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Math Help - use only four 4's to make 10

  1. #16
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Chop Suey View Post
    What he meant that this doesn't fit the requirements of four 4's. There will be a \frac{1}{2}, should it be written as an exponent.
    Yes, but at this level we can probably ignore this fact.

    -Dan
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  2. #17
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    Quote Originally Posted by Kai View Post
    Ermm, can 44 really be counted as 2 fours ??, i mean theres no operation relating 44 and 2 fours, not really sure about it
    (shrugs) I count four of them. It looks good to me.

    -Dan
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  3. #18
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    If A = \sqrt{4} = 2, B = 4, C = 4, and D = 4, so that we can tell one 4 from another, we have 186 ways:

    D+(C+(B-A)) (D+(B-A))+C (C+B)-(A-D) D+(C+(B/A)) Ax(D+(B/C))
    C+(D+(B-A)) (B+(D-A))+C (B+C)-(A-D) C+(D+(B/A)) Ax(B+(D/C))
    D+(B+(C-A)) (C+(B-A))+D (D-(A-C))+B D+(B+(C/A)) Ax(C+(B/D))
    B+(D+(C-A)) (B+(C-A))+D (C-(A-D))+B B+(D+(C/A)) Ax(B+(C/D))
    C+(B+(D-A)) ((D+C)+B)-A (D-(A-B))+C C+(B+(D/A)) Ax((D/C)+B)
    B+(C+(D-A)) ((C+D)+B)-A (B-(A-D))+C B+(C+(D/A)) Ax((C/D)+B)
    (D+C)+(B-A) ((D+B)+C)-A (C-(A-B))+D (D+C)+(B/A) Ax((D/B)+C)
    (C+D)+(B-A) ((B+D)+C)-A (B-(A-C))+D (C+D)+(B/A) Ax((B/D)+C)
    (D+B)+(C-A) ((C+B)+D)-A D-((A-C)-B) (D+B)+(C/A) Ax((C/B)+D)
    (B+D)+(C-A) ((B+C)+D)-A C-((A-D)-B) (B+D)+(C/A) Ax((B/C)+D)
    (C+B)+(D-A) ((D+C)-A)+B D-((A-B)-C) (C+B)+(D/A) (D+(C/B))xA
    (B+C)+(D-A) ((C+D)-A)+B B-((A-D)-C) (B+C)+(D/A) (C+(D/B))xA
    (D-A)+(C+B) ((D-A)+C)+B C-((A-B)-D) (D/A)+(C+B) (D+(B/C))xA
    (C-A)+(D+B) ((C-A)+D)+B B-((A-C)-D) (C/A)+(D+B) (B+(D/C))xA
    (D-A)+(B+C) ((D+B)-A)+C (DxC)-(B+A) (D/A)+(B+C) (C+(B/D))xA
    (B-A)+(D+C) ((B+D)-A)+C (CxD)-(B+A) (B/A)+(D+C) (B+(C/D))xA
    (C-A)+(B+D) ((D-A)+B)+C (DxB)-(C+A) (C/A)+(B+D) (D+(CxB))/A
    (B-A)+(C+D) ((B-A)+D)+C (BxD)-(C+A) (B/A)+(C+D) (C+(DxB))/A
    D+((C+B)-A) ((C+B)-A)+D (CxB)-(D+A) D+((C/A)+B) (D+(BxC))/A
    C+((D+B)-A) ((B+C)-A)+D (BxC)-(D+A) C+((D/A)+B) (B+(DxC))/A
    D+((B+C)-A) ((C-A)+B)+D (DxC)-(A+B) D+((B/A)+C) (C+(BxD))/A
    B+((D+C)-A) ((B-A)+C)+D (CxD)-(A+B) B+((D/A)+C) (B+(CxD))/A
    C+((B+D)-A) D-(A-(C+B)) (DxB)-(A+C) C+((B/A)+D) ((D/C)+B)xA
    B+((C+D)-A) C-(A-(D+B)) (BxD)-(A+C) B+((C/A)+D) ((C/D)+B)xA
    D+((C-A)+B) D+(C-(A-B)) (CxB)-(A+D) (D+(C/A))+B ((D/B)+C)xA
    C+((D-A)+B) C+(D-(A-B)) (BxC)-(A+D) (C+(D/A))+B ((B/D)+C)xA
    D+((B-A)+C) D-(A-(B+C)) ((DxC)-B)-A (D+(B/A))+C ((C/B)+D)xA
    B+((D-A)+C) B-(A-(D+C)) ((CxD)-B)-A (B+(D/A))+C ((B/C)+D)xA
    C+((B-A)+D) D+(B-(A-C)) ((DxB)-C)-A (C+(B/A))+D ((DxC)+B)/A
    B+((C-A)+D) B+(D-(A-C)) ((BxD)-C)-A (B+(C/A))+D ((CxD)+B)/A
    (D+(C+B))-A C-(A-(B+D)) ((CxB)-D)-A ((D/A)+C)+B ((DxB)+C)/A
    (C+(D+B))-A B-(A-(C+D)) ((BxC)-D)-A ((C/A)+D)+B ((BxD)+C)/A
    (D+(B+C))-A C+(B-(A-D)) ((DxC)-A)-B ((D/A)+B)+C ((CxB)+D)/A
    (B+(D+C))-A B+(C-(A-D)) ((CxD)-A)-B ((B/A)+D)+C ((BxC)+D)/A
    (C+(B+D))-A (D+C)-(A-B) ((DxB)-A)-C ((C/A)+B)+D
    (B+(C+D))-A (C+D)-(A-B) ((BxD)-A)-C ((B/A)+C)+D
    (D+(C-A))+B (D+B)-(A-C) ((CxB)-A)-D Ax(D+(C/B))
    (C+(D-A))+B (B+D)-(A-C) ((BxC)-A)-D Ax(C+(D/B))

    Add B = \sqrt{4} = 2 and there are 152 more ways.

