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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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 re(6): all hundred | Comment 33 of 35 |
(In reply to re(5): all hundred by Charlie)

`3!!,3!!,3-,3-,3!/-           3!!-(3!!-3-3)/3!               601 3!!,3!,3/,3!!,3!/-+          3!!+3!/3-3!!/3!                602 33,3!*,3+,3*                 (33*3!+3)*3                    603 3!!,3!,3!!-,3!/+,3+          3!!+(3!-3!!)/3!+3              604 3!,3!,3!!+,3!/,3!!--         3!-((3!+3!!)/3!-3!!)           605 3,3!^,3!!,3!/,3+-            3^3!-(3!!/3!+3)                606 3!!,3!+,3!,3!!-,3!/+         3!!+3!+(3!-3!!)/3!             607 3,3!^,3!!,3!+,3!/-           3^3!-(3!!+3!)/3!               608 3!!,333,3/-                  3!!-333/3                      609 3,3!^,3!,3!!-,3!/+           3^3!+(3!-3!!)/3!               610 33,3/,3^,3!!-                (33/3)^3-3!!                   611 3!,3,33*,3+*                 3!*(3*33+3)                    612 3,3!,3^*,33-                 3*3!^3-33                      615 3!!,3,3,33*+-                3!!-(3+3*33)                   618 3!!,3!!,3!/,3!!-,3!/+        3!!+(3!!/3!-3!!)/3!            620 3,3+!,3,33*-                 (3+3)!-3*33                    621 3,3,33*,3!!--                3-(3*33-3!!)                   624 3!,3,33*-,3!!+               3!-3*33+3!!                    627 3,3!^,3,33*-                 3^3!-3*33                      630 3,3!,3^,3!-*,3+              3*(3!^3-3!)+3                  633 3!!,3!!,3/,3/-,3!-           3!!-(3!!/3)/3-3!               634 3!,3^,3-,3*,3-               (3!^3-3)*3-3                   636 3!!,3!!,3,3*/-,3-            3!!-3!!/(3*3)-3                637 3!!,3!!,3/,3!+,3/-           3!!-(3!!/3+3!)/3               638 3,3,3+,3^,3-*                3*((3+3)^3-3)                  639 3,3+!,3!!,3/,3/-             (3+3)!-(3!!/3)/3               640 3!!,3,3!!,3/-,3/+            3!!+(3-3!!/3)/3                641 3,3,3+,3^*,3!-               3*(3+3)^3-3!                   642 3,3!!+,3!!,3/,3/-            3+3!!-(3!!/3)/3                643 3,3+,3^,3*,3-                (3+3)^3*3-3                    645 3,3!,3^*,3!,3/-              3*3!^3-3!/3                    646 3!,3^,3*,3,3/-               3!^3*3-3/3                     647 3!,3,3,33+**                 3!*3*(3+33)                    648 3,3!,3^*,3,3/+               3*3!^3+3/3                     649 3!,3^,3*,3!,3/+              3!^3*3+3!/3                    650 3,3,3,3+,3^*+                3+3*(3+3)^3                    651 3!!,3,3!,3/,3!^+-            3!!-(3+(3!/3)^3!)              653 3!!,33,33+-                  3!!-(33+33)                    654 3!!,3,3/,3+,3^-              3!!-(3/3+3)^3                  656 3,3,3+,3^,3+*                3*((3+3)^3+3)                  657 3!!,3!,3/,3!^,3--            3!!-((3!/3)^3!-3)              659 3!!,33,3,3^+-                3!!-(33+3^3)                   660 3!!,3!,3!,3/,3!^-+           3!!+3!-(3!/3)^3!               662 3!,3^,3!+,3*,3-              (3!^3+3!)*3-3                  663 3!!,3!,3/,3^!,3!!/-          3!!-(3!/3)^3!/3!!              664 3,3!^,3!,3/,3!^-             3^3!-(3!/3)^3!                 665 3!,333,3/*                   3!*333/3                       666 3!!,33-,3,3!*-               3!!-33-3*3!                    669 3,3!,3^,3!+*,3!+             3*(3!^3+3!)+3!                 672 3!!,3!,3!!,3/,3!/+-          3!!-(3!+(3!!/3)/3!)            674 3!,3^,3,3*+,3*               (3!^3+3*3)*3                   675 3!!,3,3!!,3!,3*/+-           3!!