How many smurfs are there?

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Papa Smurf gathered some berries from the forest to the village and smurfs ate all those berries:



  • The smurf who ate the most actually ate one-fourth of the berries eaten by the rest of the smurfs.

  • The smurf who ate the third-most actually ate one-ninth of the berries eaten by the rest of the smurfs

  • The smurf who ate the least actually ate one-tenth of the berries eaten by the rest of the smurfs.


How many smurfs are there?




Reference: Bilim ve Teknik Dergisi 2018-08




mathematics logical-deduction




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    up vote
    7
    down vote

    favorite
    1












    Papa Smurf gathered some berries from the forest to the village and smurfs ate all those berries:



    • The smurf who ate the most actually ate one-fourth of the berries eaten by the rest of the smurfs.

    • The smurf who ate the third-most actually ate one-ninth of the berries eaten by the rest of the smurfs

    • The smurf who ate the least actually ate one-tenth of the berries eaten by the rest of the smurfs.


    How many smurfs are there?




    Reference: Bilim ve Teknik Dergisi 2018-08




    mathematics logical-deduction




    share|improve this question





















      up vote
      7
      down vote

      favorite
      1









      up vote
      7
      down vote

      favorite
      1






      1





      Papa Smurf gathered some berries from the forest to the village and smurfs ate all those berries:



      • The smurf who ate the most actually ate one-fourth of the berries eaten by the rest of the smurfs.

      • The smurf who ate the third-most actually ate one-ninth of the berries eaten by the rest of the smurfs

      • The smurf who ate the least actually ate one-tenth of the berries eaten by the rest of the smurfs.


      How many smurfs are there?




      Reference: Bilim ve Teknik Dergisi 2018-08




      mathematics logical-deduction




      share|improve this question











      Papa Smurf gathered some berries from the forest to the village and smurfs ate all those berries:



      • The smurf who ate the most actually ate one-fourth of the berries eaten by the rest of the smurfs.

      • The smurf who ate the third-most actually ate one-ninth of the berries eaten by the rest of the smurfs

      • The smurf who ate the least actually ate one-tenth of the berries eaten by the rest of the smurfs.


      How many smurfs are there?




      Reference: Bilim ve Teknik Dergisi 2018-08





      mathematics logical-deduction





      share|improve this question










      share|improve this question




      share|improve this question









      asked 11 hours ago









      Oray

      13k433129




      13k433129




















          2 Answers
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          up vote
          11
          down vote



          accepted










          Let $N$ be the number of smurfs.



          The information we are given implies that one smurf ate $1/5$ of the total berries, and that each of the other $N-1$ smurfs ate at least $1/11$ of the total berries. This means we must have
          $$
          frac15+(N-1)frac111 leq 1,
          $$
          and so $Nleq 9$.



          The the smurf who ate the second-most at strictly less than $frac15$ of the total, and each of the $N-2$ smurfs (other than those who at the most and second most) ate at most $frac110$ of the total. This means we must have
          $$
          frac15+frac15+(N-2)frac110>1,
          $$
          and so $Ngeq 9$.



          Taken together, these bounds imply there must be $9$ smurfs.






          share|improve this answer

















          • 1




            Curious, why 1/5?
            – mascoj
            6 hours ago






          • 2




            Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5.
            – jamisans
            4 hours ago

















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          I have an answer that works, but no proof of uniqueness yet.




          There are 9 smurfs. One way this works is for there to be 1100 berries. The number of berries that each smurf eats is 220, 135, 110, 109, 108, 107, 106, 105, 100. The numbers sum to 1100. The greatest value, 220, is one-fourth of the remaining sum (1100-220=880). The third greatest value, 110, is one-ninth of the remaining sum (1100-110=990). The smallest value, 100, is one-tenth of the remaining sum (1100-100=1000).







          share|improve this answer





















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            2 Answers
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            2 Answers
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            active

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            up vote
            11
            down vote



            accepted










            Let $N$ be the number of smurfs.



            The information we are given implies that one smurf ate $1/5$ of the total berries, and that each of the other $N-1$ smurfs ate at least $1/11$ of the total berries. This means we must have
            $$
            frac15+(N-1)frac111 leq 1,
            $$
            and so $Nleq 9$.



            The the smurf who ate the second-most at strictly less than $frac15$ of the total, and each of the $N-2$ smurfs (other than those who at the most and second most) ate at most $frac110$ of the total. This means we must have
            $$
            frac15+frac15+(N-2)frac110>1,
            $$
            and so $Ngeq 9$.



            Taken together, these bounds imply there must be $9$ smurfs.






            share|improve this answer

















            • 1




              Curious, why 1/5?
              – mascoj
              6 hours ago






            • 2




              Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5.
              – jamisans
              4 hours ago














            up vote
            11
            down vote



            accepted










            Let $N$ be the number of smurfs.



            The information we are given implies that one smurf ate $1/5$ of the total berries, and that each of the other $N-1$ smurfs ate at least $1/11$ of the total berries. This means we must have
            $$
            frac15+(N-1)frac111 leq 1,
            $$
            and so $Nleq 9$.



