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Cute dice problem

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  • Chris Caldwell
    I ran across a version of the following (with fewer dice) on a middle school math contest (so we want a quick easy answer): if you roll four standard 6-sided
    Message 1 of 16 , Apr 5, 2013
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      I ran across a version of the following (with fewer dice) on a middle school math contest (so we want a quick easy answer):

      if you roll four standard 6-sided dice and add up the resulting values, what is the probability of rolling prime number?


      [Non-text portions of this message have been removed]
    • djbroadhurst
      ... 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
      Message 2 of 16 , Apr 5, 2013
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        --- In primenumbers@yahoogroups.com,
        Chris Caldwell <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
      • Jack Brennen
        Since the range of sums is 4 to 24, the sum is prime if and only if it is coprime to 6 -- we very conveniently dodge the largest prime which is not coprime to
        Message 3 of 16 , Apr 5, 2013
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          Since the range of sums is 4 to 24, the sum is prime if and
          only if it is coprime to 6 -- we very conveniently dodge
          the largest prime which is not coprime to 6 (3) and the
          smallest composite which is coprime to 6 (25).

          Once you have that insight, it's pretty easy.



          On 4/5/2013 9:04 AM, Chris Caldwell wrote:
          > I ran across a version of the following (with fewer dice) on a middle school math contest (so we want a quick easy answer):
          >
          > if you roll four standard 6-sided dice and add up the resulting values, what is the probability of rolling prime number?
          >
          >
          > [Non-text portions of this message have been removed]
          >
          >
          >
          > ------------------------------------
          >
          > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
          > The Prime Pages : http://primes.utm.edu/
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          >
        • djbroadhurst
          ... So the sides can be taken as {-2,-1,0,1,2,3} and we want a total of 0, from 4 throws. ... I still had to some work to get the very simple answer. David
          Message 4 of 16 , Apr 5, 2013
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            --- In primenumbers@yahoogroups.com,
            Jack Brennen <jfb@...> wrote:

            > Since the range of sums is 4 to 24, the sum is prime if and
            > only if it is coprime to 6

            So the sides can be taken as {-2,-1,0,1,2,3}
            and we want a total of 0, from 4 throws.

            > Once you have that insight, it's pretty easy.

            I still had to some work to get the very simple answer.

            David
          • Jack Brennen
            Throw the first three dice and figure your sum. How many sides of the fourth die will give you a sum congruent to 1 or 5 modulo 6?
            Message 5 of 16 , Apr 5, 2013
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              Throw the first three dice and figure your sum.

              How many sides of the fourth die will give you a sum congruent
              to 1 or 5 modulo 6?

              On 4/5/2013 10:11 AM, djbroadhurst wrote:
              >
              >
              > --- In primenumbers@yahoogroups.com,
              > Jack Brennen <jfb@...> wrote:
              >
              >> Since the range of sums is 4 to 24, the sum is prime if and
              >> only if it is coprime to 6
              >
              > So the sides can be taken as {-2,-1,0,1,2,3}
              > and we want a total of 0, from 4 throws.
              >
              >> Once you have that insight, it's pretty easy.
              >
              > I still had to some work to get the very simple answer.
              >
              > David
              >
              >
              >
              > ------------------------------------
              >
              > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
              > The Prime Pages : http://primes.utm.edu/
              >
              > Yahoo! Groups Links
              >
              >
              >
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              >
            • Chris Caldwell
              From: Jack Brennen ... Yes! So four dice is a trivial problem! Sadly five does not divide six, so five dice is much harder (it is 137/432 for five dice and
              Message 6 of 16 , Apr 5, 2013
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                From: Jack Brennen
                > Throw the first three dice and figure your sum.
                > How many sides of the fourth die will give you a sum congruent to 1 or 5 modulo 6?

                Yes! So four dice is a trivial problem! Sadly five does not divide six, so five dice is much harder (it is 137/432 for five dice and 235/864 for six). The possibility of rolling 2 or 3 makes 2 or 3 dice a little harder. But the 4 dice case can be done instantly.

                But why would a middle school child recognize that? Surely they did not want a long calculation. (One of my colleagues said generating functions give you the answer very quickly--definitely not a middle school solution!)

                CC

                -----Original Message-----
                From: primenumbers@yahoogroups.com [mailto:primenumbers@yahoogroups.com] On Behalf Of Jack Brennen
                Sent: Friday, April 05, 2013 12:22 PM
                To: djbroadhurst
                Cc: primenumbers@yahoogroups.com
                Subject: Re: [PrimeNumbers] Re: Cute dice problem

                Throw the first three dice and figure your sum.

