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Re: Cute dice problem

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  • woodhodgson@xtra.co.nz
    ... values, ... work out the number of permutations for n=5,7,11,13,17,19,23 and add them up, and divide by 1296. ... It seems to me there is a very short
    Message 1 of 16 , May 16, 2013
      --- In primenumbers@yahoogroups.com, "John" wrote:
      >
      >
      >
      > --- In primenumbers@yahoogroups.com, "djbroadhurst" d.broadhurst@
      wrote:
      > >
      > >
      > >
      > > --- In primenumbers@yahoogroups.com,
      > > Chris Caldwell wrote:
      > >
      > > > if you roll four standard 6-sided dice and add up the resulting
      values,
      > > > what is the probability of rolling prime number?
      > >
      > > Help! I did this a dumb way, but got a simple answer:
      > > (4+20+104+140+104+56+4)/6^4 = 1/3
      > > Is there a smarter way, please?
      > >
      > > David
      > >
      >
      > Not all that dumb in my book. In fact, it is the most logical. Just
      work out the number of permutations for n=5,7,11,13,17,19,23 and add
      them up, and divide by 1296.
      >

      It seems to me there is a very short proof available. The prime numbers
      in the range (4,24) are precisely those equivalent to 1 or 5, modulo 6.

      Now it can very easily be shown by mathematical induction (**) that the
      6^N values (modulo 6) obtained by adding N numbers selected "one from
      each" of N copies of the 6-moduli {0,1,2,3,4,5} are represented the same
      number of times {i.e. 6^(N-1) times}.

      It then follows trivially that the required probability is 2/6 = 1/3.

      I have only run this "through my head", but it seems correct to me.


      (**) I suspect it is a trivial consequence of some algebraic or
      number-theoretic theorem.



      [Non-text portions of this message have been removed]
    • woodhodgson@xtra.co.nz
      ... I have worded part of that clumsily and possibly misleadingly: it should read ... the set of 6^N results (modulo 6) obtained by adding N numbers
      Message 2 of 16 , May 16, 2013
        --- In primenumbers@yahoogroups.com, "woodhodgson@..." <rupert.weather@...> wrote:
        >
        >
        >
        > --- In primenumbers@yahoogroups.com, "John" wrote:
        > >
        > >
        > >
        > > --- In primenumbers@yahoogroups.com, "djbroadhurst" d.broadhurst@
        > wrote:
        > > >
        > > >
        > > >
        > > > --- In primenumbers@yahoogroups.com,
        > > > Chris Caldwell wrote:
        > > >
        > > > > if you roll four standard 6-sided dice and add up the resulting
        > values,
        > > > > what is the probability of rolling prime number?
        > > >
        > > > Help! I did this a dumb way, but got a simple answer:
        > > > (4+20+104+140+104+56+4)/6^4 = 1/3
        > > > Is there a smarter way, please?
        > > >
        > > > David
        > > >
        > >
        > > Not all that dumb in my book. In fact, it is the most logical. Just
        > work out the number of permutations for n=5,7,11,13,17,19,23 and add
        > them up, and divide by 1296.
        > >
        >
        > It seems to me there is a very short proof available. The prime numbers
        > in the range (4,24) are precisely those equivalent to 1 or 5, modulo 6.
        >
        > Now it can very easily be shown by mathematical induction (**) that the
        > 6^N values (modulo 6) obtained by adding N numbers selected "one from
        > each" of N copies of the 6-moduli {0,1,2,3,4,5} are represented the same
        > number of times {i.e. 6^(N-1) times}.
        >
        > It then follows trivially that the required probability is 2/6 = 1/3.
        >
        > I have only run this "through my head", but it seems correct to me.
        >
        >
        > (**) I suspect it is a trivial consequence of some algebraic or
        > number-theoretic theorem.
        >
        >
        >
        > [Non-text portions of this message have been removed]
        >

        I have worded part of that clumsily and possibly misleadingly: it should read " ... the set of 6^N results (modulo 6) obtained by adding N numbers selected "one from each" of N copies of the 6-moduli {0,1,2,3,4,5} contains each of the values 0,1,2,3,4,5 the same number of times {i.e. 6^(N-1) times}."
        >
      • djbroadhurst
        ... Indeed: http://tech.groups.yahoo.com/group/primenumbers/message/25015 David
        Message 3 of 16 , May 16, 2013
          --- In primenumbers@yahoogroups.com,
          "woodhodgson@..." <rupert.weather@...> wrote:

          > It seems to me there is a very short proof available ....
          > by mathematical induction ....

          Indeed:
          http://tech.groups.yahoo.com/group/primenumbers/message/25015

          David
        • John
          ... In this little interesting problem, David s dumb answer was the dumbness of ordinary arithmetic compared with the elegance of modular arithmetic and
          Message 4 of 16 , May 19, 2013
            --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
            >
            > --- In primenumbers@yahoogroups.com,
            > "woodhodgson@" <rupert.weather@> wrote:
            >
            > > It seems to me there is a very short proof available ....
            > > by mathematical induction ....
            >
            > Indeed:
            > http://tech.groups.yahoo.com/group/primenumbers/message/25015
            >
            > David
            >

            In this little interesting problem, David's "dumb" answer was the dumbness of ordinary arithmetic compared with the elegance of modular arithmetic and induction.

            But, though both give the right answer, I have to "dip the lid" to elegance, even if it requires a bit of effort to bring it about.
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