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Phantasmagoria: a hidden mathematical puzzle?

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  • hagaysc
    Hello, I was wondering if the next quote from Carroll s Phantasmagoria is a mathematical puzzle. If so, then I d have to come up with a similiar one to
    Message 1 of 5 , Nov 1, 2005
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      Hello, I was wondering if the next quote from Carroll's Phantasmagoria
      is a mathematical puzzle. If so, then I'd have to come up with a
      similiar one to translate it into my language, as I am in the midst of
      translating Phantasmagoria:

      And I remember nothing more
      That I can clearly fix,
      Till I was sitting on the floor,
      Repeating "Two and five are four,
      But five and two are six."

      Any help, telling me whether it's a puzzle or not, and if so, helping
      me solve it, will be much appreciated.

      Thanks a lot,

      Hagay
    • Jim Buch
      ... Just make it into a pair of equations..... 2X + 5Y = 4 5X + 2Y = 6 You can solve these by the use of determinates, and CLD wrote a book on the theory of
      Message 2 of 5 , Nov 1, 2005
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        --- In lewiscarroll@yahoogroups.com, "hagaysc" <hschurr@g...> wrote:
        >
        > Hello, I was wondering if the next quote from Carroll's Phantasmagoria
        > is a mathematical puzzle. If so, then I'd have to come up with a
        > similiar one to translate it into my language, as I am in the midst of
        > translating Phantasmagoria:
        >
        > And I remember nothing more
        > That I can clearly fix,
        > Till I was sitting on the floor,
        > Repeating "Two and five are four,
        > But five and two are six."
        >
        > Any help, telling me whether it's a puzzle or not, and if so, helping
        > me solve it, will be much appreciated.
        >
        > Thanks a lot,
        >

        Just make it into a pair of equations.....

        2X + 5Y = 4
        5X + 2Y = 6


        You can solve these by the use of determinates, and CLD wrote a book
        on the theory of determinates as well as a paper of his was read
        before the Royal Society of London.


        You can pretend that you are CLD and "play" with the equations.

        Add them...... and you get

        7X + 7Y = 10 ....... which is kind of cute or you can get

        X + Y = (10/7)

        You can subtract the equations and you can then get

        3X - 3Y = 2 ....... which is kind of cute also or you can get

        X - Y = (2/3)

        or

        X + Y = A .......[A = 10/7]
        X - Y = B .......[B = 2/3]

        And by adding the equations you solve for X and by subtracting the
        equations you solve for Y.

        Adding gives:

        2X = (A + B)
        X = (A + B)/2

        Subtracting gives:

        2Y = (A - B)

        and finally

        Y = (A - B)/2
        X = (A + B)/2


        This is even more cute, or pretty or might have a little algebraic
        beauty.....

        I am arithmetically challenged, and usually mess up the fractional
        reduction of sticking in the known values of A and B... so I leave it
        to the reader......

        Leaving it to the reader is a common ploy used in mathematical
        writings to avoid doing what the author find tedious, and shifts the
        burden to the reader......

        When the fractions are done, the actual fractional numbers may suggest
        physical things in Dodgson's life, or may not do so.

        He could have just been fascinated that the simple rhyme led to such
        an interesting form of mathematical solution as....

        Y = (A - B)/2
        X = (A + B)/2

        Now you can work backwards, maybe, and find out how to invent more
        rhymes that lead to pretty equations, or maybe not (which is what I
        think is the result).

        Play like Dodgson, and you may think like Dodgson. Or maybe not.

        Jim





        > Hagay
        >

        Try solving the simultaneous algebra equations....

        2X + 5Y = 4
        5X + 2Y = 6

        Then maybe given X and Y you could puzzle out something physical or
        logical that corresponds to those two numbers.

        Or just leave them alone as an intrepretation of the mathematical
        phrasing of simultaneous linear algebraic equations.