            The the smurf who ate the second-most at strictly less than $frac15$ of the total, and each of the $N-2$ smurfs (other than those who at the most and second most) ate at most $frac110$ of the total. This means we must have
            $$
            frac15+frac15+(N-2)frac110>1,
            $$
            and so $Ngeq 9$.



            Taken together, these bounds imply there must be $9$ smurfs.






            share|improve this answer

















            • 1




              Curious, why 1/5?
              – mascoj
              6 hours ago






            • 2




              Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5.
              – jamisans
              4 hours ago












            up vote
            11
            down vote



            accepted







            up vote
            11
            down vote



            accepted






            Let $N$ be the number of smurfs.



            The information we are given implies that one smurf ate $1/5$ of the total berries, and that each of the other $N-1$ smurfs ate at least $1/11$ of the total berries. This means we must have
            $$
            frac15+(N-1)frac111 leq 1,
            $$
            and so $Nleq 9$.



            The the smurf who ate the second-most at strictly less than $frac15$ of the total, and each of the $N-2$ smurfs (other than those who at the most and second most) ate at most $frac110$ of the total. This means we must have
            $$
            frac15+frac15+(N-2)frac110>1,
            $$
            and so $Ngeq 9$.



            Taken together, these bounds imply there must be $9$ smurfs.






            share|improve this answer













            Let $N$ be the number of smurfs.



            The information we are given implies that one smurf ate $1/5$ of the total berries, and that each of the other $N-1$ smurfs ate at least $1/11$ of the total berries. This means we must have
            $$
            frac15+(N-1)frac111 leq 1,
            $$
            and so $Nleq 9$.



            The the smurf who ate the second-most at strictly less than $frac15$ of the total, and each of the $N-2$ smurfs (other than those who at the most and second most) ate at most $frac110$ of the total. This means we must have
            $$
            frac15+frac15+(N-2)frac110>1,
            $$
            and so $Ngeq 9$.



            Taken together, these bounds imply there must be $9$ smurfs.







            share|improve this answer













            share|improve this answer



            share|improve this answer











            answered 10 hours ago









            Julian Rosen

            11.7k13879




            11.7k13879







            • 1




              Curious, why 1/5?
              – mascoj
              6 hours ago






            • 2




              Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5.
              – jamisans
              4 hours ago












            • 1




              Curious, why 1/5?
              – mascoj
              6 hours ago






            • 2




              Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5.
              – jamisans
              4 hours ago







            1




            1




            Curious, why 1/5?
            – mascoj
            6 hours ago




            Curious, why 1/5?
            – mascoj
            6 hours ago




            2




            2




            Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5.
            – jamisans
            4 hours ago




            Let T be the total number of berries and S be number that the hungriest smurf ate. From the first clue we know that S = .25(T-S). Solving for S gives T/5.
            – jamisans
            4 hours ago










            up vote
            5
            down vote













            I have an answer that works, but no proof of uniqueness yet.




            There are 9 smurfs. One way this works is for there to be 1100 berries. The number of berries that each smurf eats is 220, 135, 110, 109, 108, 107, 106, 105, 100. The numbers sum to 1100. The greatest value, 220, is one-fourth of the remaining sum (1100-220=880). The third greatest value, 110, is one-ninth of the remaining sum (1100-110=990). The smallest value, 100, is one-tenth of the remaining sum (1100-100=1000).







            share|improve this answer

























              up vote
              5
              down vote













              I have an answer that works, but no proof of uniqueness yet.




              There are 9 smurfs. One way this works is for there to be 1100 berries. The number of berries that each smurf eats is 220, 135, 110, 109, 108, 107, 106, 105, 100. The numbers sum to 1100. The greatest value, 220, is one-fourth of the remaining sum (1100-220=880). The third greatest value, 110, is one-ninth of the remaining sum (1100-110=990). The smallest value, 100, is one-tenth of the remaining sum (1100-100=1000).







              share|improve this answer























                up vote
                5
                down vote










                up vote
                5
                down vote









                I have an answer that works, but no proof of uniqueness yet.




                There are 9 smurfs. One way this works is for there to be 1100 berries. The number of berries that each smurf eats is 220, 135, 110, 109, 108, 107, 106, 105, 100. The numbers sum to 1100. The greatest value, 220, is one-fourth of the remaining sum (1100-220=880). The third greatest value, 110, is one-ninth of the remaining sum (1100-110=990). The smallest value, 100, is one-tenth of the remaining sum (1100-100=1000).







                share|improve this answer













                I have an answer that works, but no proof of uniqueness yet.




                There are 9 smurfs. One way this works is for there to be 1100 berries. The number of berries that each smurf eats is 220, 135, 110, 109, 108, 107, 106, 105, 100. The numbers sum to 1100. The greatest value, 220, is one-fourth of the remaining sum (1100-220=880). The third greatest value, 110, is one-ninth of the remaining sum (1100-110=990). The smallest value, 100, is one-tenth of the remaining sum (1100-100=1000).








                share|improve this answer













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                answered 11 hours ago









                jamisans

                463111




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