (3!+3!^3)/(3!-3)               74 
3!,3,3+,3^+,3/               (3!+(3+3)^3)/3                 74 
3!,3-!!,3!,3+/,3!-           (3!-3)!!/(3!+3)-3!             74 
3!,3-!!,3,3!+/,3!-           (3!-3)!!/(3+3!)-3!             74 
3!,3-!!,3,3*/,3!-            (3!-3)!!/(3*3)-3!              74 
3!,3-!!,3/,3/,3!-            ((3!-3)!!/3)/3-3!              74 
3!,3-!,3!,3^+,3/             ((3!-3)!+3!^3)/3               74 
3!,3-!,3^,3!+,3/             ((3!-3)!^3+3!)/3               74 
3!,3/,3!,3!+,3!*+            3!/3+(3!+3!)*3!                74 
3!,3/,3!,3!,3!+*+            3!/3+3!*(3!+3!)                74 
3!,3/,3!,3^,3/+              3!/3+3!^3/3                    74 
3!,3^,3!+,3!,3-/             (3!^3+3!)/(3!-3)               74 
3!,3^,3!,3-!+,3/             (3!^3+(3!-3)!)/3               74 
3!,3^,3+,3+,3/               (3!^3+3+3)/3                   74 
3!,3^,3,3++,3/               (3!^3+3+3)/3                   74 
3!,3^,3-,3/,3+               (3!^3-3)/3+3                   74 
3!,3^,3/,3!,3/+              3!^3/3+3!/3                    74 
3,3!!*,3,3^/,3!-             3*3!!/3^3-3!                   74 
3,3!!,3,3^/*,3!-             3*3!!/3^3-3!                   74 
3,3!,3^+,3+,3/               (3+3!^3+3)/3                   74 
3,3!,3^,3++,3/               (3+3!^3+3)/3                   74 
3,3!,3^,3-,3/+               3+(3!^3-3)/3                   74 
3,3+!,3!,3+/,3!-             (3+3)!/(3!+3)-3!               74 
3,3+!,3,3!+/,3!-             (3+3)!/(3+3!)-3!               74 
3,3+!,3,3*/,3!-              (3+3)!/(3*3)-3!                74 
3,3+!,3/,3/,3!-              ((3+3)!/3)/3-3!                74 
3,3+,3!,3^+,3/               (3+3+3!^3)/3                   74 
3,3+,3^,3!+,3/               ((3+3)^3+3!)/3                 74 
3,3,3!,3^++,3/               (3+3+3!^3)/3                   74 
3,3,3!,3^-,3/-               3-(3-3!^3)/3                   74 
3,3,3^,3!!//,3!-             3/(3^3/3!!)-3!                 74 
3,3,3^/,3!!*,3!-             (3/3^3)*3!!-3!                 74 

3!!,3!,3!*-,3!,3+/           (3!!-3!*3!)/(3!+3)             76 
3!!,3!,3!*-,3,3!+/           (3!!-3!*3!)/(3+3!)             76 
3!!,3!,3!*-,3,3*/            (3!!-3!*3!)/(3*3)              76 
3!!,3!,3!*-,3/,3/            ((3!!-3!*3!)/3)/3              76 
3!!,3!,3*/,3!,3!*+           3!!/(3!*3)+3!*3!               76 
3!!,3!/,3/,3!,3!*+           (3!!/3!)/3+3!*3!               76 
3!!,3,3!*/,3!,3!*+           3!!/(3*3!)+3!*3!               76 
3!!,3/,3!+,3/,3!-            (3!!/3+3!)/3-3!                76 
3!!,3/,3!,3!+-,3/            (3!!/3-(3!+3!))/3              76 
3!!,3/,3!,3^+,3!/            (3!!/3+3!^3)/3!                76 
3!!,3/,3!-,3!-,3/            (3!!/3-3!-3!)/3                76 
3!!,3/,3!/,3!,3!*+           (3!!/3)/3!+3!*3!               76 
3!!,3/,3-,3/,3-              (3!!/3-3)/3-3                  76 
3!,3!!,3/+,3/,3!-            (3!+3!!/3)/3-3!                76 
3!,3!*,3!!,3!,3*/+           3!*3!+3!!/(3!*3)               76 
3!,3!*,3!!,3!/,3/+           3!*3!+(3!!/3!)/3               76 
3!,3!*,3!!,3,3!*/+           3!*3!+3!!/(3*3!)               76 
3!,3!*,3!!,3/,3!/+           3!*3!+(3!!/3)/3!               76 
3!,3!+,3!,3/,3!^+            3!+3!+(3!/3)^3!                76 
3!,3!+,3!,3^+,3/             (3!+3!+3!^3)/3                 76 
3!,3!,3!,3/,3!^++            3!+3!+(3!/3)^3!                76 
3!,3!,3!,3^++,3/             (3!+3!+3!^3)/3                 76 
3!,3!,3!,3^-,3/-             3!-(3!-3!^3)/3                 76 
3!,3!,3/,3!^+,3!+            3!+(3!/3)^3!+3!                76 
3!,3!,3/,3!^,3!++            3!+(3!/3)^3!+3!                76 
3!,3!,3^+,3!+,3/             (3!+3!^3+3!)/3                 76 
3!,3!,3^,3!++,3/             (3!+3!^3+3!)/3                 76 
3!,3!,3^,3!-,3/+             3!+(3!^3-3!)/3                 76 
3!,3/,3!^,3!+,3!+            (3!/3)^3!+3!+3!                76 
3!,3/,3!^,3!,3!++            (3!/3)^3!+3!+3!                76 
3!,3^,3!!,3/+,3!/            (3!^3+3!!/3)/3!                76 
3!,3^,3!+,3!+,3/             (3!^3+3!+3!)/3                 76 
3!,3^,3!,3!++,3/             (3!^3+3!+3!)/3                 76 
3!,3^,3!-,3/,3!+             (3!^3-3!)/3+3!                 76 
3!,3^,3+,3/,3+               (3!^3+3)/3+3                   76 
3,3!,3^+,3/,3+               (3+3!^3)/3+3                   76 
3,3!,3^,3+,3/+               3+(3!^3+3)/3                   76 
3,3,3!,3^+,3/+               3+(3+3!^3)/3                   76 

