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(2): all the asked-for numbers (spoilers) by Charlie)
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!^ (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
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Posted by Charlie
on 2008-07-08 11:46:47 |
re(3): all the asked-for numbers (spoilers)
