15. (15 points) Five children, A-Chun, A-Tian, A-Zhen, A-Mei, and A-Li, took candies from a box in order. After they finished, they said the following in sequence:
A-Chun: "Everyone took a different number of candies."
A-Tian: "I took half of the remaining candies."

A-Zhen: "I took $\frac{2}{3}$ of the remaining candies."
A-Mei: "I took all the remaining candies."
A-Li: "I took half of the remaining candies."
Please answer:
(1) Which one did A-Zhen take the candies?
(2) Given that each person took at least one candy, what is the minimum number of candies in the box?
【Answer】Solution: (1) According to the problem, Achun is the 1st to take the candies, since Amei took all the remaining candies,

so Amei is the last 1 to take the candies;
since Atian and Ali cannot be in the second-to-last position, otherwise it would be the same as the last one, so Azhen is the second-to-last to take the candies,

so Azhen is the 4th to take the candies.
(2) To make the box of candies the least,
then the last person takes 1 candy,
then the second-to-last person takes: $1 \times\left(\frac{2}{3} \div \frac{1}{3}\right)=2$ (candies)
$$
\begin{array}{l}
1+2+(1+2)+(1+2+3)+4 \\
=3+3+6+4 \\
=16 \text { (candies) }
\end{array}
$$

Answer: The box of candies has at least 16 candies.