    B+(Dx(C-A)) A+((C-B)xD) ((CxA)-B)+D Dx(A+(B/C)) (Dx(C/B))+A
    B+(Cx(D-A)) (Dx(C-B))+A ((AxC)-B)+D Cx(B+(A/D)) (Cx(D/B))+A
    A+(Dx(C-B)) (Cx(D-B))+A D-(B-(CxA)) Cx(A+(B/D)) (Dx(C/A))+B
    A+(Cx(D-B)) (D+(CxB))-A C-(B-(DxA)) (D/B)+(CxA) (Cx(D/A))+B
    (DxB)+(C-A) (C+(DxB))-A D-(A-(CxB)) (C/B)+(DxA) (B+(A/D))xC
    (BxD)+(C-A) (D+(BxC))-A C-(A-(DxB)) (DxB)+(C/A) (A+(B/D))xC
    (CxB)+(D-A) (C+(BxD))-A B-(Dx(A-C)) (BxD)+(C/A) (B+(A/C))xD
    (BxC)+(D-A) (Dx(C-A))+B A-(Dx(B-C)) (CxB)+(D/A) (A+(B/C))xD
    (D-B)+(CxA) (Cx(D-A))+B D-(B-(AxC)) (BxC)+(D/A) ((DxC)/B)+A
    (C-B)+(DxA) (D+(CxA))-B D-(A-(BxC)) (D/A)+(CxB) ((CxD)/B)+A
    (DxA)+(C-B) (C+(DxA))-B B-(Cx(A-D)) (C/A)+(DxB) ((D/B)xC)+A
    (AxD)+(C-B) (D+(AxC))-B A-(Cx(B-D)) (DxA)+(C/B) ((C/B)xD)+A
    (CxA)+(D-B) (C+(AxD))-B C-(B-(AxD)) (AxD)+(C/B) ((DxC)/A)+B
    (AxC)+(D-B) ((D-B)xC)+A C-(A-(BxD)) (CxA)+(D/B) ((CxD)/A)+B
    (D-A)+(CxB) ((C-B)xD)+A (DxB)-(A-C) (AxC)+(D/B) ((D/A)xC)+B
    (C-A)+(DxB) ((DxB)+C)-A (BxD)-(A-C) (D/B)+(AxC) ((C/A)xD)+B
    (D-B)+(AxC) ((BxD)+C)-A (DxA)-(B-C) (D/A)+(BxC) ((B/D)+A)xC
    (D-A)+(BxC) ((CxB)+D)-A (AxD)-(B-C) (C/B)+(AxD) ((A/D)+B)xC
    (C-B)+(AxD) ((BxC)+D)-A (CxB)-(A-D) (C/A)+(BxD) ((B/C)+A)xD
    (C-A)+(BxD) ((D-A)xC)+B (BxC)-(A-D) Dx((B/C)+A) ((A/C)+B)xD
    D+((CxB)-A) ((C-A)xD)+B (CxA)-(B-D) Cx((B/D)+A) B+(D/(A/C))
    C+((DxB)-A) ((DxA)+C)-B (AxC)-(B-D) B+((DxC)/A) A+(D/(B/C))
    D+((BxC)-A) ((AxD)+C)-B B-((A-D)xC) B+((CxD)/A) B+(C/(A/D))
    C+((BxD)-A) ((CxA)+D)-B A-((B-D)xC) Dx((A/C)+B) A+(C/(B/D))
    D+((CxA)-B) ((AxC)+D)-B B-((A-C)xD) Cx((A/D)+B) (D/(B/C))+A
    C+((DxA)-B) ((DxB)-A)+C A-((B-C)xD) A+((DxC)/B) (C/(B/D))+A
    D+((AxC)-B) ((BxD)-A)+C B+(Dx(C/A)) A+((CxD)/B) (D/(A/C))+B
    C+((AxD)-B) ((DxA)-B)+C B+(Cx(D/A)) B+((D/A)xC) (C/(A/D))+B
    B+((D-A)xC) ((AxD)-B)+C A+(Dx(C/B)) A+((D/B)xC)
    A+((D-B)xC) ((CxB)-A)+D A+(Cx(D/B)) B+((C/A)xD)
    B+((C-A)xD) ((BxC)-A)+D Dx(B+(A/C)) A+((C/B)xD)

    Change C = \sqrt{4} = 2 and there are 125 more ways:

    D+(C+(B+A)) (D+C)+(AxB) A+((CxB)+D) ((BxA)+C)+D ((A+D)xB)-C
    (D+C)+(B+A) (C+D)+(AxB) B+((AxC)+D) ((AxB)+C)+D Cx(D+(B/A))
    D+((C+B)+A) (D+A)+(CxB) A+((BxC)+D) (Cx(D+B))-A Bx(D+(C/A))
    (D+(C+B))+A (A+D)+(CxB) (D+(CxB))+A (Bx(D+C))-A Cx(D+(A/B))
    ((D+C)+B)+A (BxA)+(D+C) (D+(BxC))+A (Cx(B+D))-A Ax(D+(C/B))
    D+(C+(BxA)) (AxB)+(D+C) (D+(CxA))+B (Bx(C+D))-A Bx(D+(A/C))
    C+(D+(BxA)) (D+B)+(AxC) (D+(AxC))+B (Cx(D+A))-B Ax(D+(B/C))
    D+(B+(CxA)) (B+D)+(AxC) (D+(BxA))+C (Ax(D+C))-B Cx((B/A)+D)
    B+(D+(CxA)) (D+A)+(BxC) (D+(AxB))+C (Cx(A+D))-B Bx((C/A)+D)
    D+(C+(AxB)) (A+D)+(BxC) (C+(BxA))+D (Ax(C+D))-B Cx((A/B)+D)
    C+(D+(AxB)) (CxB)+(A+D) (B+(CxA))+D (Bx(D+A))-C Ax((C/B)+D)
    D+(A+(CxB)) (BxC)+(A+D) (C+(AxB))+D (Ax(D+B))-C Bx((A/C)+D)
    A+(D+(CxB)) (CxA)+(B+D) (A+(CxB))+D (Bx(A+D))-C Ax((B/C)+D)
    D+(B+(AxC)) (AxC)+(B+D) (B+(AxC))+D (Ax(B+D))-C (D+(C/B))xA
    B+(D+(AxC)) (BxA)+(C+D) (A+(BxC))+D ((D+C)xB)-A (D+(B/C))xA
    D+(A+(BxC)) (AxB)+(C+D) ((CxB)+D)+A ((C+D)xB)-A (D+(C/A))xB
    A+(D+(BxC)) D+((CxB)+A) ((BxC)+D)+A ((D+B)xC)-A (D+(A/C))xB
    (CxB)+(D+A) D+((BxC)+A) ((CxA)+D)+B ((B+D)xC)-A (D+(B/A))xC
    (BxC)+(D+A) D+((CxA)+B) ((AxC)+D)+B ((D+C)xA)-B (D+(A/B))xC
    (D+C)+(BxA) D+((AxC)+B) ((BxA)+D)+C ((C+D)xA)-B ((C/B)+D)xA
    (C+D)+(BxA) D+((BxA)+C) ((AxB)+D)+C ((D+A)xC)-B ((B/C)+D)xA
    (D+B)+(CxA) D+((AxB)+C) ((CxB)+A)+D ((A+D)xC)-B ((C/A)+D)xB
    (B+D)+(CxA) C+((BxA)+D) ((BxC)+A)+D ((D+B)xA)-C ((A/C)+D)xB
    (CxA)+(D+B) B+((CxA)+D) ((CxA)+B)+D ((B+D)xA)-C ((B/A)+D)xC
    (AxC)+(D+B) C+((AxB)+D) ((AxC)+B)+D ((D+A)xB)-C ((A/B)+D)xC

    Finally, make all four the same, D = \sqrt{4} = 2 and there are 192 more ways.