-(3+3!!/(3!*3))             677 3!!,3,3*,33+-                3!!-(3*3+33)                   678 3!!,3,3!!,3!/+,3/-           3!!-(3+3!!/3!)/3               679 3!!,3!,3/,3+!,3/-            3!!-(3!/3+3)!/3                680 33,3!,3^,3*+                 33+3!^3*3                      681 3!!,3!,3/-,3!,3!*-           3!!-3!/3-3!*3!                 682 3!!,3!,3!*-,3,3/-            3!!-3!*3!-3/3                  683 3,3+!,3,33+-                 (3+3)!-(3+33)                  684 3!!,33,3!,3/+-               3!!-(33+3!/3)                  685 3!!,33-,3,3/-                3!!-33-3/3                     686 3,3*,3-!,33-                 (3*3-3)!-33                    687 3,3/,33,3!!--                3/3-(33-3!!)                   688 3!!,3!,3/,33-+               3!!+3!/3-33                    689 3,33,3,3+!--                 3-(33-(3+3)!)                  690 3!!,3,3^,3!,3/+-             3!!-(3^3+3!/3)                 691 3!!,3,3^-,3,3/-              3!!-3^3-3/3                    692 33,3!,3*,3+*                 33*(3!*3+3)                    693 3!!,3,3^-,3,3/+              3!!-3^3+3/3                    694 3!!,3,3^,3!,3/--             3!!-(3^3-3!/3)                 695 3,3*,3^,33-                  (3*3)^3-33                     696 3!!,3!!,3!/,3!/,3+-          3!!-((3!!/3!)/3!+3)            697 3!!,3!!,3!+,33/-             3!!-(3!!+3!)/33                698 3,33-,3,3!^+                 3-33+3^3!                      699 3!!,3!!,33,3+/-              3!!-3!!/(33+3)                 700 3!!,3,3/,3,3!*+-             3!!-(3/3+3*3!)                 701 3,3!,3!,33+**                3*3!*(3!+33)                   702 3!!,3!-,33,3/-               3!!-3!-33/3                    703 3!,3/,3,3!*-,3!!+            3!/3-3*3!+3!!                  704 3,3,3^-,3,3!^+               3-3^3+3^3!                     705 3!!,33,3/-,3-                3!!-33/3-3                     706 3!!,3!,33+,3/-               3!!-(3!+33)/3                  707 3!!,33,3+,3/-                3!!-(33+3)/3                   708 3,3+!,33,3/-                 (3+3)!-33/3                    709 3!!,3,33-,3/+                3!!+(3-33)/3                   710 3,3+!,3,3+,3+-               (3+3)!-(3+3+3)                 711 3!!,33,3/,3--                3!!-(33/3-3)                   712 3!!,3,3,3,3/++-              3!!-(3+3+3/3)                  713 3!!,3,3^,33-+                3!!+3^3-33                     714 3!!,3!+,33,3/-               3!!+3!-33/3                    715 3,3+!,3,3/-,3-               (3+3)!-3/3-3                   716 33,3,3^-!,3-                 (33-3^3)!-3                    717 3,3!^,33,3/-                 3^3!-33/3                      718 3!!,33,33/-                  3!!-33/33                      719 33,3!!,33-+                  33+3!!-33                      720 33,33/,3!!+                  33/33+3!!                      721 3,3+!,3+,3,3/-               (3+3)!+3-3/3                   722 33,3,3^-!,3+                 (33-3^3)!+3                    723 3,3/,3,3,3+!++               3/3+3+(3+3)!                   724 3!!,3!-,33,3/+               3!!-3!+33/3                    725 33,3!!+,3,3^-                33+3!!-3^3                     726 3,3*,3^,3!,3/-               (3*3)^3-3!/3                   727 3,3,3+^,3,3/-                3^(3+3)-3/3                    728 3,33,3,3^-^                  3^(33-3^3)                     729 3,3*,3^,3,3/+                (3*3)^3+3/3                    730 33,3/,3,3+!+                 33/3+(3+3)!                    