                How many sides of the fourth die will give you a sum congruent to 1 or 5 modulo 6?

                On 4/5/2013 10:11 AM, djbroadhurst wrote:
                >
                >
                > --- In primenumbers@yahoogroups.com,
                > Jack Brennen <jfb@...> wrote:
                >
                >> Since the range of sums is 4 to 24, the sum is prime if and only if
                >> it is coprime to 6
                >
                > So the sides can be taken as {-2,-1,0,1,2,3} and we want a total of 0,
                > from 4 throws.
                >
                >> Once you have that insight, it's pretty easy.
                >
                > I still had to some work to get the very simple answer.
                >
                > David
                >
                >
                >
                > ------------------------------------
                >
                > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                > The Prime Pages : http://primes.utm.edu/
                >
                > Yahoo! Groups Links
                >
                >
                >
                >
                >



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              • djbroadhurst
                ... Jack s first three seems to be a red herring. Solution: First n give equipartition mod 6, by induction. So the probability of (1 or 5) mod 6 is /always/
                Message 7 of 16 , Apr 5, 2013
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                  --- In primenumbers@yahoogroups.com,
                  Jack Brennen <jfb@...> wrote:

                  > Throw the first three dice and figure your sum.

                  Jack's "first three" seems to be a red herring.

                  Solution: First n give equipartition mod 6, by induction.
                  So the probability of (1 or 5) mod 6 is /always/ 1/3,
                  no matter how may dice you throw.

                  Doh!

                  David (not taught induction until high school)
                • Jack Brennen
                  ... Sort of a red herring. You could replace throw the first three dice and figure your sum with pick any integer . :) ... Most wouldn t recognize it. I d
                  Message 8 of 16 , Apr 5, 2013
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                    On 4/5/2013 11:20 AM, djbroadhurst wrote:
                    >
                    > --- In primenumbers@yahoogroups.com,
                    > Jack Brennen <jfb@...> wrote:
                    >
                    >> Throw the first three dice and figure your sum.
                    >
                    > Jack's "first three" seems to be a red herring.
                    >

                    Sort of a red herring.

                    You could replace "throw the first three dice and figure your sum"
                    with "pick any integer". :)


                    On 4/5/2013 10:28 AM, Chris Caldwell wrote:
                    >
                    > But why would a middle school child recognize that? Surely they did
                    > not want a long calculation. (One of my colleagues said generating
                    > functions give you the answer very quickly--definitely not a middle
                    > school solution!)

                    Most wouldn't recognize it. I'd like to think I would have recognized
                    it at that age, but I can't be sure. In seventh grade, I scored a 105
                    out of 150 on the contest that is now called the AMC-12, which was a
                    pretty decent score. If you believe what you read online, Noam Elkies,
                    who is three months younger than me, scored a 112 on that test, and then
                    he proceeded to beat my math competition scores routinely over the next
                    three years.

                    All of which is to say... a decent math competition should have some
                    problems which are well beyond the capabilities of the average
                    competitor. In ninth grade, I think I was a national co-champion of
                    the Continental Math League competition, because I didn't get a single
                    question wrong all year. It wasn't a particularly satisfying result.
                    That league was fairly new at the time, and I hope they learned that
                    you need to put some "stretch" questions in with the routine ones.
                  • Chris Caldwell
                    I also enjoyed that test (the ACM-12) as a child, but of course that test is for students 4 years older than this test for Tennessee youth (only) is. Also
                    Message 9 of 16 , Apr 5, 2013
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                      I also enjoyed that test (the ACM-12) as a child, but of course that test is for students 4 years older than this test for Tennessee youth (only) is. Also this test is supposed to be limited to the state standards, so should not be the level of the ACM-8 either . But you are right in that they are designed to not allow the dreaded perfect scores (or even tied high values).

                      -----Original Message-----
                      From: Jack Brennen [mailto:jfb@...]
                      Sent: Friday, April 05, 2013 2:30 PM
                      To: primenumbers@yahoogroups.com
                      Cc: djbroadhurst; Chris Caldwell
                      Subject: Re: [PrimeNumbers] Re: Cute dice problem [by induction]

                      On 4/5/2013 11:20 AM, djbroadhurst wrote:
                      >
                      > --- In primenumbers@yahoogroups.com,
                      > Jack Brennen <jfb@...> wrote:
                      >
                      >> Throw the first three dice and figure your sum.
                      >
                      > Jack's "first three" seems to be a red herring.
                      >

                      Sort of a red herring.