    D+(Cx(B+A)) A+((D+B)xC) ((D+C)xA)+B A+(Cx(BxD)) (Dx(BxA))+C
    C+(Dx(B+A)) B+((A+D)xC) ((C+D)xA)+B B+(Ax(CxD)) (Bx(DxA))+C
    D+(Bx(C+A)) A+((B+D)xC) ((D+A)xC)+B A+(Bx(CxD)) (Dx(AxB))+C
    B+(Dx(C+A)) C+((B+A)xD) ((A+D)xC)+B D+((CxB)xA) (Ax(DxB))+C
    C+(Bx(D+A)) B+((C+A)xD) ((C+A)xD)+B C+((DxB)xA) (Bx(AxD))+C
    B+(Cx(D+A)) C+((A+B)xD) ((A+C)xD)+B D+((BxC)xA) (Ax(BxD))+C
    D+(Cx(A+B)) A+((C+B)xD) ((D+B)xA)+C B+((DxC)xA) (Cx(BxA))+D
    C+(Dx(A+B)) B+((A+C)xD) ((B+D)xA)+C C+((BxD)xA) (Bx(CxA))+D
    D+(Ax(C+B)) A+((B+C)xD) ((D+A)xB)+C B+((CxD)xA) (Cx(AxB))+D
    A+(Dx(C+B)) (Dx(C+B))+A ((A+D)xB)+C D+((CxA)xB) (Ax(CxB))+D
    C+(Ax(D+B)) (Cx(D+B))+A ((B+A)xD)+C C+((DxA)xB) (Bx(AxC))+D
    A+(Cx(D+B)) (Dx(B+C))+A ((A+B)xD)+C D+((AxC)xB) (Ax(BxC))+D
    D+(Bx(A+C)) (Bx(D+C))+A ((C+B)xA)+D A+((DxC)xB) ((DxC)xB)+A
    B+(Dx(A+C)) (Cx(B+D))+A ((B+C)xA)+D C+((AxD)xB) ((CxD)xB)+A
    D+(Ax(B+C)) (Bx(C+D))+A ((C+A)xB)+D A+((CxD)xB) ((DxB)xC)+A
    A+(Dx(B+C)) (Dx(C+A))+B ((A+C)xB)+D D+((BxA)xC) ((BxD)xC)+A
    B+(Ax(D+C)) (Cx(D+A))+B ((B+A)xC)+D B+((DxA)xC) ((CxB)xD)+A
    A+(Bx(D+C)) (Dx(A+C))+B ((A+B)xC)+D D+((AxB)xC) ((BxC)xD)+A
    C+(Bx(A+D)) (Ax(D+C))+B D+(Cx(BxA)) A+((DxB)xC) ((DxC)xA)+B
    B+(Cx(A+D)) (Cx(A+D))+B C+(Dx(BxA)) B+((AxD)xC) ((CxD)xA)+B
    C+(Ax(B+D)) (Ax(C+D))+B D+(Bx(CxA)) A+((BxD)xC) ((DxA)xC)+B
    A+(Cx(B+D)) (Dx(B+A))+C B+(Dx(CxA)) C+((BxA)xD) ((AxD)xC)+B
    B+(Ax(C+D)) (Bx(D+A))+C C+(Bx(DxA)) B+((CxA)xD) ((CxA)xD)+B
    A+(Bx(C+D)) (Dx(A+B))+C B+(Cx(DxA)) C+((AxB)xD) ((AxC)xD)+B
    D+((C+B)xA) (Ax(D+B))+C D+(Cx(AxB)) A+((CxB)xD) ((DxB)xA)+C
    C+((D+B)xA) (Bx(A+D))+C C+(Dx(AxB)) B+((AxC)xD) ((BxD)xA)+C
    D+((B+C)xA) (Ax(B+D))+C D+(Ax(CxB)) A+((BxC)xD) ((DxA)xB)+C
    B+((D+C)xA) (Cx(B+A))+D A+(Dx(CxB)) (Dx(CxB))+A ((AxD)xB)+C
    C+((B+D)xA) (Bx(C+A))+D C+(Ax(DxB)) (Cx(DxB))+A ((BxA)xD)+C
    B+((C+D)xA) (Cx(A+B))+D A+(Cx(DxB)) (Dx(BxC))+A ((AxB)xD)+C
    D+((C+A)xB) (Ax(C+B))+D D+(Bx(AxC)) (Bx(DxC))+A ((CxB)xA)+D
    C+((D+A)xB) (Bx(A+C))+D B+(Dx(AxC)) (Cx(BxD))+A ((BxC)xA)+D
    D+((A+C)xB) (Ax(B+C))+D D+(Ax(BxC)) (Bx(CxD))+A ((CxA)xB)+D
    A+((D+C)xB) ((D+C)xB)+A A+(Dx(BxC)) (Dx(CxA))+B ((AxC)xB)+D
    C+((A+D)xB) ((C+D)xB)+A B+(Ax(DxC)) (Cx(DxA))+B ((BxA)xC)+D
    A+((C+D)xB) ((D+B)xC)+A A+(Bx(DxC)) (Dx(AxC))+B ((AxB)xC)+D
    D+((B+A)xC) ((B+D)xC)+A C+(Bx(AxD)) (Ax(DxC))+B
    B+((D+A)xC) ((C+B)xD)+A B+(Cx(AxD)) (Cx(AxD))+B
    D+((A+B)xC) ((B+C)xD)+A C+(Ax(BxD)) (Ax(CxD))+B

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  4. #19
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    If D = 4! and B = C = A = 4, there are 18 more.

    (D+(CxB))/A (D+(BxA))/C ((CxA)+D)/B (CxB)-(D/A) (BxA)-(D/C)
    (D+(BxC))/A (D+(AxB))/C ((AxC)+D)/B (BxC)-(D/A) (AxB)-(D/C)
    (D+(CxA))/B ((CxB)+D)/A ((BxA)+D)/C (CxA)-(D/B)
    (D+(AxC))/B ((BxC)+D)/A ((AxB)+D)/C (AxC)-(D/B)

    We can start adding square roots, too, but I'm tired.
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  5. #20
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by TKHunny View Post
    If A = \sqrt{4} = 2, B = 4, C = 4, and D = 4, so that we can tell one 4 from another, we have 186 ways:

    D+(C+(B-A)) (D+(B-A))+C (C+B)-(A-D) D+(C+(B/A)) Ax(D+(B/C))
    C+(D+(B-A)) (B+(D-A))+C (B+C)-(A-D) C+(D+(B/A)) Ax(B+(D/C))
    D+(B+(C-A)) (C+(B-A))+D (D-(A-C))+B D+(B+(C/A)) Ax(C+(B/D))
    B+(D+(C-A)) (B+(C-A))+D (C-(A-D))+B B+(D+(C/A)) Ax(B+(C/D))
    C+(B+(D-A)) ((D+C)+B)-A (D-(A-B))+C C+(B+(D/A)) Ax((D/C)+B)
    B+(C+(D-A)) ((C+D)+B)-A (B-(A-D))+C B+(C+(D/A)) Ax((C/D)+B)
    (D+C)+(B-A) ((D+B)+C)-A (C-(A-B))+D (D+C)+(B/A) Ax((D/B)+C)
    (C+D)+(B-A) ((B+D)+C)-A (B-(A-C))+D (C+D)+(B/A) Ax((B/D)+C)
    (D+B)+(C-A) ((C+B)+D)-A D-((A-C)-B) (D+B)+(C/A) Ax((C/B)+D)
    (B+D)+(C-A) ((B+C)+D)-A C-((A-D)-B) (B+D)+(C/A) Ax((B/C)+D)
    (C+B)+(D-A) ((D+C)-A)+B D-((A-B)-C) (C+B)+(D/A) (D+(C/B))xA
    (B+C)+(D-A) ((C+D)-A)+B B-((A-D)-C) (B+C)+(D/A) (C+(D/B))xA
    (D-A)+(C+B) ((D-A)+C)+B C-((A-B)-D) (D/A)+(C+B) (D+(B/C))xA
    (C-A)+(D+B) ((C-A)+D)+B B-((A-C)-D) (C/A)+(D+B) (B+(D/C))xA
    (D-A)+(B+C) ((D+B)-A)+C (DxC)-(B+A) (D/A)+(B+C) (C+(B/D))xA
    (B-A)+(D+C) ((B+D)-A)+C (CxD)-(B+A) (B/A)+(D+C) (B+(C/D))xA
    (C-A)+(B+D) ((D-A)+B)+C (DxB)-(C+A) (C/A)+(B+D) (D+(CxB))/A
    (B-A)+(C+D) ((B-A)+D)+C (BxD)-(C+A) (B/A)+(C+D) (C+(DxB))/A
    D+((C+B)-A) ((C+B)-A)+D (CxB)-(D+A) D+((C/A)+B) (D+(BxC))/A
    C+((D+B)-A) ((B+C)-A)+D (BxC)-(D+A) C+((D/A)+B) (B+(DxC))/A
    D+((B+C)-A) ((C-A)+B)+D (DxC)-(A+B) D+((B/A)+C) (C+(BxD))/A
    B+((D+C)-A) ((B-A)+C)+D (CxD)-(A+B) B+((D/A)+C) (B+(CxD))/A
    C+((B+D)-A) D-(A-(C+B)) (DxB)-(A+C) C+((B/A)+D) ((D/C)+B)xA
    B+((C+D)-A) C-(A-(D+B)) (BxD)-(A+C) B+((C/A)+D) ((C/D)+B)xA
    D+((C-A)+B) D+(C-(A-B)) (CxB)-(A+D) (D+(C/A))+B ((D/B)+C)xA
    C+((D-A)+B) C+(D-(A-B)) (BxC)-(A+D) (C+(D/A))+B ((B/D)+C)xA
    D+((B-A)+C) D-(A-(B+C)) ((DxC)-B)-A (D+(B/A))+C ((C/B)+D)xA
    B+((D-A)+C) B-(A-(D+C)) ((CxD)-B)-A (B+(D/A))+C ((B/C)+D)xA
    C+((B-A)+D) D+(B-(A-C)) ((DxB)-C)-A (C+(B/A))+D ((DxC)+B)/A
    B+((C-A)+D) B+(D-(A-C)) ((BxD)-C)-A (B+(C/A))+D ((CxD)+B)/A
    (D+(C+B))-A C-(A-(B+D)) ((CxB)-D)-A ((D/A)+C)+B ((DxB)+C)/A
    (C+(D+B))-A B-(A-(C+D)) ((BxC)-D)-A ((C/A)+D)+B ((BxD)+C)/A
    (D+(B+C))-A C+(B-(A-D)) ((DxC)-A)-B ((D/A)+B)+C ((CxB)+D)/A
    (B+(D+C))-A B+(C-(A-D)) ((CxD)-A)-B ((B/A)+D)+C ((BxC)+D)/A
    (C+(B+D))-A (D+C)-(A-B) ((DxB)-A)-C ((C/A)+B)+D
    (B+(C+D))-A (C+D)-(A-B) ((BxD)-A)-C ((B/A)+C)+D
    (D+(C-A))+B (D+B)-(A-C) ((CxB)-A)-D Ax(D+(C/B))
    (C+(D-A))+B (B+D)-(A-C) ((BxC)-A)-D Ax(C+(D/B))

    Add B = \sqrt{4} = 2 and there are 152 more ways.

