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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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 still not all, even with exponentiation (spoilers present) | Comment 8 of 35 |
(In reply to geeky computer solution for some values (spoilers present) by Charlie)

`3,3!!,3!/+,3/,3!+           (3+3!!/3!)/3+3!              47 3,3^,3!!,3!,3!*/+           3^3+3!!/(3!*3!)              47 3,3^,3!!,3!/,3!/+           3^3+(3!!/3!)/3!              47 33,3/,3!,3!*+               33/3+3!*3!                   47 `
`3,3!*,3*,3!,3/+             3*3!*3+3!/3                  56 3,3!+,3!*,3!,3/+            (3+3!)*3!+3!/3               56 3,3!,3**,3!,3/+             3*3!*3+3!/3                  56 3,3*,3!*,3!,3/+             3*3*3!+3!/3                  56 3,3,3!**,3!,3/+             3*3*3!+3!/3                  56 3,333+,3!/                  (3+333)/3!                   56 3,3^,3!*,3!+,3/             (3^3*3!+3!)/3                56 333,3+,3!/                  (333+3)/3!                   56 `
`3,3!!,3!!-!+,3^,3!-         (3+(3!!-3!!)!)^3-3!          58 3,3!!,3!!-!-,3!^,3!-        (3-(3!!-3!!)!)^3!-3!         58 3,3!!,3!!/+,3^,3!-          (3+3!!/3!!)^3-3!             58 3,3!!,3!!/-,3!^,3!-         (3-3!!/3!!)^3!-3!            58 3,3!*,3!!,3!,3*/+           3*3!+3!!/(3!*3)              58 3,3!*,3!!,3!/,3/+           3*3!+(3!!/3!)/3              58 3,3!*,3!!,3,3!*/+           3*3!+3!!/(3*3!)              58 3,3!*,3!!,3/,3!/+           3*3!+(3!!/3)/3!              58 3,3!,3!-!+,3^,3!-           (3+(3!-3!)!)^3-3!            58 3,3!,3!-!-,3!^,3!-          (3-(3!-3!)!)^3!-3!           58 3,3!,3!/+,3^,3!-            (3+3!/3!)^3-3!               58 3,3!,3!/-,3!^,3!-           (3-3!/3!)^3!-3!              58 3,3+,3/,3!^,3!-             ((3+3)/3)^3!-3!              58 3,3,3-!+,3^,3!-             (3+(3-3)!)^3-3!              58 3,3,3-!-,3!^,3!-            (3-(3-3)!)^3!-3!             58 3,3,3/+,3^,3!-              (3+3/3)^3-3!                 58 3,3,3/-,3!^,3!-             (3-3/3)^3!-3!                58 3,3/,3+,3^,3!-              (3/3+3)^3-3!                 58 `
```3,3!!*,3!!+,3!!/,3^         ((3*3!!+3!!)/3!!)^3          64 3,3!!*,3!!-,3!!/,3!^        ((3*3!!-3!!)/3!!)^3!         64 3,3!!,3!!,3!!-*!+,3^        (3+(3!!*(3!!-3!!))!)^3       64 3,3!!,3!!,3!!-*!-,3!^       (3-(3!!*(3!!-3!!))!)^3!      64 3,3!!,3!!,3!!-^+,3^         (3+3!!^(3!!-3!!))^3          64 3,3!!,3!!,3!!-^-,3!^        (3-3!!^(3!!-3!!))^3!         64 3,3!!,3!!-!+,3!,3-^         (3+(3!!-3!!)!)^(3!-3)        64 3,3!!,3!!-!,3!!^+,3^        (3+(3!!-3!!)!^3!!)^3         64 3,3!!,3!!-!,3!!^-,3!^       (3-(3!!-3!!)!^3!!)^3!        64 3,3!!,3!!-!,3!^+,3^         (3+(3!!-3!!)!^3!)^3          64 3,3!!,3!!-!,3!^-,3!^        (3-(3!!-3!!)!^3!)^3!         64 3,3!!,3!!-!,3^+,3^          (3+(3!!-3!!)!^3)^3           64 3,3!!,3!!-!,3^-,3!^         (3-(3!!-3!!)!^3)^3!          64 3,3!!,3!!-!-,3!,3-!^        (3-(3!!-3!!)!)^(3!