        You can formally solve these equations with the application of
        determinates, and the mathematician Dodgson wrote a book on
        determinates and a mathematical paper which was read (by his member
        correspondent) before the Royal Society of London --- usually a mark
        of significant technical merit.

        So, we made this into a mathematical thing.... and then what?

        We can play further.

        By adding the two equations we obtain a third equation...

        7X + 7Y = = 10

        or X + Y = 10/7

        and by subtracting the first equation from the second we obtain

        3X - 3Y = 2

        or X - Y = 2/3

        Or.. X + Y = A [A = 10/7]
        And. X - Y = B [B = 2/3]


        Adding the equations gives...
        2X = (A + B)
        or
        X = (A + B)/2

        Subtracting the equations gives...
        2Y = (A - B)
        or
        Y = (A - B)/2
        X = (A + B)/2

        Which is very pretty, isn't it. Some would call this a kind of
        mathematical beauty, of the algebraic sort.

        I am arithmetically challanged, so I won't put in the fractions for A
        and B.

        Use of a calculator is cheating.

        Jim
      • hagaysc
        ... Thank you so very much Jim. That s very helpful. I should have figured out myself I have to put them into a pair of equations.
        Message 3 of 5 , Nov 2, 2005
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          --- In lewiscarroll@yahoogroups.com, "Jim Buch" <jbuch@r...> wrote:
          >
          > --- In lewiscarroll@yahoogroups.com, "hagaysc" <hschurr@g...> wrote:
          > >
          > > Hello, I was wondering if the next quote from Carroll's Phantasmagoria
          > > is a mathematical puzzle. If so, then I'd have to come up with a
          > > similiar one to translate it into my language, as I am in the midst of
          > > translating Phantasmagoria:
          > >
          > > And I remember nothing more
          > > That I can clearly fix,
          > > Till I was sitting on the floor,
          > > Repeating "Two and five are four,
          > > But five and two are six."
          > >
          > > Any help, telling me whether it's a puzzle or not, and if so, helping
          > > me solve it, will be much appreciated.
          > >
          > > Thanks a lot,
          > >
          >
          > Just make it into a pair of equations.....
          >
          > 2X + 5Y = 4
          > 5X + 2Y = 6
          >
          >
          > You can solve these by the use of determinates, and CLD wrote a book
          > on the theory of determinates as well as a paper of his was read
          > before the Royal Society of London.
          >
          >
          > You can pretend that you are CLD and "play" with the equations.
          >
          > Add them...... and you get
          >
          > 7X + 7Y = 10 ....... which is kind of cute or you can get
          >
          > X + Y = (10/7)
          >
          > You can subtract the equations and you can then get
          >
          > 3X - 3Y = 2 ....... which is kind of cute also or you can get
          >
          > X - Y = (2/3)
          >
          > or
          >
          > X + Y = A .......[A = 10/7]
          > X - Y = B .......[B = 2/3]
          >
          > And by adding the equations you solve for X and by subtracting the
          > equations you solve for Y.
          >
          > Adding gives:
          >
          > 2X = (A + B)
          > X = (A + B)/2
          >
          > Subtracting gives:
          >
          > 2Y = (A - B)
          >
          > and finally
          >
          > Y = (A - B)/2
          > X = (A + B)/2
          >
          >
          > This is even more cute, or pretty or might have a little algebraic
          > beauty.....
          >
          > I am arithmetically challenged, and usually mess up the fractional
          > reduction of sticking in the known values of A and B... so I leave it
          > to the reader......
          >
          > Leaving it to the reader is a common ploy used in mathematical
          > writings to avoid doing what the author find tedious, and shifts the
          > burden to the reader......
          >
          > When the fractions are done, the actual fractional numbers may suggest
          > physical things in Dodgson's life, or may not do so.
          >
          > He could have just been fascinated that the simple rhyme led to such
          > an interesting form of mathematical solution as....
          >
          > Y = (A - B)/2
          > X = (A + B)/2
          >
          > Now you can work backwards, maybe, and find out how to invent more
          > rhymes that lead to pretty equations, or maybe not (which is what I
          > think is the result).
          >
          > Play like Dodgson, and you may think like Dodgson. Or maybe not.
          >
          > Jim
          >
          >
          >
          >
          >
          > > Hagay
          > >
          >
          > Try solving the simultaneous algebra equations....
          >
          > 2X + 5Y = 4
          > 5X + 2Y = 6
          >
          > Then maybe given X and Y you could puzzle out something physical or
          > logical that corresponds to those two numbers.
          >
          > Or just leave them alone as an intrepretation of the mathematical
          > phrasing of simultaneous linear algebraic equations.
          >
          > You can formally solve these equations with the application of
          > determinates, and the mathematician Dodgson wrote a book on
          > determinates and a mathematical paper which was read (by his member
          > correspondent) before the Royal Society of London --- usually a mark
          > of significant technical merit.
          >
          > So, we made this into a mathematical thing.... and then what?
          >
          > We can play further.
          >
          > By adding the two equations we obtain a third equation...
          >
          > 7X + 7Y = = 10
          >
          > or X + Y = 10/7
          >
          > and by subtracting the first equation from the second we obtain
          >
          > 3X - 3Y = 2
          >
          > or X - Y = 2/3
          >
          > Or.. X + Y = A [A = 10/7]
          > And. X - Y = B [B = 2/3]
          >
          >
          > Adding the equations gives...
          > 2X = (A + B)
          > or
          > X = (A + B)/2
          >
          > Subtracting the equations gives...
          > 2Y = (A - B)
          > or
          > Y = (A - B)/2
          > X = (A + B)/2
          >
          > Which is very pretty, isn't it. Some would call this a kind of
          > mathematical beauty, of the algebraic sort.
          >
          > I am arithmetically challanged, so I won't put in the fractions for A
          > and B.
          >
          > Use of a calculator is cheating.
          >
          > Jim





