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First post, any help would be greatly appreciated

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  • Dave
    Let me start by saying I m far from a mathematician, however this question was posed to me many years ago in high school and I was never able to find a
    Message 1 of 7 , Aug 7, 2003
      Let me start by saying I'm far from a mathematician, however this
      question was posed to me many years ago in high school and I was never
      able to find a solution, furthermore I've asked over 30 people with
      widely varying math experience and none have been able to answer it, and
      to top it off I once physically modeled over 30 iterations of this
      problem using coins of varying denominations, measuring with a ruler and
      finally fed the resultant answer table into a genetic algorithm based
      program which could not generate better than a 90~% fit for the answer
      givens it.

      So Here Goes:

      Given an equilateral triangle with a base of X, what is the maximum
      number(Z) of circles of radius Y that can be inscribed within the
      boundaries of the triangle?

      An answer to this one would totally rock.

      TIA

      -Dave Pratt
      Berkeley, CA


      [Non-text portions of this message have been removed]
    • Michael Lambrou
      ... Either I don t understand the question or it is trivial: The largest circle you can inscribe in an equilateral triangle is the incircle. Call its radius r.
      Message 2 of 7 , Aug 7, 2003
        >
        > Given an equilateral triangle with a base of X, what is the maximum
        > number(Z) of circles of radius Y that can be inscribed within the
        > boundaries of the triangle?
        >

        Either I don't understand the question or it is trivial:

        The largest circle you can inscribe in an equilateral triangle is
        the incircle. Call its radius r. So for the above question,

        If Y>r then Z is zero
        If Y=r then Z is one
        If Y < r then Z is infinity.
      • Darij Grinberg
        ... I think he means: how many circles can be inscribed simultaneously without overlapping each other? I don t know the answer, but it must be an interesting
        Message 3 of 7 , Aug 7, 2003
          In Hyacinthos message #7445, Michael Lambrou wrote:

          >> > Given an equilateral triangle with a base of X, what
          >> > is the maximum number(Z) of circles of radius Y that
          >> > can be inscribed within the boundaries of the
          >> > triangle?
          >> >
          >>
          >> Either I don't understand the question or it is trivial:

          I think he means: how many circles can be inscribed
          simultaneously without overlapping each other?

          I don't know the answer, but it must be an interesting
          question!

          Darij Grinberg
        • Dave
          Darij, your interpretation of the question is correct. -Dave ... From: Darij Grinberg [mailto:darij_grinberg@web.de] Sent: Thursday, August 07, 2003 7:55 AM
          Message 4 of 7 , Aug 7, 2003
            Darij, your interpretation of the question is correct.

            -Dave

            -----Original Message-----
            From: Darij Grinberg [mailto:darij_grinberg@...]
            Sent: Thursday, August 07, 2003 7:55 AM
            To: Hyacinthos@yahoogroups.com
            Subject: [ipod] RE: [EMHL] First post, any help would be greatly
            appreciated

            In Hyacinthos message #7445, Michael Lambrou wrote:

            >> > Given an equilateral triangle with a base of X, what
            >> > is the maximum number(Z) of circles of radius Y that
            >> > can be inscribed within the boundaries of the
            >> > triangle?
            >> >
            >>
            >> Either I don't understand the question or it is trivial:

            I think he means: how many circles can be inscribed
            simultaneously without overlapping each other?

            I don't know the answer, but it must be an interesting
            question!

            Darij Grinberg




            [Non-text portions of this message have been removed]
          • Nikolaos Dergiades
            Dear all, [Dave] ... [Michael] ... [Darij] ... ******* If Darij, is right then I think that if we could put n of these circles on the base of the triangle then
            Message 5 of 7 , Aug 7, 2003
              Dear all,

              [Dave]
              >>> > Given an equilateral triangle with a base of X, what
              >>> > is the maximum number(Z) of circles of radius Y that
              >>> > can be inscribed within the boundaries of the
              >>> > triangle?


              [Michael]
              >>> Either I don't understand the question or it is trivial:

              [Darij]
              >I think he means: how many circles can be inscribed
              >simultaneously without overlapping each other?
              >
              >I don't know the answer, but it must be an interesting
              >question!