-3)!       64 3,3!!,3!!-!-,3,3+^          (3-(3!!-3!!)!)^(3+3)         64 3,3!!,3!!-,3!!*!+,3^        (3+((3!!-3!!)*3!!)!)^3       64 3,3!!,3!!-,3!!*!-,3!^       (3-((3!!-3!!)*3!!)!)^3!      64 3,3!!,3!!-,3!!/!+,3^        (3+((3!!-3!!)/3!!)!)^3       64 3,3!!,3!!-,3!!/!-,3!^       (3-((3!!-3!!)/3!!)!)^3!      64 3,3!!,3!!-,3!*!+,3^         (3+((3!!-3!!)*3!)!)^3        64 3,3!!,3!!-,3!*!-,3!^        (3-((3!!-3!!)*3!)!)^3!       64 3,3!!,3!!-,3!/!+,3^         (3+((3!!-3!!)/3!)!)^3        64 3,3!!,3!!-,3!/!-,3!^        (3-((3!!-3!!)/3!)!)^3!       64 3,3!!,3!!-,3*!+,3^          (3+((3!!-3!!)*3)!)^3         64 3,3!!,3!!-,3*!-,3!^         (3-((3!!-3!!)*3)!)^3!        64 3,3!!,3!!-,3/!+,3^          (3+((3!!-3!!)/3)!)^3         64 3,3!!,3!!-,3/!-,3!^         (3-((3!!-3!!)/3)!)^3!        64 3,3!!,3!!-^,3+,3^           (3^(3!!-3!!)+3)^3            64 3,3!!,3!!/+,3!,3-^          (3+3!!/3!!)^(3!-3)           64 3,3!!,3!!/,3!!^+,3^         (3+(3!!/3!!)^3!!)^3          64 3,3!!,3!!/,3!!^-,3!^        (3-(3!!/3!!)^3!!)^3!         64 3,3!!,3!!/,3!^+,3^          (3+(3!!/3!!)^3!)^3           64 3,3!!,3!!/,3!^-,3!^         (3-(3!!/3!!)^3!)^3!          64 3,3!!,3!!/,3^+,3^           (3+(3!!/3!!)^3)^3            64 3,3!!,3!!/,3^-,3!^          (3-(3!!/3!!)^3)^3!           64 3,3!!,3!!/-,3!,3-!^         (3-3!!/3!!)^(3!-3)!          64 3,3!!,3!!/-,3,3+^           (3-3!!/3!!)^(3+3)            64 3,3!!,3!,3!!-^!+,3^         (3+3!!^(3!-3!!)!)^3          64 3,3!!,3!,3!!-^!-,3!^        (3-3!!^(3!-3!!)!)^3!         64 3,3!!,3!,3!-*!+,3^          (3+(3!!*(3!-3!))!)^3         64 3,3!!,3!,3!-*!-,3!^         (3-(3!!*(3!-3!))!)^3!        64 3,3!!,3!,3!-^+,3^           (3+3!!^(3!-3!))^3            64 3,3!!,3!,3!-^-,3!^          (3-3!!^(3!-3!))^3!           64 3,3!!,3!,3-!!-!+,3^         (3+(3!!-(3!-3)!!)!)^3        64 3,3!!,3!,3-!!-!-,3!^        (3-(3!!-(3!-3)!!)!)^3!       64 3,3!!,3!,3-!!/+,3^          (3+3!!/(3!-3)!!)^3           64 3,3!!,3!,3-!!/-,3!^         (3-3!!/(3!-3)!!)^3!          64 3,3!!,3,3!!-^!+,3^          (3+3!!^(3-3!!)!)^3           64 3,3!!,3,3!!-^!-,3!^         (3-3!!^(3-3!!)!)^3!          64 3,3!!,3,3+!-!+,3^           (3+(3!!-(3+3)!)!)^3          64 3,3!!,3,3+!-!-,3!^          (3-(3!!-(3+3)!)!)^3!         64 3,3!!,3,3+!/+,3^            (3+3!!/(3+3)!)^3             64 3,3!!,3,3+!/-,3!^           (3-3!!/(3+3)!)^3!            64 3,3!!,3,3-*!+,3^            (3+(3!!*(3-3))!)^3           64 3,3!!,3,3-*!-,3!^           (3-(3!!*(3-3))!)^3!          64 3,3!!,3,3-^+,3^             (3+3!!^(3-3))^3              64 3,3!!,3,3-^-,3!^            (3-3!!^(3-3))^3!             64 3,3!*,3!+,3!/,3^            ((3*3!+3!)/3!)^3             64 3,3!*,3!,3+/,3!^            (3*3!/(3!+3))^3!             64 3,3!*,3!-,3!/,3!^           ((3*3!-3!)/3!)^3!            64 3,3!*,3!-,3/,3^             ((3*3!-3!)/3)^3              64 3,3!*,3,3!+/,3!^            (3*3!/(3+3!))^3!             64 3,3!*,3,3*/,3!^             (3*3!/(3*3))^3!              64 3,3!*,3/,3/,3!