    B+(Dx(C-A)) A+((C-B)xD) ((CxA)-B)+D Dx(A+(B/C)) (Dx(C/B))+A
    B+(Cx(D-A)) (Dx(C-B))+A ((AxC)-B)+D Cx(B+(A/D)) (Cx(D/B))+A
    A+(Dx(C-B)) (Cx(D-B))+A D-(B-(CxA)) Cx(A+(B/D)) (Dx(C/A))+B
    A+(Cx(D-B)) (D+(CxB))-A C-(B-(DxA)) (D/B)+(CxA) (Cx(D/A))+B
    (DxB)+(C-A) (C+(DxB))-A D-(A-(CxB)) (C/B)+(DxA) (B+(A/D))xC
    (BxD)+(C-A) (D+(BxC))-A C-(A-(DxB)) (DxB)+(C/A) (A+(B/D))xC
    (CxB)+(D-A) (C+(BxD))-A B-(Dx(A-C)) (BxD)+(C/A) (B+(A/C))xD
    (BxC)+(D-A) (Dx(C-A))+B A-(Dx(B-C)) (CxB)+(D/A) (A+(B/C))xD
    (D-B)+(CxA) (Cx(D-A))+B D-(B-(AxC)) (BxC)+(D/A) ((DxC)/B)+A
    (C-B)+(DxA) (D+(CxA))-B D-(A-(BxC)) (D/A)+(CxB) ((CxD)/B)+A
    (DxA)+(C-B) (C+(DxA))-B B-(Cx(A-D)) (C/A)+(DxB) ((D/B)xC)+A
    (AxD)+(C-B) (D+(AxC))-B A-(Cx(B-D)) (DxA)+(C/B) ((C/B)xD)+A
    (CxA)+(D-B) (C+(AxD))-B C-(B-(AxD)) (AxD)+(C/B) ((DxC)/A)+B
    (AxC)+(D-B) ((D-B)xC)+A C-(A-(BxD)) (CxA)+(D/B) ((CxD)/A)+B
    (D-A)+(CxB) ((C-B)xD)+A (DxB)-(A-C) (AxC)+(D/B) ((D/A)xC)+B
    (C-A)+(DxB) ((DxB)+C)-A (BxD)-(A-C) (D/B)+(AxC) ((C/A)xD)+B
    (D-B)+(AxC) ((BxD)+C)-A (DxA)-(B-C) (D/A)+(BxC) ((B/D)+A)xC
    (D-A)+(BxC) ((CxB)+D)-A (AxD)-(B-C) (C/B)+(AxD) ((A/D)+B)xC
    (C-B)+(AxD) ((BxC)+D)-A (CxB)-(A-D) (C/A)+(BxD) ((B/C)+A)xD
    (C-A)+(BxD) ((D-A)xC)+B (BxC)-(A-D) Dx((B/C)+A) ((A/C)+B)xD
    D+((CxB)-A) ((C-A)xD)+B (CxA)-(B-D) Cx((B/D)+A) B+(D/(A/C))
    C+((DxB)-A) ((DxA)+C)-B (AxC)-(B-D) B+((DxC)/A) A+(D/(B/C))
    D+((BxC)-A) ((AxD)+C)-B B-((A-D)xC) B+((CxD)/A) B+(C/(A/D))
    C+((BxD)-A) ((CxA)+D)-B A-((B-D)xC) Dx((A/C)+B) A+(C/(B/D))
    D+((CxA)-B) ((AxC)+D)-B B-((A-C)xD) Cx((A/D)+B) (D/(B/C))+A
    C+((DxA)-B) ((DxB)-A)+C A-((B-C)xD) A+((DxC)/B) (C/(B/D))+A
    D+((AxC)-B) ((BxD)-A)+C B+(Dx(C/A)) A+((CxD)/B) (D/(A/C))+B
    C+((AxD)-B) ((DxA)-B)+C B+(Cx(D/A)) B+((D/A)xC) (C/(A/D))+B
    B+((D-A)xC) ((AxD)-B)+C A+(Dx(C/B)) A+((D/B)xC)
    A+((D-B)xC) ((CxB)-A)+D A+(Cx(D/B)) B+((C/A)xD)
    B+((C-A)xD) ((BxC)-A)+D Dx(B+(A/C)) A+((C/B)xD)

    Change C = \sqrt{4} = 2 and there are 125 more ways:

    D+(C+(B+A)) (D+C)+(AxB) A+((CxB)+D) ((BxA)+C)+D ((A+D)xB)-C
    (D+C)+(B+A) (C+D)+(AxB) B+((AxC)+D) ((AxB)+C)+D Cx(D+(B/A))
    D+((C+B)+A) (D+A)+(CxB) A+((BxC)+D) (Cx(D+B))-A Bx(D+(C/A))
    (D+(C+B))+A (A+D)+(CxB) (D+(CxB))+A (Bx(D+C))-A Cx(D+(A/B))
    ((D+C)+B)+A (BxA)+(D+C) (D+(BxC))+A (Cx(B+D))-A Ax(D+(C/B))
    D+(C+(BxA)) (AxB)+(D+C) (D+(CxA))+B (Bx(C+D))-A Bx(D+(A/C))
    C+(D+(BxA)) (D+B)+(AxC) (D+(AxC))+B (Cx(D+A))-B Ax(D+(B/C))
    D+(B+(CxA)) (B+D)+(AxC) (D+(BxA))+C (Ax(D+C))-B Cx((B/A)+D)
    B+(D+(CxA)) (D+A)+(BxC) (D+(AxB))+C (Cx(A+D))-B Bx((C/A)+D)
    D+(C+(AxB)) (A+D)+(BxC) (C+(BxA))+D (Ax(C+D))-B Cx((A/B)+D)
    C+(D+(AxB)) (CxB)+(A+D) (B+(CxA))+D (Bx(D+A))-C Ax((C/B)+D)
    D+(A+(CxB)) (BxC)+(A+D) (C+(AxB))+D (Ax(D+B))-C Bx((A/C)+D)
    A+(D+(CxB)) (CxA)+(B+D) (A+(CxB))+D (Bx(A+D))-C Ax((B/C)+D)
    D+(B+(AxC)) (AxC)+(B+D) (B+(AxC))+D (Ax(B+D))-C (D+(C/B))xA
    B+(D+(AxC)) (BxA)+(C+D) (A+(BxC))+D ((D+C)xB)-A (D+(B/C))xA
    D+(A+(BxC)) (AxB)+(C+D) ((CxB)+D)+A ((C+D)xB)-A (D+(C/A))xB
    A+(D+(BxC)) D+((CxB)+A) ((BxC)+D)+A ((D+B)xC)-A (D+(A/C))xB
    (CxB)+(D+A) D+((BxC)+A) ((CxA)+D)+B ((B+D)xC)-A (D+(B/A))xC
    (BxC)+(D+A) D+((CxA)+B) ((AxC)+D)+B ((D+C)xA)-B (D+(A/B))xC
    (D+C)+(BxA) D+((AxC)+B) ((BxA)+D)+C ((C+D)xA)-B ((C/B)+D)xA
    (C+D)+(BxA) D+((BxA)+C) ((AxB)+D)+C ((D+A)xC)-B ((B/C)+D)xA
    (D+B)+(CxA) D+((AxB)+C) ((CxB)+A)+D ((A+D)xC)-B ((C/A)+D)xB
    (B+D)+(CxA) C+((BxA)+D) ((BxC)+A)+D ((D+B)xA)-C ((A/C)+D)xB
    (CxA)+(D+B) B+((CxA)+D) ((CxA)+B)+D ((B+D)xA)-C ((B/A)+D)xC
    (AxC)+(D+B) C+((AxB)+D) ((AxC)+B)+D ((D+A)xB)-C ((A/B)+D)xC

