What is the total numbers of chocolates belonging to three of them?












0












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Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates, Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than
Zunayed. What is the total numbers of chocolates belonging to three of them?




Source: Bangladesh Math Olympiad 2018 junior category.



I can not understand the question properly. "Zunayed has more chocolates"- does it mean he has more chocolate than both Pial and Bindu or only Pial. Is there any valid solution of this question?






number-theory elementary-number-theory contest-math puzzle







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  • 2




    $begingroup$
    In my opinion, it means "Zunayed has more chocolates than Pial."
    $endgroup$
    – angryavian
    Jan 12 at 6:47






  • 1




    $begingroup$
    But don't you think that this makes the puzzle with unlimited values. Like if Pial have 20, Zunayed has 30 and Bindu has 60 the answer will be correct if any values for Pial < Zunayed < Bindu
    $endgroup$
    – Shromi
    Jan 12 at 6:50
















0












$begingroup$



Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates, Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than
Zunayed. What is the total numbers of chocolates belonging to three of them?




Source: Bangladesh Math Olympiad 2018 junior category.



I can not understand the question properly. "Zunayed has more chocolates"- does it mean he has more chocolate than both Pial and Bindu or only Pial. Is there any valid solution of this question?






number-theory elementary-number-theory contest-math puzzle







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    In my opinion, it means "Zunayed has more chocolates than Pial."
    $endgroup$
    – angryavian
    Jan 12 at 6:47






  • 1




    $begingroup$
    But don't you think that this makes the puzzle with unlimited values. Like if Pial have 20, Zunayed has 30 and Bindu has 60 the answer will be correct if any values for Pial < Zunayed < Bindu
    $endgroup$
    – Shromi
    Jan 12 at 6:50














0












0








0





$begingroup$



Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates, Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than
Zunayed. What is the total numbers of chocolates belonging to three of them?




Source: Bangladesh Math Olympiad 2018 junior category.



I can not understand the question properly. "Zunayed has more chocolates"- does it mean he has more chocolate than both Pial and Bindu or only Pial. Is there any valid solution of this question?






number-theory elementary-number-theory contest-math puzzle







share|cite|improve this question











$endgroup$





Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates, Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than
Zunayed. What is the total numbers of chocolates belonging to three of them?




Source: Bangladesh Math Olympiad 2018 junior category.



I can not understand the question properly. "Zunayed has more chocolates"- does it mean he has more chocolate than both Pial and Bindu or only Pial. Is there any valid solution of this question?






number-theory elementary-number-theory contest-math puzzle




number-theory elementary-number-theory contest-math puzzle






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 12 at 6:51







Shromi

















asked Jan 12 at 6:45









ShromiShromi

3289




3289








  • 2




    $begingroup$
    In my opinion, it means "Zunayed has more chocolates than Pial."
    $endgroup$
    – angryavian
    Jan 12 at 6:47






  • 1




    $begingroup$
    But don't you think that this makes the puzzle with unlimited values. Like if Pial have 20, Zunayed has 30 and Bindu has 60 the answer will be correct if any values for Pial < Zunayed < Bindu
    $endgroup$
    – Shromi
    Jan 12 at 6:50














  • 2




    $begingroup$
    In my opinion, it means "Zunayed has more chocolates than Pial."
    $endgroup$
    – angryavian
    Jan 12 at 6:47






  • 1




    $begingroup$
    But don't you think that this makes the puzzle with unlimited values. Like if Pial have 20, Zunayed has 30 and Bindu has 60 the answer will be correct if any values for Pial < Zunayed < Bindu
    $endgroup$
    – Shromi
    Jan 12 at 6:50








2




2




$begingroup$
In my opinion, it means "Zunayed has more chocolates than Pial."
$endgroup$
– angryavian
Jan 12 at 6:47




$begingroup$
In my opinion, it means "Zunayed has more chocolates than Pial."
$endgroup$
– angryavian
Jan 12 at 6:47




1




1




$begingroup$
But don't you think that this makes the puzzle with unlimited values. Like if Pial have 20, Zunayed has 30 and Bindu has 60 the answer will be correct if any values for Pial < Zunayed < Bindu
$endgroup$
– Shromi
Jan 12 at 6:50




$begingroup$
But don't you think that this makes the puzzle with unlimited values. Like if Pial have 20, Zunayed has 30 and Bindu has 60 the answer will be correct if any values for Pial < Zunayed < Bindu
$endgroup$
– Shromi
Jan 12 at 6:50










2 Answers
2






active

oldest

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1












$begingroup$

Assuming Zunayed has more chocolates than Pial and at the end it is not required to have $P<Z<B$, but only the intermediate steps. (Pial is not to blame if Zunayed decided to give away chocolates to Bindu).




Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates (than Pial).




Can Zunayed have 3 chocolates? No, because Pial can give 9 chocolates so that $Z=12>11=P$. (According to the condition, Pial must give 10 and keep $color{red}{10}$).



Can Zunayed have 2 chocolates? Yes, Zunayed has $color{red}{2}$ chocolates.



Can Zunayed have 1 chocolate? No, because Zunayed gets 10 from Pial to have 11 in total, however, Zunayed can not give half of it to Bindu (it is assumed a chocolate is indivisible).




Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than Zunayed.




First note that Zunayed has 12 chocolates after receiving 10 from Pial.



Can Bindu have 3 chocolates? No, because Zunayed has 12 chocolates and Zunayed can give 5 to Bindu, because $B=8>7=Z$. (According to the condition, Zunayed must give 6).



Can Bindu have 2 or 1 chocolates? Yes, Bindu has $color{red}{2}$ or $color{red}{1}$ chocolates.



Hence, the total number of chocolates is $color{red}{24}$ or $color{red}{23}$. That is:
$$P=20,Z=2,B=2 text{or} P=20,Z=2,B=1$$
in the beginning.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I am sorry, but don't you think that only Pial have 20 chocolates, then how the total number of chocolates are 14 or 13?
    $endgroup$
    – Shromi
    Jan 12 at 12:29






  • 1




    $begingroup$
    You are right, I mistakenly took $P=10$, fixed.
    $endgroup$
    – farruhota
    Jan 12 at 12:33










  • $begingroup$
    Could you please explain why Zynayed's number of chocolates is not more than Pial? Actually I am struggling at that part.
    $endgroup$
    – Shromi
    Jan 12 at 13:36






  • 1




    $begingroup$
    It states "Pial gives half of his so that Zynayed has more than Pial" implying initially Pial had more than Zynayed. Similarly, once Zynayed gets chocolates from Pial, Zynayed will have more than Bindu, therefore Zynayed is giving to Bindu. So, there are several assumptions I am using. Even these restrictive assumptions does not produce unique solution.
    $endgroup$
    – farruhota
    Jan 12 at 14:08






  • 1




    $begingroup$
    matholympiad.org.bd/resources/all-questions/category/… Just download the zip file, extract it and then in set-01 you will find this question in page 03. In every question it is translated to English from Bangla (not Hindi!).
    $endgroup$
    – Shromi
    Jan 13 at 3:44





















0












$begingroup$

I read it to mean that after the first transfer Zunayed has more than Pial has left.
In that case the problem is not well posed. As long as Zunayed starts with a positive even number of chocolates and Bindu starts with at least one the transfers will satisfy the requirements. After the first Pial has $10$ and Zunayed has however many he started with plus $10$, so more than Pial. After the second transfer Bindu has more than Zunayed by the number he started with.






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    2 Answers
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    2 Answers
    2






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    1












    $begingroup$

    Assuming Zunayed has more chocolates than Pial and at the end it is not required to have $P<Z<B$, but only the intermediate steps. (Pial is not to blame if Zunayed decided to give away chocolates to Bindu).




    Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates (than Pial).




    Can Zunayed have 3 chocolates? No, because Pial can give 9 chocolates so that $Z=12>11=P$. (According to the condition, Pial must give 10 and keep $color{red}{10}$).



    Can Zunayed have 2 chocolates? Yes, Zunayed has $color{red}{2}$ chocolates.



    Can Zunayed have 1 chocolate? No, because Zunayed gets 10 from Pial to have 11 in total, however, Zunayed can not give half of it to Bindu (it is assumed a chocolate is indivisible).




    Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than Zunayed.




    First note that Zunayed has 12 chocolates after receiving 10 from Pial.



    Can Bindu have 3 chocolates? No, because Zunayed has 12 chocolates and Zunayed can give 5 to Bindu, because $B=8>7=Z$. (According to the condition, Zunayed must give 6).



    Can Bindu have 2 or 1 chocolates? Yes, Bindu has $color{red}{2}$ or $color{red}{1}$ chocolates.