^             ((3*3!/3)/3)^3!              64 3,3!+,3+,3!/,3!^            ((3+3!+3)/3!)^3!             64 3,3!+,3+,3/,3^              ((3+3!+3)/3)^3               64 3,3!+,3-,3/,3!^             ((3+3!-3)/3)^3!              64 3,3!,3!!,3!!-*!+,3^         (3+(3!*(3!!-3!!))!)^3        64 3,3!,3!!,3!!-*!-,3!^        (3-(3!*(3!!-3!!))!)^3!       64 3,3!,3!!,3!!-^+,3^          (3+3!^(3!!-3!!))^3           64 3,3!,3!!,3!!-^-,3!^         (3-3!^(3!!-3!!))^3!          64 3,3!,3!!/,3!!^!+,3^         (3+(3!/3!!)^3!!!)^3          64 3,3!,3!!/,3!!^!-,3!^        (3-(3!/3!!)^3!!!)^3!         64 3,3!,3!+/,3,3!-^            (3/(3!+3!))^(3-3!)           64 3,3!,3!,3!!-^!+,3^          (3+3!^(3!-3!!)!)^3           64 3,3!,3!,3!!-^!-,3!^         (3-3!^(3!-3!!)!)^3!          64 3,3!,3!,3!-*!+,3^           (3+(3!*(3!-3!))!)^3          64 3,3!,3!,3!-*!-,3!^          (3-(3!*(3!-3!))!)^3!         64 3,3!,3!,3!-^+,3^            (3+3!^(3!-3!))^3             64 3,3!,3!,3!-^-,3!^           (3-3!^(3!-3!))^3!            64 3,3!,3!,3-!-!+,3^           (3+(3!-(3!-3)!)!)^3          64 3,3!,3!,3-!-!-,3!^          (3-(3!-(3!-3)!)!)^3!         64 3,3!,3!,3-!/+,3^            (3+3!/(3!-3)!)^3             64 3,3!,3!,3-!/-,3!^           (3-3!/(3!-3)!)^3!            64 3,3!,3!-!+,3!,3-^           (3+(3!-3!)!)^(3!-3)          64 3,3!,3!-!,3!!^+,3^          (3+(3!-3!)!^3!!)^3           64 3,3!,3!-!,3!!^-,3!^         (3-(3!-3!)!^3!!)^3!          64 3,3!,3!-!,3!^+,3^           (3+(3!-3!)!^3!)^3            64 3,3!,3!-!,3!^-,3!^          (3-(3!-3!)!^3!)^3!           64 3,3!,3!-!,3^+,3^            (3+(3!-3!)!^3)^3             64 3,3!,3!-!,3^-,3!^           (3-(3!-3!)!^3)^3!            64 3,3!,3!-!-,3!,3-!^          (3-(3!-3!)!)^(3!-3)!         64 3,3!,3!-!-,3,3+^            (3-(3!-3!)!)^(3+3)           64 3,3!,3!-,3!!*!+,3^          (3+((3!-3!)*3!!)!)^3         64 3,3!,3!-,3!!*!-,3!^         (3-((3!-3!)*3!!)!)^3!        64 3,3!,3!-,3!!/!+,3^          (3+((3!-3!)/3!!)!)^3         64 3,3!,3!-,3!!/!-,3!^         (3-((3!-3!)/3!!)!)^3!        64 3,3!,3!-,3!*!+,3^           (3+((3!-3!)*3!)!)^3          64 3,3!,3!-,3!*!-,3!^          (3-((3!-3!)*3!)!)^3!         64 3,3!,3!-,3!/!+,3^           (3+((3!-3!)/3!)!)^3          64 3,3!,3!-,3!/!-,3!^          (3-((3!-3!)/3!)!)^3!         64 3,3!,3!-,3*!+,3^            (3+((3!-3!)*3)!)^3           64 3,3!,3!-,3*!-,3!^           (3-((3!-3!)*3)!)^3!          64 3,3!,3!-,3/!+,3^            (3+((3!-3!)/3)!)^3           64 3,3!,3!-,3/!-,3!^           (3-((3!-3!)/3)!)^3!          64 3,3!,3!-^,3+,3^             (3^(3!-3!)+3)^3              64 3,3!,3!/+,3!,3-^            (3+3!/3!)^(3!-3)             64 3,3!,3!/,3!!^+,3^           (3+(3!/3!)^3!!)^3            64 3,3!,3!/,3!!^-,3!^          (3-(3!/3!)^3!!)^3!           64 3,3!,3!/,3!^+,3^            (3+(3!/3!)^3!)^3             64 3,3!,3!/,3!^-,3!^           (3-(3!/3!)^3!)^3!            64 3,3!,3!/,3^+,3^             (3+(3!/3!)^3)^3              64 3,3!,3!/,3^-,3!^            (3-(3!