3!!,3!!+,3!,3*/,3-           (3!!+3!!)/(3!*3)-3             77 
3!!,3!!+,3!/,3/,3-           ((3!!+3!!)/3!)/3-3             77 
3!!,3!!+,3,3!*/,3-           (3!!+3!!)/(3*3!)-3             77 
3!!,3!!+,3/,3!/,3-           ((3!!+3!!)/3)/3!-3             77 
3!!,3!!,3/-,3!/,3-           (3!!-3!!/3)/3!-3               77 
3!!,3!,3!+,3-/,3-            3!!/(3!+3!-3)-3                77 
3!!,3!,3!,3-+/,3-            3!!/(3!+3!-3)-3                77 
3!!,3!,3+/,3!,3--            3!!/(3!+3)-(3!-3)              77 
3!!,3!,3+/,3!-,3+            3!!/(3!+3)-3!+3                77 
3!!,3!,3+/,3+,3!-            3!!/(3!+3)+3-3!                77 
3!!,3!,3+/,3,3!-+            3!!/(3!+3)+3-3!                77 
3!!,3!,3,3!--/,3-            3!!/(3!-(3-3!))-3              77 
3!!,3!,3-!,3+/,3-            3!!/((3!-3)!+3)-3              77 
3!!,3!,3-,3!+/,3-            3!!/(3!-3+3!)-3                77 
3!!,3!,3-,3*/,3-             3!!/((3!-3)*3)-3               77 
3!!,3!,3-/,3/,3-             (3!!/(3!-3))/3-3               77 
3!!,3*,3,3^/,3-              3!!*3/3^3-3                    77 
3!!,3,3!+/,3!,3--            3!!/(3+3!)-(3!-3)              77 
3!!,3,3!+/,3!-,3+            3!!/(3+3!)-3!+3                77 
3!!,3,3!+/,3+,3!-            3!!/(3+3!)+3-3!                77 
3!!,3,3!+/,3,3!-+            3!!/(3+3!)+3-3!                77 
3!!,3,3!,3-!+/,3-            3!!/(3+(3!-3)!)-3              77 
3!!,3,3!,3-*/,3-             3!!/(3*(3!-3))-3               77 
3!!,3,3!,3/^/,3-             3!!/3^(3!/3)-3                 77 
3!!,3,3!^,3!!-/,3-           3!!/(3^3!-3!!)-3               77 
3!!,3,3*/,3!,3--             3!!/(3*3)-(3!-3)               77 
3!!,3,3*/,3!-,3+             3!!/(3*3)-3!+3                 77 
3!!,3,3*/,3+,3!-             3!!/(3*3)+3-3!                 77 
3!!,3,3*/,3,3!-+             3!!/(3*3)+3-3!                 77 
3!!,3,3+,3+/,3-              3!!/(3+3+3)-3                  77 
3!!,3,3,3++/,3-              3!!/(3+3+3)-3                  77 
3!!,3,3,3^/*,3-              3!!*3/3^3-3                    77 
3!!,3,3^,3//,3-              3!!/(3^3/3)-3                  77 
3!!,3,3^-,3!,3+/             (3!!-3^3)/(3!+3)               77 
3!!,3,3^-,3,3!+/             (3!!-3^3)/(3+3!)               77 
3!!,3,3^-,3,3*/              (3!!-3^3)/(3*3)                77 
3!!,3,3^-,3/,3/              ((3!!-3^3)/3)/3                77 
3!!,3,3^/,3*,3-              (3!!/3^3)*3-3                  77 
3!!,3/,3!,3+-,3/             (3!!/3-(3!+3))/3               77 
3!!,3/,3!,3-/,3-             (3!!/3)/(3!-3)-3               77 
3!!,3/,3!-,3-,3/             (3!!/3-3!-3)/3                 77 
3!!,3/,3,3!+-,3/             (3!!/3-(3+3!))/3               77 
3!!,3/,3,3*-,3/              (3!!/3-3*3)/3                  77 
3!!,3/,3-,3!-,3/             (3!!/3-3-3!)/3                 77 
3!!,3/,3/,3!,3--             (3!!/3)/3-(3!-3)               77 
3!!,3/,3/,3!-,3+             (3!!/3)/3-3!+3                 77 
3!!,3/,3/,3+,3!-             (3!!/3)/3+3-3!                 77 
3!!,3/,3/,3,3!-+             (3!!/3)/3+3-3!                 77 
3!,3!,3^+,3/,3+              (3!+3!^3)/3+3                  77 
3!,3!,3^,3-,3/+              3!+(3!^3-3)/3                  77 
3!,3,3!,3^-,3/-              3!-(3-3!^3)/3                  77 
3!,3-!!,3!,3+/,3-            (3!-3)!!/(3!+3)-3              77 
3!,3-!!,3,3!+/,3-            (3!-3)!!/(3+3!)-3              77 
3!,3-!!,3,3*/,3-             (3!-3)!!/(3*3)-3               77 
3!,3-!!,3/,3/,3-             ((3!-3)!!/3)/3-3               77 
3!,3^,3!+,3/,3+              (3!^3+3!)/3+3                  77 
3!,3^,3-,3/,3!+              (3!^3-3)/3+3!                  77 
3,3!!*,3,3^/,3-              3*3!!/3^3-3                    77 
3,3!!,3!,3+/+,3!-            3+3!!/(3!+3)-3!                77 
3,3!!,3!,3+/,3!-+            3+3!!/(3!+3)-3!                77 
3,3!!,3,3!+/+,3!-            3+3!!/(3+3!)-3!                77 
3,3!!,3,3!+/,3!-+            3+3!!/(3+3!)-3!                77 
3,3!!,3,3*/+,3!-             3+3!!/(3*3)-3!                 77 
3,3!!,3,3*/,3!-+             3+3!!/(3*3)-3!                 77 
3,3!!,3,3^/*,3-              3*3!!/3^3-3                    77 
3,3!!,3/,3/+,3!-             3+(3!!/3)/3-3!                 77 
3,3!!,3/,3/,3!-+             3+(3!!/3)/3-3!                 77 
3,3!,3!!,3!,3+/--            3-(3!-3!!/(3!+3))              77 
3,3!,3!!,3,3!+/--            3-(3!-3!!/(3+3!))              77 
3,3!,3!!,3,3*/--             3-(3!-3!!/(3*3))               77 
3,3!,3!!,3/,3/--             3-(3!-(3!!/3)/3)               77 
3,3!,3!,3^+,3/+              3+(3!+3!^3)/3                  77 
3,3!,3^,3!+,3/+              3+(3!^3+3!)/3                  77 
3,3!-,3!!,3!,3+/+            3-3!+3!!/(3!+3)                77 
3,3!-,3!!,3,3!+/+            3-3!+3!!/(3+3!)                77 
3,3!-,3!!,3,3*/+             3-3!+3!!/(3*3)                 77 
3,3!-,3!!,3/,3/+             3-3!+(3!!/3)/3                 77 
3,3+!,3!,3+/,3-              (3+3)!/(3!+3)-3                77 
3,3+!,3,3!+/,3-              (3+3)!/(3+3!)-3                77 
3,3+!,3,3*/,3-               (3+3)!/(3*3)-3                 77 
3,3+!,3/,3/,3-               ((3+3)!/3)/3-3                 77 
3,3,3^,3!!//,3-              3/(3^3/3!!)-3                  77 
3,3,3^/,3!!*,3-              (3/3^3)*3!!-3                  77 
33,3/!,3!!,3!!*/             (33/3)!/(3!!*3!!)              77 
33,3/!,3!!/,3!!/             ((33/3)!/3!!)/3!!              77 