731 3,3,3*,3-^,3+                3^(3*3-3)+3                    732 3,3,3/+,3,3!^+               3+3/3+3^3!                     733 3,33,3/+,3!!+                3+33/3+3!!                     734 3,3,3+^,3+,3+                3^(3+3)+3+3                    735 3,3!^,3!,3,3/++              3^3!+3!+3/3                    736 3!,33,3/,3!!++               3!+33/3+3!!                    737 3,3*,3^,3,3*+                (3*3)^3+3*3                    738 3!,3*,3!!+,3,3/+             3!*3+3!!+3/3                   739 3,3!^,33,3/+                 3^3!+33/3                      740 3,3,3*,3,3!^++               3+3*3+3^3!                     741 3!!,3!!,3!+,33/+             3!!+(3!!+3!)/33                742 3,3!!,3!!,3!/,3!/++          3+3!!+(3!!/3!)/3!              743 3!!,3,3*-,33+                3!!-3*3+33                     744 3,3^,3!!+,3!,3/-             3^3+3!!-3!/3                   745 3,3^,3,3/,3!!--              3^3-(3/3-3!!)                  746 3,33,3!,3^+*                 3*(33+3!^3)                    747 3,3^,3!!,3,3/++              3^3+3!!+3/3                    748 3!,3/,3!!+,3,3^+             3!/3+3!!+3^3                   749 33,3,3,3+!--                 33-(3-(3+3)!)                  750 3!!,33+,3!,3/-               3!!+33-3!/3                    751 33,3,3/,3!!--                33-(3/3-3!!)                   752 33,3,3*,3-!+                 33+(3*3-3)!                    753 33,3,3/+,3!!+                33+3/3+3!!                     754 3!,3/,3!!+,33+               3!/3+3!!+33                    755 33,3+,3,3+!+                 33+3+(3+3)!                    756 3,3/,3!,3!*,3!!++            3/3+3!*3!+3!!                  757 3!,3!*,3!!+,3!,3/+           3!*3!+3!!+3!/3                 758 33,3,3,3!^--                 33-(3-3^3!)                    759 3,3+!,3,3!*/,3!!+            (3+3)!/(3*3!)+3!!              760 3!!,3,3!!,3!/+,3/+           3!!+(3+3!!/3!)/3               761 33,3,3,3+^+                  33+3^(3+3)                     762 3!!,3!,3*/,3,3!!++           3!!/(3!*3)+3+3!!               763 33,3+,3,3!^+                 33+3+3^3!                      765 3!!,3!!,3/,3!/+,3!+          3!!+(3!!/3)/3!+3!              766 3,3!^,33+,3!+                3^3!+33+3!                     768 3!!,3!,3*/,3,3!^+            3!!/(3!*3)+3^3!                769 33,3!!+,3,3!*+               33+3!!+3*3!                    771 3,3,3+*,3*,3!!+              3*(3+3)*3+3!!                  774 3!!,3!,3/,3^!,3!!/+          3!!+(3!/3)^3!/3!!              776 3,3,3,3!**+,3!!+             3+3*3*3!+3!!                   777 3!,3/,3!^,3!,3!!--           (3!/3)^3!-(3!-3!!)             778 3,3^,3!!+,33+                3^3+3!!+33                     780 3!!,3!,3/,3!^+,3-            3!!+(3!/3)^3!-3                781 3,3,3!**,3,3!^+              3*3*3!+3^3!                    783 3!!,3,3/,3+,3^+              3!!+(3/3+3)^3                  784 33,3!!+,33+                  33+3!!+33                      786 3!!,3,3!,3/,3!^++            3!!+3+(3!/3)^3!                787 33,3!!+,3!,3!*+              33+3!!+3!*3!                   789 3!!,3!,3^,3!-,3/+            3!!+(3!^3-3!)/3                790 3!!,3!,3^,3-,3/+             3!!+(3!^3-3)/3                 791 3,3^,3-,33*                  (3^3-3)*33                     792 3!!,3!,3^,3+,3/+             3!!+(3!^3+3)/3                 793 3!!,3!,3!,3^+,3/+            3!!+(3!+3!^3)/3                794 3!!,3,3^,3*+,3!-             3!!+3^3*3-3!                   795 3!!,3/,3/,3!!,3-+            (3!!/3)/3+3!!