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(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
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Posted by Charlie
on 2008-07-09 11:40:10 |
re(6): all hundred
