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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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 the shortest representation of each | Comment 25 of 35 |

The following are the shortest representations of each in the algebraic column. Remember that x!! is to be interpreted as (x!)! and that ! binds closer than ^.

(whoops; I see the conversion from RPN to Alg in some instances did not put needed parentheses around raising to a power before the factorial. But you can catch the drift and see where needed).

`3!,3!*,33,3/+                3!*3!+33/3                     47 33,3/,3!,3!*+                33/3+3!*3!                     47 `
`3!,3/,3^!,3!!/,3!-           (3!/3)^3!/3!!-3!               50 `
`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                   64 3!,3,3/,3-+,3^               (3!+3/3-3)^3                   64 3!,3-,3,3/+,3^               (3!-3+3/3)^3                   64 3!,33*,3!-,3/                (3!*33-3!)/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 33,3!*,3!-,3/                (33*3!-3!)/3                   64 `
`3!,33*,3-,3/                 (3!*33-3)/3                    65 33,3!*,3-,3/                 (33*3!-3)/3                    65 `
`3!,3!,33*+,3/                (3!+3!*33)/3                   68 3!,33*,3!+,3/                (3!*33+3!)/3                   68 3!,33,3!*+,3/                (3!+33*3!)/3                   68 3!,3^,3-,3/,3-               (3!^3-3)/3-3                   68 33,3!*,3!+,3/                (33*3!+3!)/3                   68 `
`3!,3^,3/,3!,3/-              3!^3/3-3!/3                    70 `
`3!,3^,3/,3,3/-               3!^3/3-3/3                     71 `
`3!,3^,3/,3,3/+               3!^3/3+3/3                     73 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/,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                    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                 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!/,33-              (3!+3!!)/3!-33                 88 `
`3!!,3/,33+,3/                (3!!/3+33)/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!               95 3!,3^,3!,3!!+,3!/-           3!^3-(3!+3!!)/3!               95 `

Edited on July 8, 2008, 12:31 pm
 Posted by Charlie on 2008-07-08 12:15:01

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