3!!,3!+,3!,3^-,3!/           (3!!+3!-3!^3)/3!               85 
3!!,3!+,3!/,3!,3!*-          (3!!+3!)/3!-3!*3!              85 
3!!,3!,3!,3^-+,3!/           (3!!+3!-3!^3)/3!               85 
3!!,3!,3+!,3!/+,3!!/         (3!!+(3!+3)!/3!)/3!!           85 
3!!,3!,3^,3!--,3!/           (3!!-(3!^3-3!))/3!             85 
3!!,3!,3^-,3!+,3!/           (3!!-3!^3+3!)/3!               85 
3!!,3,3!+!,3!/+,3!!/         (3!!+(3+3!)!/3!)/3!!           85 
3!!,3,3*!,3!/+,3!!/          (3!!+(3*3)!/3!)/3!!            85 
3!!,3/,3!+,3/,3+             (3!!/3+3!)/3+3                 85 
3!!,3/,3-,3/,3!+             (3!!/3-3)/3+3!                 85 
3!,3!!+,3!,3^-,3!/           (3!+3!!-3!^3)/3!               85 
3!,3!!+,3!/,3!,3!*-          (3!+3!!)/3!-3!*3!              85 
3!,3!!,3!,3^-+,3!/           (3!+3!!-3!^3)/3!               85 
3!,3!!,3/+,3/,3+             (3!+3!!/3)/3+3                 85 
3!,3!!,3/,3-,3/+             3!+(3!!/3-3)/3                 85 
3!,3!,3+!,3!!/+,3!/          (3!+(3!+3)!/3!!)/3!            85 
3!,3!,3^,3!!--,3!/           (3!-(3!^3-3!!))/3!             85 
3!,3!,3^-,3!!+,3!/           (3!-3!^3+3!!)/3!               85 
3!,3+!,3!!/,3!+,3!/          ((3!+3)!/3!!+3!)/3!            85 
3!,3+!,3!/,3!!+,3!!/         ((3!+3)!/3!+3!!)/3!!           85 
3!,3,3!!,3/-,3/-             3!-(3-3!!/3)/3                 85 
3!,3,3!+!,3!!/+,3!/          (3!+(3+3!)!/3!!)/3!            85 
3!,3,3*!,3!!/+,3!/           (3!+(3*3)!/3!!)/3!             85 
3,3!!,3/,3!+,3/+             3+(3!!/3+3!)/3                 85 
3,3!+!,3!!/,3!+,3!/          ((3+3!)!/3!!+3!)/3!            85 
3,3!+!,3!/,3!!+,3!!/         ((3+3!)!/3!+3!!)/3!!           85 
3,3!,3!!,3/+,3/+             3+(3!+3!!/3)/3                 85 
3,3*!,3!!/,3!+,3!/           ((3*3)!/3!!+3!)/3!             85 
3,3*!,3!/,3!!+,3!!/          ((3*3)!/3!+3!!)/3!!            85 