    Finally, make all four the same, D = \sqrt{4} = 2 and there are 192 more ways.

    D+(Cx(B+A)) A+((D+B)xC) ((D+C)xA)+B A+(Cx(BxD)) (Dx(BxA))+C
    C+(Dx(B+A)) B+((A+D)xC) ((C+D)xA)+B B+(Ax(CxD)) (Bx(DxA))+C
    D+(Bx(C+A)) A+((B+D)xC) ((D+A)xC)+B A+(Bx(CxD)) (Dx(AxB))+C
    B+(Dx(C+A)) C+((B+A)xD) ((A+D)xC)+B D+((CxB)xA) (Ax(DxB))+C
    C+(Bx(D+A)) B+((C+A)xD) ((C+A)xD)+B C+((DxB)xA) (Bx(AxD))+C
    B+(Cx(D+A)) C+((A+B)xD) ((A+C)xD)+B D+((BxC)xA) (Ax(BxD))+C
    D+(Cx(A+B)) A+((C+B)xD) ((D+B)xA)+C B+((DxC)xA) (Cx(BxA))+D
    C+(Dx(A+B)) B+((A+C)xD) ((B+D)xA)+C C+((BxD)xA) (Bx(CxA))+D
    D+(Ax(C+B)) A+((B+C)xD) ((D+A)xB)+C B+((CxD)xA) (Cx(AxB))+D
    A+(Dx(C+B)) (Dx(C+B))+A ((A+D)xB)+C D+((CxA)xB) (Ax(CxB))+D
    C+(Ax(D+B)) (Cx(D+B))+A ((B+A)xD)+C C+((DxA)xB) (Bx(AxC))+D
    A+(Cx(D+B)) (Dx(B+C))+A ((A+B)xD)+C D+((AxC)xB) (Ax(BxC))+D
    D+(Bx(A+C)) (Bx(D+C))+A ((C+B)xA)+D A+((DxC)xB) ((DxC)xB)+A
    B+(Dx(A+C)) (Cx(B+D))+A ((B+C)xA)+D C+((AxD)xB) ((CxD)xB)+A
    D+(Ax(B+C)) (Bx(C+D))+A ((C+A)xB)+D A+((CxD)xB) ((DxB)xC)+A
    A+(Dx(B+C)) (Dx(C+A))+B ((A+C)xB)+D D+((BxA)xC) ((BxD)xC)+A
    B+(Ax(D+C)) (Cx(D+A))+B ((B+A)xC)+D B+((DxA)xC) ((CxB)xD)+A
    A+(Bx(D+C)) (Dx(A+C))+B ((A+B)xC)+D D+((AxB)xC) ((BxC)xD)+A
    C+(Bx(A+D)) (Ax(D+C))+B D+(Cx(BxA)) A+((DxB)xC) ((DxC)xA)+B
    B+(Cx(A+D)) (Cx(A+D))+B C+(Dx(BxA)) B+((AxD)xC) ((CxD)xA)+B
    C+(Ax(B+D)) (Ax(C+D))+B D+(Bx(CxA)) A+((BxD)xC) ((DxA)xC)+B
    A+(Cx(B+D)) (Dx(B+A))+C B+(Dx(CxA)) C+((BxA)xD) ((AxD)xC)+B
    B+(Ax(C+D)) (Bx(D+A))+C C+(Bx(DxA)) B+((CxA)xD) ((CxA)xD)+B
    A+(Bx(C+D)) (Dx(A+B))+C B+(Cx(DxA)) C+((AxB)xD) ((AxC)xD)+B
    D+((C+B)xA) (Ax(D+B))+C D+(Cx(AxB)) A+((CxB)xD) ((DxB)xA)+C
    C+((D+B)xA) (Bx(A+D))+C C+(Dx(AxB)) B+((AxC)xD) ((BxD)xA)+C
    D+((B+C)xA) (Ax(B+D))+C D+(Ax(CxB)) A+((BxC)xD) ((DxA)xB)+C
    B+((D+C)xA) (Cx(B+A))+D A+(Dx(CxB)) (Dx(CxB))+A ((AxD)xB)+C
    C+((B+D)xA) (Bx(C+A))+D C+(Ax(DxB)) (Cx(DxB))+A ((BxA)xD)+C
    B+((C+D)xA) (Cx(A+B))+D A+(Cx(DxB)) (Dx(BxC))+A ((AxB)xD)+C
    D+((C+A)xB) (Ax(C+B))+D D+(Bx(AxC)) (Bx(DxC))+A ((CxB)xA)+D
    C+((D+A)xB) (Bx(A+C))+D B+(Dx(AxC)) (Cx(BxD))+A ((BxC)xA)+D
    D+((A+C)xB) (Ax(B+C))+D D+(Ax(BxC)) (Bx(CxD))+A ((CxA)xB)+D
    A+((D+C)xB) ((D+C)xB)+A A+(Dx(BxC)) (Dx(CxA))+B ((AxC)xB)+D
    C+((A+D)xB) ((C+D)xB)+A B+(Ax(DxC)) (Cx(DxA))+B ((BxA)xC)+D
    A+((C+D)xB) ((D+B)xC)+A A+(Bx(DxC)) (Dx(AxC))+B ((AxB)xC)+D
    D+((B+A)xC) ((B+D)xC)+A C+(Bx(AxD)) (Ax(DxC))+B
    B+((D+A)xC) ((C+B)xD)+A B+(Cx(AxD)) (Cx(AxD))+B
    D+((A+B)xC) ((B+C)xD)+A C+(Ax(BxD)) (Ax(CxD))+B