              *******
              If Darij, is right then I think that if we could put
              n of these circles on the base of the triangle then
              2Y(n -1)+2Y*sqrt(3) <= X < 2Yn+2Y*sqrt(3) or that
              n = [(X + 2Y - 2Y*sqrt(3))/2Y]
              where [ ] denotes integer part.
              Over these circles we could put n - 1 circles and so on.
              Hence the total maximum number of circles is Z = n(n+1)/2
              and n the number given above.

              Best regards
              Nikolaos Dergiades
            • Dave
              Wow Nikolaos, that was a very fast response. However, the last math class I took was about 5 years ago and it was a very basic high school level geometry
              Message 6 of 7 , Aug 7, 2003
                Wow Nikolaos, that was a very fast response. However, the last math
                class I took was about 5 years ago and it was a very basic high school
                level geometry class. Do you think you could walk me through your
                solution with a little more detail?

                Thanks in advance

                -Dave

                -----Original Message-----
                From: Nikolaos Dergiades [mailto:ndergiades@...]
                Sent: Thursday, August 07, 2003 11:42 AM
                To: Hyacinthos@yahoogroups.com
                Subject: [mathdorks] Re: [EMHL] First post, any help would be greatly
                appreciated

                Dear all,

                [Dave]
                >>> > Given an equilateral triangle with a base of X, what
                >>> > is the maximum number(Z) of circles of radius Y that
                >>> > can be inscribed within the boundaries of the
                >>> > triangle?



                *******
                If Darij, is right then I think that if we could put
                n of these circles on the base of the triangle then
                2Y(n -1)+2Y*sqrt(3) <= X < 2Yn+2Y*sqrt(3) or that
                n = [(X + 2Y - 2Y*sqrt(3))/2Y]
                where [ ] denotes integer part.
                Over these circles we could put n - 1 circles and so on.
                Hence the total maximum number of circles is Z = n(n+1)/2
                and n the number given above.

                Best regards
                Nikolaos Dergiades
              • Richard Guy
                I don t think it s quite as simple. There s a fairly large literature on this packing problem. If n is large and you can t quite get n circles in the
                Message 7 of 7 , Aug 7, 2003
                  I don't think it's quite as simple. There's a fairly
                  large literature on this packing problem.

                  If n is large and you can't quite get n circles
                  in the bottom row, then you can get more than

                  (n-1) + (n-2) + ... + 2 + 1

                  by spacing the bottom row so that the 2nd, 3rd, ...
                  rows don't need to be quite so high. Eventually,
                  there'll be room for more circles at the top.

                  See, for example R L Graham & B D Lubachevsky, Dense
                  packings of equal disks in an equilateral triangle:
                  from 22 to 34 and beyond, {\it Electron. J. Combin.},
                  {\bf2}(1995) Article 1, 39pp; MR 96a:52027. R.

                  On Thu, 7 Aug 2003, Nikolaos Dergiades wrote:

                  > Dear all,
                  >
                  > [Dave]
                  > >>> > Given an equilateral triangle with a base of X, what
                  > >>> > is the maximum number(Z) of circles of radius Y that
                  > >>> > can be inscribed within the boundaries of the
                  > >>> > triangle?
                  >
                  >
                  > [Michael]
                  > >>> Either I don't understand the question or it is trivial:
                  >
                  > [Darij]
                  > >I think he means: how many circles can be inscribed
                  > >simultaneously without overlapping each other?
                  > >
                  > >I don't know the answer, but it must be an interesting
                  > >question!
                  >
                  >
                  > *******
                  > If Darij, is right then I think that if we could put
                  > n of these circles on the base of the triangle then
                  > 2Y(n -1)+2Y*sqrt(3) <= X < 2Yn+2Y*sqrt(3) or that
                  > n = [(X + 2Y - 2Y*sqrt(3))/2Y]
                  > where [ ] denotes integer part.
                  > Over these circles we could put n - 1 circles and so on.
                  > Hence the total maximum number of circles is Z = n(n+1)/2
                  > and n the number given above.
                  >
                  > Best regards
                  > Nikolaos Dergiades
                  >
                  >
                  >
                  >
                  >
                  > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
                  >
                  >
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