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

Question

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?

Answer 1

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.

Answer 2

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.