    Hence, the total number of chocolates is $color{red}{24}$ or $color{red}{23}$. That is:
    $$P=20,Z=2,B=2 text{or} P=20,Z=2,B=1$$
    in the beginning.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      I am sorry, but don't you think that only Pial have 20 chocolates, then how the total number of chocolates are 14 or 13?
      $endgroup$
      – Shromi
      Jan 12 at 12:29






    • 1




      $begingroup$
      You are right, I mistakenly took $P=10$, fixed.
      $endgroup$
      – farruhota
      Jan 12 at 12:33










    • $begingroup$
      Could you please explain why Zynayed's number of chocolates is not more than Pial? Actually I am struggling at that part.
      $endgroup$
      – Shromi
      Jan 12 at 13:36






    • 1




      $begingroup$
      It states "Pial gives half of his so that Zynayed has more than Pial" implying initially Pial had more than Zynayed. Similarly, once Zynayed gets chocolates from Pial, Zynayed will have more than Bindu, therefore Zynayed is giving to Bindu. So, there are several assumptions I am using. Even these restrictive assumptions does not produce unique solution.
      $endgroup$
      – farruhota
      Jan 12 at 14:08






    • 1




      $begingroup$
      matholympiad.org.bd/resources/all-questions/category/… Just download the zip file, extract it and then in set-01 you will find this question in page 03. In every question it is translated to English from Bangla (not Hindi!).
      $endgroup$
      – Shromi
      Jan 13 at 3:44


















    1












    $begingroup$

    Assuming Zunayed has more chocolates than Pial and at the end it is not required to have $P<Z<B$, but only the intermediate steps. (Pial is not to blame if Zunayed decided to give away chocolates to Bindu).




    Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates (than Pial).




    Can Zunayed have 3 chocolates? No, because Pial can give 9 chocolates so that $Z=12>11=P$. (According to the condition, Pial must give 10 and keep $color{red}{10}$).



    Can Zunayed have 2 chocolates? Yes, Zunayed has $color{red}{2}$ chocolates.



    Can Zunayed have 1 chocolate? No, because Zunayed gets 10 from Pial to have 11 in total, however, Zunayed can not give half of it to Bindu (it is assumed a chocolate is indivisible).




    Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than Zunayed.




    First note that Zunayed has 12 chocolates after receiving 10 from Pial.



    Can Bindu have 3 chocolates? No, because Zunayed has 12 chocolates and Zunayed can give 5 to Bindu, because $B=8>7=Z$. (According to the condition, Zunayed must give 6).



    Can Bindu have 2 or 1 chocolates? Yes, Bindu has $color{red}{2}$ or $color{red}{1}$ chocolates.



    Hence, the total number of chocolates is $color{red}{24}$ or $color{red}{23}$. That is:
    $$P=20,Z=2,B=2 text{or} P=20,Z=2,B=1$$
    in the beginning.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      I am sorry, but don't you think that only Pial have 20 chocolates, then how the total number of chocolates are 14 or 13?
      $endgroup$
      – Shromi
      Jan 12 at 12:29






    • 1




      $begingroup$
      You are right, I mistakenly took $P=10$, fixed.
      $endgroup$
      – farruhota
      Jan 12 at 12:33










    • $begingroup$
      Could you please explain why Zynayed's number of chocolates is not more than Pial? Actually I am struggling at that part.
      $endgroup$
      – Shromi
      Jan 12 at 13:36






    • 1




      $begingroup$
      It states "Pial gives half of his so that Zynayed has more than Pial" implying initially Pial had more than Zynayed. Similarly, once Zynayed gets chocolates from Pial, Zynayed will have more than Bindu, therefore Zynayed is giving to Bindu. So, there are several assumptions I am using. Even these restrictive assumptions does not produce unique solution.
      $endgroup$
      – farruhota
      Jan 12 at 14:08






    • 1




      $begingroup$
      matholympiad.org.bd/resources/all-questions/category/… Just download the zip file, extract it and then in set-01 you will find this question in page 03. In every question it is translated to English from Bangla (not Hindi!).
      $endgroup$
      – Shromi
      Jan 13 at 3:44
















    1












    1








    1





    $begingroup$

    Assuming Zunayed has more chocolates than Pial and at the end it is not required to have $P<Z<B$, but only the intermediate steps. (Pial is not to blame if Zunayed decided to give away chocolates to Bindu).




    Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates (than Pial).




    Can Zunayed have 3 chocolates? No, because Pial can give 9 chocolates so that $Z=12>11=P$. (According to the condition, Pial must give 10 and keep $color{red}{10}$).



    Can Zunayed have 2 chocolates? Yes, Zunayed has $color{red}{2}$ chocolates.



    Can Zunayed have 1 chocolate? No, because Zunayed gets 10 from Pial to have 11 in total, however, Zunayed can not give half of it to Bindu (it is assumed a chocolate is indivisible).




    Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than Zunayed.