/3!)^3)^3!             64 3,3!,3!/-,3!,3-!^           (3-3!/3!)^(3!-3)!            64 3,3!,3!/-,3,3+^             (3-3!/3!)^(3+3)              64 3,3!,3++,3!/,3!^            ((3+3!+3)/3!)^3!             64 3,3!,3++,3/,3^              ((3+3!+3)/3)^3               64 3,3!,3+,3!//,3!^            (3/((3!+3)/3!))^3!           64 3,3!,3+/,3!*,3!^            ((3/(3!+3))*3!)^3!           64 3,3!,3,3!!-^!+,3^           (3+3!^(3-3!!)!)^3            64 3,3!,3,3!!-^!-,3!^          (3-3!^(3-3!!)!)^3!           64 3,3!,3,3+-!+,3^             (3+(3!-(3+3))!)^3            64 3,3!,3,3+-!-,3!^            (3-(3!-(3+3))!)^3!           64 3,3!,3,3+/+,3^              (3+3!/(3+3))^3               64 3,3!,3,3+/-,3!^             (3-3!/(3+3))^3!              64 3,3!,3,3-*!+,3^             (3+(3!*(3-3))!)^3            64 3,3!,3,3-*!-,3!^            (3-(3!*(3-3))!)^3!           64 3,3!,3,3-^+,3^              (3+3!^(3-3))^3               64 3,3!,3,3-^-,3!^             (3-3!^(3-3))^3!              64 3,3!,3-!!,3!!-!+,3^         (3+((3!-3)!!-3!!)!)^3        64 3,3!,3-!!,3!!-!-,3!^        (3-((3!-3)!!-3!!)!)^3!       64 3,3!,3-!!,3!!/+,3^          (3+(3!-3)!!/3!!)^3           64 3,3!,3-!!,3!!/-,3!^         (3-(3!-3)!!/3!!)^3!          64 3,3!,3-!,3!-!+,3^           (3+((3!-3)!-3!)!)^3          64 3,3!,3-!,3!-!-,3!^          (3-((3!-3)!-3!)!)^3!         64 3,3!,3-!,3!/+,3^            (3+(3!-3)!/3!)^3             64 3,3!,3-!,3!/-,3!^           (3-(3!-3)!/3!)^3!            64 3,3!,3-+,3/,3!^             ((3+3!-3)/3)^3!              64 3,3!,3-,3-!+,3^             (3+(3!-3-3)!)^3              64 3,3!,3-,3-!-,3!^            (3-(3!-3-3)!)^3!             64 3,3!,3-,3/+,3^              (3+(3!-3)/3)^3               64 3,3!,3-,3/-,3!^             (3-(3!-3)/3)^3!              64 3,3!,3-/,3+,3^              (3/(3!-3)+3)^3               64 3,3!,3/*,3/,3!^             ((3*3!/3)/3)^3!              64 3,3!,3/+,3-,3!^             (3+3!/3-3)^3!     64 3,3!,3/,3!^*,3/             3*(3!/3)^3!/3     64 3,3!,3/,3!^+,3-             3+(3!/3)^3!-3     64 3,3!,3/,3!^,3-+             3+(3!/3)^3!-3     64 3,3!,3/,3!^,3/*             3*(3!/3)^3!/3     64 3,3!,3/,3-+,3!^             (3+3!/3-3)^3!     64 3,3!,3/,3--,3^              (3-(3!/3-3))^3               64 3,3!,3/-,3+,3^              (3-3!/3+3)^3                 64 3,3!-,3/,3+,3!^             ((3-3!)/3+3)^3!              64 3,3!/,3!!,3!!,3!+-^         (3/3!)^(3!!-(3!!+3!))        64 3,3!/,3!!,3!!-,3!-^         (3/3!)^(3!!-3!!-3!)          64 3,3!/,3!!,3!,3!!+-^         (3/3!)^(3!!-(3!+3!!))        64 3,3!/,3!!,3!-,3!!-^         (3/3!)^(3!!-3!-3!!)          64 3,3!/,3!,3!,3!+-^           (3/3!)^(3!-(3!+3!))          64 3,3!/,3!,3!-,3!-^           (3/3!)^(3!-3!-3!)            64 3,3!/,3,3!,3+-^             (3/3!)^(3-(3!+3))            64 3,3!/,3,3!-,3-^             (3/3!)^(3-3!-3)              64 3,3!/,3,3,3!+-^         &nbs

Posted by Charlie
on 2008-07-07 15:51:06

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