                      You could replace "throw the first three dice and figure your sum"
                      with "pick any integer". :)


                      On 4/5/2013 10:28 AM, Chris Caldwell wrote:
                      >
                      > But why would a middle school child recognize that? Surely they did > not want a long calculation. (One of my colleagues said generating > functions give you the answer very quickly--definitely not a middle > school solution!)

                      Most wouldn't recognize it. I'd like to think I would have recognized it at that age, but I can't be sure. In seventh grade, I scored a 105 out of 150 on the contest that is now called the AMC-12, which was a pretty decent score. If you believe what you read online, Noam Elkies, who is three months younger than me, scored a 112 on that test, and then he proceeded to beat my math competition scores routinely over the next three years.

                      All of which is to say... a decent math competition should have some problems which are well beyond the capabilities of the average competitor. In ninth grade, I think I was a national co-champion of the Continental Math League competition, because I didn't get a single question wrong all year. It wasn't a particularly satisfying result.
                      That league was fairly new at the time, and I hope they learned that you need to put some "stretch" questions in with the routine ones.
                    • Jack Brennen
                      The math required to understand the easy solution to the dice problem is easily within the realm of 8th grade mathematics. The insight required to come up
                      Message 10 of 16 , Apr 5, 2013
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                        The math required to understand the "easy" solution to the dice problem
                        is easily within the realm of 8th grade mathematics.

                        The insight required to come up with the easy solution, while under time
                        pressure... That's another issue entirely. To get to the insight
                        requires recognizing that for 4 dice (and for 4 dice only!), you can
                        replace "prime" with "coprime with 6". Transforming a problem into
                        an equivalent easier problem -- that's something that is really hard
                        to master. (And almost certainly something that most 8th graders have
                        not been taught in school.)



                        On 4/5/2013 12:52 PM, Chris Caldwell wrote:
                        > I also enjoyed that test (the ACM-12) as a child, but of course that
                        > test is for students 4 years older than this test for Tennessee youth
                        > (only) is. Also this test is supposed to be limited to the state
                        > standards, so should not be the level of the ACM-8 either . But you
                        > are right in that they are designed to not allow the dreaded perfect
                        > scores (or even tied high values).
                        >
                      • John
                        ... 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
                        Message 11 of 16 , Apr 15, 2013
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                          --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
                          >
                          >
                          >
                          > --- In primenumbers@yahoogroups.com,
                          > Chris Caldwell <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.
                        • Jack Brennen
                          Show the values that you can have after rolling the first 3 dice, and then for each of those values, show how many primes can be reached with the last die:
                          Message 12 of 16 , Apr 15, 2013
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                            Show the values that you can have after rolling the first 3 dice, and
                            then for each of those values, show how many primes can be reached with
                            the last die:

                            First three == 3: 2 primes (5 and 7)
                            First three == 4: 2 primes (5 and 7)
                            First three == 5: 2 primes (7 and 11)
                            First three == 6: 2 primes (7 and 11)
                            First three == 7: 2 primes (11 and 13)
                            First three == 8: 2 primes (11 and 13)
                            First three == 9: 2 primes (11 and 13)
                            First three == 10: 2 primes (11 and 13)
                            First three == 11: 2 primes (13 and 17)
                            First three == 12: 2 primes (13 and 17)
                            First three == 13: 2 primes (17 and 19)
                            First three == 14: 2 primes (17 and 19)
                            First three == 15: 2 primes (17 and 19)
                            First three == 16: 2 primes (17 and 19)
                            First three == 17: 2 primes (19 and 23)
                            First three == 18: 2 primes (19 and 23)

                            The rest is left as an exercise for the reader in use of the
                            distributive property.



                            On 4/15/2013 9:30 AM, John wrote:
                            >
                            >
                            > --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
                            >>
                            >>
                            >>
                            >> --- In primenumbers@yahoogroups.com,
                            >> Chris Caldwell <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.
                            >
                            >
                            >
                            > ------------------------------------
                            >
                            > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                            > The Prime Pages : http://primes.utm.edu/
                            >
                            > Yahoo! Groups Links
                            >
                            >
                            >
                            >
                            >
                          • 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 13 of 16 , May 16, 2013
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                              --- 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 14 of 16 , May 16, 2013
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                                --- 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 15 of 16 , May 16, 2013
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                                  --- 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 16 of 16 , May 19, 2013
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                                    --- 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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