    That's a lot of ways....I hope you didn't drive yourself insane figuring all of these out

    Now you know why no one invites me to parties.


    It makes more sense now

    --Chris
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  6. #21
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    I'm hoping he wrong a simple software script to brute force identify all the ways to do that, otherwise u just makin me feel lazy !
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  7. #22
    MHF Contributor
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    Program
    Brute Force (It takes 0.14 seconds to produce the first listing of 186 solutions, but that includes some progress updates on the screen.)
    That would describe it.

    I never much liked math games. I always found them irritating, rather than challenging or entertaining.

    When I first met "The 24 Challenge" I was delighted. It was the first math game I enjoyed. However, in my usual style, I decide to create a list of every possible solution. When they came out with the Three-Whole-Numbers-and-One-Fraction version, I generalized the program a bit and created that list, too. ALL solutions for one fraction consisting of a single digit in the numerator and in the denominator. It was a little longer than the Natural number less than 10 version.

    Since then, I generalized it a bit to handle other scenarios. That is what you have before you. I'm sure there are other generalizations. Anyway, it fits me since it so easily discards any worrying or fretting about finding solutions to such problems.
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  8. #23
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    Quote Originally Posted by darryl950 View Post
    hi its my second day at high school and they given me homework due in tomorrow morning plz help

    ive got to get four 4's into 10

    for example if i had to get four 4's into 8 i could do

    4 x 4 -4-4 = 8
    can use no other numbers but 4 and only 4 times

    for the number 7 i got 4 + 4 - (4/4) = 7




    plz help !!
    very grateful ty
    =4 \times 4 - (4 + \sqrt {4})

    =10
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  9. #24
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    Quote Originally Posted by Shyam View Post
    =4 \times 4 - (4 + \sqrt {4})

    =10
    Quote Originally Posted by damadama View Post
    4*4 -4 - sqrt(4)
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  10. #25
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    See, I managed that solution six different ways.

    (D*C)-(B+A)
    (C*D)-(B+A)
    (D*B)-(C+A)
    (B*D)-(C+A)
    (C*B)-(D+A)
    (B*C)-(D+A)

    Again, A = \sqrt{4} = 2 and B = C = D = 4

    Okay, I'm done goofing off. It's become almost fun and we can't have that, can we?
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  11. #26
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    darry, I would go with (44-4)/4.
    But when you get the answer to this can you post it on this thread, since I cant seen any other way of doing it, and it feels a bit 'unethical' using 44 or sqr.roots
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  12. #27
    Forum Admin topsquark's Avatar
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    Just as a weird thought, you can do 4 + 4 + 4 + 4 in base 12...

    -Dan
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  13. #28
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    If the 4's are the only numerals allowed,

    . . then we can use this formula:


    . . n \;=\;-\log_{\frac{4}{\sqrt{4}}} \left[\log_4\;\left( \underbrace{\sqrt{\sqrt{\sqrt{\hdots\sqrt{4}} }}}  \right) \right]
    . . . . . . . . . . . . . . . .
    n radicals

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  14. #29
    Super Member Matt Westwood's Avatar
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    Drat, and I was gonna be the smartyboots to mention that you can also use \Gamma (4) = 6 ...
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  15. #30
    Moo
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    \int_{\sqrt{4}}^4 ~dx+4+4 ?
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