    First note that Zunayed has 12 chocolates after receiving 10 from Pial.



    Can Bindu have 3 chocolates? No, because Zunayed has 12 chocolates and Zunayed can give 5 to Bindu, because $B=8>7=Z$. (According to the condition, Zunayed must give 6).



    Can Bindu have 2 or 1 chocolates? Yes, Bindu has $color{red}{2}$ or $color{red}{1}$ chocolates.



    Hence, the total number of chocolates is $color{red}{24}$ or $color{red}{23}$. That is:
    $$P=20,Z=2,B=2 text{or} P=20,Z=2,B=1$$
    in the beginning.






    share|cite|improve this answer











    $endgroup$



    Assuming Zunayed has more chocolates than Pial and at the end it is not required to have $P<Z<B$, but only the intermediate steps. (Pial is not to blame if Zunayed decided to give away chocolates to Bindu).




    Pial gives half of his 20 chocolates to Zunayed so that Zunayed has more chocolates (than Pial).




    Can Zunayed have 3 chocolates? No, because Pial can give 9 chocolates so that $Z=12>11=P$. (According to the condition, Pial must give 10 and keep $color{red}{10}$).



    Can Zunayed have 2 chocolates? Yes, Zunayed has $color{red}{2}$ chocolates.



    Can Zunayed have 1 chocolate? No, because Zunayed gets 10 from Pial to have 11 in total, however, Zunayed can not give half of it to Bindu (it is assumed a chocolate is indivisible).




    Zunayed gives half of his chocolates to Bindu so that Bindu has more chocolates than Zunayed.




    First note that Zunayed has 12 chocolates after receiving 10 from Pial.



    Can Bindu have 3 chocolates? No, because Zunayed has 12 chocolates and Zunayed can give 5 to Bindu, because $B=8>7=Z$. (According to the condition, Zunayed must give 6).



    Can Bindu have 2 or 1 chocolates? Yes, Bindu has $color{red}{2}$ or $color{red}{1}$ chocolates.



    Hence, the total number of chocolates is $color{red}{24}$ or $color{red}{23}$. That is:
    $$P=20,Z=2,B=2 text{or} P=20,Z=2,B=1$$
    in the beginning.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Jan 12 at 12:32

























    answered Jan 12 at 11:20









    farruhotafarruhota

    22.1k2942




    22.1k2942












    • $begingroup$
      I am sorry, but don't you think that only Pial have 20 chocolates, then how the total number of chocolates are 14 or 13?
      $endgroup$
      – Shromi
      Jan 12 at 12:29






    • 1




      $begingroup$
      You are right, I mistakenly took $P=10$, fixed.
      $endgroup$
      – farruhota
      Jan 12 at 12:33










    • $begingroup$
      Could you please explain why Zynayed's number of chocolates is not more than Pial? Actually I am struggling at that part.
      $endgroup$
      – Shromi
      Jan 12 at 13:36






    • 1




      $begingroup$
      It states "Pial gives half of his so that Zynayed has more than Pial" implying initially Pial had more than Zynayed. Similarly, once Zynayed gets chocolates from Pial, Zynayed will have more than Bindu, therefore Zynayed is giving to Bindu. So, there are several assumptions I am using. Even these restrictive assumptions does not produce unique solution.
      $endgroup$
      – farruhota
      Jan 12 at 14:08






    • 1




      $begingroup$
      matholympiad.org.bd/resources/all-questions/category/… Just download the zip file, extract it and then in set-01 you will find this question in page 03. In every question it is translated to English from Bangla (not Hindi!).
      $endgroup$
      – Shromi
      Jan 13 at 3:44




















    • $begingroup$
      I am sorry, but don't you think that only Pial have 20 chocolates, then how the total number of chocolates are 14 or 13?
      $endgroup$
      – Shromi
      Jan 12 at 12:29






    • 1




      $begingroup$
      You are right, I mistakenly took $P=10$, fixed.
      $endgroup$
      – farruhota
      Jan 12 at 12:33










    • $begingroup$
      Could you please explain why Zynayed's number of chocolates is not more than Pial? Actually I am struggling at that part.
      $endgroup$
      – Shromi
      Jan 12 at 13:36






    • 1




      $begingroup$
      It states "Pial gives half of his so that Zynayed has more than Pial" implying initially Pial had more than Zynayed. Similarly, once Zynayed gets chocolates from Pial, Zynayed will have more than Bindu, therefore Zynayed is giving to Bindu. So, there are several assumptions I am using. Even these restrictive assumptions does not produce unique solution.
      $endgroup$
      – farruhota
      Jan 12 at 14:08