3!!,3!+,3!/,33-              (3!!+3!)/3!-33                 88 
3!!,3/,3!+,3/,3!+            (3!!/3+3!)/3+3!                88 
3!,3!!+,3!/,33-              (3!+3!!)/3!-33                 88 
3!,3!!,3/+,3/,3!+            (3!+3!!/3)/3+3!                88 
3!,3!!,3/,3!+,3/+            3!+(3!!/3+3!)/3                88 
3!,3!,3!!,3/+,3/+            3!+(3!+3!!/3)/3                88 

3!!,3/,33+,3/                (3!!/3+33)/3                   91 
3!,3/,3!^,3,3^+              (3!/3)^3!+3^3                  91 
3,3^,3!,3/,3!^+              3^3+(3!/3)^3!                  91 
33,3!!,3/+,3/                (33+3!!/3)/3                   91 

3!!,3!!,3!/-,3!/,3!-         (3!!-3!!/3!)/3!-3!             94 
3!!,3!+,3!/,3,3^-            (3!!+3!)/3!-3^3                94 
3!,3!!+,3!/,3,3^-            (3!+3!!)/3!-3^3                94 

3!,3^,3!!,3!+,3!/-           3!^3-(3!!+3!)/3!               95 
3!,3^,3!,3!!+,3!/-           3!^3-(3!+3!!)/3!               95 

Posted by Charlie on 2008-07-08 11:48:29