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
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
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Posted by Charlie
on 2008-07-07 15:51:06 |
still not all, even with exponentiation (spoilers present)