    • 1




      $begingroup$
      matholympiad.org.bd/resources/all-questions/category/… Just download the zip file, extract it and then in set-01 you will find this question in page 03. In every question it is translated to English from Bangla (not Hindi!).
      $endgroup$
      – Shromi
      Jan 13 at 3:44


















    $begingroup$
    I am sorry, but don't you think that only Pial have 20 chocolates, then how the total number of chocolates are 14 or 13?
    $endgroup$
    – Shromi
    Jan 12 at 12:29




    $begingroup$
    I am sorry, but don't you think that only Pial have 20 chocolates, then how the total number of chocolates are 14 or 13?
    $endgroup$
    – Shromi
    Jan 12 at 12:29




    1




    1




    $begingroup$
    You are right, I mistakenly took $P=10$, fixed.
    $endgroup$
    – farruhota
    Jan 12 at 12:33




    $begingroup$
    You are right, I mistakenly took $P=10$, fixed.
    $endgroup$
    – farruhota
    Jan 12 at 12:33












    $begingroup$
    Could you please explain why Zynayed's number of chocolates is not more than Pial? Actually I am struggling at that part.
    $endgroup$
    – Shromi
    Jan 12 at 13:36




    $begingroup$
    Could you please explain why Zynayed's number of chocolates is not more than Pial? Actually I am struggling at that part.
    $endgroup$
    – Shromi
    Jan 12 at 13:36




    1




    1




    $begingroup$
    It states "Pial gives half of his so that Zynayed has more than Pial" implying initially Pial had more than Zynayed. Similarly, once Zynayed gets chocolates from Pial, Zynayed will have more than Bindu, therefore Zynayed is giving to Bindu. So, there are several assumptions I am using. Even these restrictive assumptions does not produce unique solution.
    $endgroup$
    – farruhota
    Jan 12 at 14:08




    $begingroup$
    It states "Pial gives half of his so that Zynayed has more than Pial" implying initially Pial had more than Zynayed. Similarly, once Zynayed gets chocolates from Pial, Zynayed will have more than Bindu, therefore Zynayed is giving to Bindu. So, there are several assumptions I am using. Even these restrictive assumptions does not produce unique solution.
    $endgroup$
    – farruhota
    Jan 12 at 14:08




    1




    1




    $begingroup$
    matholympiad.org.bd/resources/all-questions/category/… Just download the zip file, extract it and then in set-01 you will find this question in page 03. In every question it is translated to English from Bangla (not Hindi!).
    $endgroup$
    – Shromi
    Jan 13 at 3:44






    $begingroup$
    matholympiad.org.bd/resources/all-questions/category/… Just download the zip file, extract it and then in set-01 you will find this question in page 03. In every question it is translated to English from Bangla (not Hindi!).
    $endgroup$
    – Shromi
    Jan 13 at 3:44













    0












    $begingroup$

    I read it to mean that after the first transfer Zunayed has more than Pial has left.
    In that case the problem is not well posed. As long as Zunayed starts with a positive even number of chocolates and Bindu starts with at least one the transfers will satisfy the requirements. After the first Pial has $10$ and Zunayed has however many he started with plus $10$, so more than Pial. After the second transfer Bindu has more than Zunayed by the number he started with.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      I read it to mean that after the first transfer Zunayed has more than Pial has left.
      In that case the problem is not well posed. As long as Zunayed starts with a positive even number of chocolates and Bindu starts with at least one the transfers will satisfy the requirements. After the first Pial has $10$ and Zunayed has however many he started with plus $10$, so more than Pial. After the second transfer Bindu has more than Zunayed by the number he started with.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        I read it to mean that after the first transfer Zunayed has more than Pial has left.
        In that case the problem is not well posed. As long as Zunayed starts with a positive even number of chocolates and Bindu starts with at least one the transfers will satisfy the requirements. After the first Pial has $10$ and Zunayed has however many he started with plus $10$, so more than Pial. After the second transfer Bindu has more than Zunayed by the number he started with.






        share|cite|improve this answer









        $endgroup$



        I read it to mean that after the first transfer Zunayed has more than Pial has left.
        In that case the problem is not well posed. As long as Zunayed starts with a positive even number of chocolates and Bindu starts with at least one the transfers will satisfy the requirements. After the first Pial has $10$ and Zunayed has however many he started with plus $10$, so more than Pial. After the second transfer Bindu has more than Zunayed by the number he started with.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 12 at 7:00









        Ross MillikanRoss Millikan

        301k24200375




        301k24200375






























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