-3                797 3,3,3^*,3!!,3-+              3*3^3+3!!-3                    798 3!!,3!!,3/,3-,3/+            3!!+(3!!/3-3)/3                799 3,3+!,3!!,3/,3/+             (3+3)!+(3!!/3)/3               800 3!!,3,33,3!-*+               3!!+3*(33-3!)                  801 3!!,3/,3!+,3/,3!!+           (3!!/3+3!)/3+3!!               802 3!!,3,3*/,3,3!!++            3!!/(3*3)+3+3!!                803 3,3,3^*,3+,3!!+              3*3^3+3+3!!                    804 3!,3!!,3/,3/+,3!!+           3!+(3!!/3)/3+3!!               806 3!!,3!+,3,3^,3*+             3!!+3!+3^3*3                   807 3!!,3/,3/,3,3!^+             (3!!/3)/3+3^3!                 809 33,3-,3,3^*                  (33-3)*3^3                     810 3!!,3!,3,33*--               3!!-(3!-3*33)                  813 33,3*,3,3!!--                33*3-(3-3!!)                   816 3,33*,3,3+!+                 3*33+(3+3)!                    819 3!!,3!!,3!/-,3!/,3!!+        (3!!-3!!/3!)/3!+3!!            820 3,3!!,3,33*++                3+3!!+3*33                     822 3!!,33,3*,3!++               3!!+33*3+3!                    825 3,3!^,33,3*+                 3^3!+33*3                      828 333,3/,3!!+                  333/3+3!!                      831 3!!,3!,3!,3!!-,3!/+-         3!!-(3!+(3!-3!!)/3!)           833 3!,3,3/+!,3!/,3!-            (3!+3/3)!/3!-3!                834 3!,3!!+,3!/,3!-,3!!+         (3!+3!!)/3!-3!+3!!             835 3!!,3!,3!!-,3!/-,3-          3!!-(3!-3!!)/3!-3              836 33,3!+,3*,3!!+               (33+3!)*3+3!!                  837 3!!,3!/,3!,3/-,3!!+          3!!/3!-3!/3+3!!                838 3!,3,3/+!,3!-,3!/            ((3!+3/3)!-3!)/3!              839 33,3-,3!/!,3!!+              ((33-3)/3!)!+3!!               840 3!,3,3/,3!+!+,3!/            (3!+(3/3+3!)!)/3!              841 3!!,3,3,3!^+,3!/+            3!!+(3+3^3!)/3!                842 3,3!,3,3/+!,3!/+             3+(3!+3/3)!/3!                 843 3,3!,3!!+,3!/,3!!++          3+(3!+3!!)/3!+3!!              844 3!!,3!,3/,3+,3^+             3!!+(3!/3+3)^3                 845 3!!,3!,3!,3*,3+*+            3!!+3!*(3!*3+3)                846 3!!,3!,3!!+,3!/,3!++         3!!+(3!+3!!)/3!+3!             847 3!!,3!-,3!/,3,3!^+           (3!!-3!)/3!+3^3!               848 3,3!^,3,3+!,3!/+             3^3!+(3+3)!/3!                 849 3!,3!!+,3!/,3,3!^+           (3!+3!!)/3!+3^3!               850 3!!,3!/,3+,3,3!^+            3!!/3!+3+3^3!                  852 3!,3!!,3!/+,3,3!^+           3!+3!!/3!+3^3!                 855 3!!,3!,3*,3!!,3!/++          3!!+3!*3+3!!/3!                858 3!!,3!!,3!!,3!/+,3!/+        3!!+(3!!+3!!/3!)/3!            860 3,3,3/+,3!,3^*               (3+3/3)*3!^3                   864 3,3^,3!!+,3!!,3!/+           3^3+3!!+3!!/3!                 867 33,3!!+,3!!,3!/+             33+3!!+3!!/3!                  873 3!!,3!,3,3^*+,3!-            3!!+3!*3^3-3!                  876 3!,3,3^*,3-,3!!+             3!*3^3-3+3!!                   879 3!!,3,3^,33//                3!!/(3^3/33)                   880 3!!,3!,33,3!-*+              3!!+3!*(33-3!)                 882 3,3^,33*,3!-                 3^3*33-3!                      885 3,3^,33*,3-                  3^3*33-3                       888 33,33,3!-*                   33*(33-3!)                     891 33,3,3^*,3+                  33*3^3+3                       894 33,3,3^*,3!+                 33*3^3+3!                      897 33,3-,3!,3/^                 (33-3)^(3!/3)                  900 3!!,3!,3^,33-+               3!!+3!