          Thank you so very much Jim. That's very helpful. I should have figured
          out myself I have to put them into a pair of equations.
          >
        • Jim Buch
          ... of translating Phantasmagoria: And I remember nothing more That I can clearly fix, Till I was sitting on the floor, Repeating Two and five are four, But
          Message 4 of 5 , Nov 2, 2005
          • 0 Attachment
            > --- In lewiscarroll@yahoogroups.com, "hagaysc" <hschurr@g...> wrote:

            > Hello, I was wondering if the next quote from Carroll's Phantasmagoria
            > is a mathematical puzzle. If so, then I'd have to come up with a
            > similiar one to translate it into my language, as I am in the midst
            of> translating Phantasmagoria:

            And I remember nothing more
            That I can clearly fix,
            Till I was sitting on the floor,
            Repeating "Two and five are four,
            But five and two are six."
            > > >
            > > > Any help, telling me whether it's a puzzle or not, and if so,
            helping
            > > > me solve it, will be much appreciated.

            And I remember from two men
            some clever arithmetic tricks.
            And wrote them down with my pen.
            "How can five and one be ten,
            but one and five are twenty six."

            If I were American, what would I be talking about, using common
            everyday items?


            This has a solution, but I don't think it is "mathematically pretty"
            as was the one by CLD.

            If you can play like Dodgson, it might help you think like Dodgson, or
            maybe not.

            Jim
          • BRYAN TALBOT
            Hi, I m on the scrounge again. If any member could email me a jpeg of the cover to Rhyme?And Reason? and one of a sample interior illustration to my personal
            Message 5 of 5 , Nov 4, 2005
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              Hi, I'm on the scrounge again. If any member could
              email me a jpeg of the cover to "Rhyme?And Reason?"
              and one of a sample interior illustration to my
              personal address, I would be very grateful (and I'd
              give you an acknowledgement in the book).

              Bryan
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