^3-33                    903 3!,3^,3!!,3,3^-+             3!^3+3!!-3^3                   909 3!!,3!-,3!,33*+              3!!-3!+3!*33                   912 3!!,3,3!,33*--               3!!-(3-3!*33)                  915 3!!,33,3,3+*+                3!!+33*(3+3)                   918 3!!,3!!,3!!,3!/-,3/+         3!!+(3!!-3!!/3!)/3             920 3,3!!,3!,33*++               3+3!!+3!*33                    921 3!,3!+!,3!!/,3!!-,3!!/       ((3!+3!)!/3!!-3!!)/3!!         923 3!,33*,3!!,3!++              3!*33+3!!+3!                   924 3!!,3!,3!+!,3!!/+,3!!/       (3!!+(3!+3!)!/3!!)/3!!         925 3,3!^,33,3!*+                3^3!+33*3!                     927 3!,3^,3,3!!,3---             3!^3-(3-(3!!-3))               930 3,3+,3^,3!!+,3-              (3+3)^3+3!!-3                  933 3!!,3!,3/,3!,3^--            3!!-(3!/3-3!^3)                934 3!!,3,3/-,3!,3^+             3!!-3/3+3!^3                   935 33,3+,3!*,3!!+               (33+3)*3!+3!!                  936 3,3/,3!!+,3!,3^+             3/3+3!!+3!^3                   937 3!,3^,3!,3/+,3!!+            3!^3+3!/3+3!!                  938 3,3+!,3!,3^,3++              (3+3)!+3!^3+3                  939 3,3!^,3-,3!,3^+              3^3!-3+3!^3                    942 3,3,3+^,3!,3^+               3^(3+3)+3!^3                   945 3,3!,3^+,3,3!^+              3+3!^3+3^3!                    948 3!!,3!!,33-,3/+              3!!+(3!!-33)/3                 949 3!,3!,3^,3,3!^++             3!+3!^3+3^3!                   951 3!!,3!-,3/,3!!,3!-+          (3!!-3!)/3+3!!-3!              952 3!!,3!,3!!,3-,3/--           3!!-(3!-(3!!-3)/3)             953 3!!,33,3!+,3!*+              3!!+(33+3!)*3!                 954 3!!,3!-,3/,3,3!!--           (3!!-3!)/3-(3-3!!)             955 3!!,3!!,3-,3/,3-+            3!!+(3!!-3)/3-3                956 3!!,3/,3,3+!+,3-             3!!/3+(3+3)!-3                 957 3!!,3!!,3-,3-,3/+            3!!+(3!!-3-3)/3                958 3,3+!,3!!,3-,3/+             (3+3)!+(3!!-3)/3               959 3!!,3,3,3/+*,3/              3!!*(3+3/3)/3                  960 3,3+!,3,3!!+,3/+             (3+3)!+(3+3!!)/3               961 3!!,3,3,3!^-,3/-             3!!-(3-3^3!)/3                 962 3,3*,3,3^*,3!!+              3*3*3^3+3!!                    963 3,3!^,3+,3/,3!!+             (3^3!+3)/3+3!!                 964 3!!,3,3!^,3!+,3/+            3!!+(3^3!+3!)/3                965 3!,3,3^*,3!*,3!-             3!*3^3*3!-3!                   966 3!!,3!-,3/,3,3!^+            (3!!-3!)/3+3^3!                967 3!!,3-,3/,3,3!^+             (3!!-3)/3+3^3!                 968 33,3!!+,3!,3^+               33+3!!+3!^3                    969 3!!,3+,3/,3,3!^+             (3!!+3)/3+3^3!                 970 33,3!!+,3/,3!!+              (33+3!!)/3+3!!                 971 3,3^,33,3+*                  3^3*(33+3)                     972 3,3!*,3^,3!+,3!/             ((3*3!)^3+3!)/3!               973 3,3,3^,3!*,3!*+              3+3^3*3!*3!                    975 3!,3,3^*,3!*,3!+             3!*3^3*3!+3!                   978 333,3!-,3*                   (333-3!)*3                     981 3!!,3/,3!!,3,3^++            3!!/3+3!!+3^3                  987 33,33,3-*                    33*(33-3)                      990 3,333*,3!-                   3*333-3!                       993 3,333*,3-                    3*333-3                        996 333,3!,3-*                   333*(3!-3)                     999 `

 Posted by Charlie on 2008-07-09 11:40:10

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