$\textbf{Poisonous Wine}$
A king has $1000$ bottles of expensive wine. One assassin is just able to poison one bottle of wine before being killed. The king does not want to discard all the bottles, so he decides to force some prisoners to taste these wines in order to find out the poisonous one. While he is ruthless, the king is also intelligent. He figures that it is not necessary to use $1000$ prisoners because it is known that this type poison will take effect and kill an in-taker after $24$ hours. Is it possible for the king to use no more than $10$ prisoners to identify the poisonous bottle?
$\textbf{Solution}$
Yes, it is possible. First, label the bottles from $1$ to $1000$ and write these numbers down using binaries i.e.,
- $1=0000000001$
- $2=0000000010$
- $3=0000000011$
- $\cdots$
- $999=1111100011$
- $1000=1111100100$
Because, $1000 < 2^{10}$, therefore there are at most $10$ digits required. Next, label the $10$ prisoners $1$ to $10$. The $1^{st}$ prisoner will be forced to taste all the bottles whose binary labels have digit $1$ at the $1^{st}$ position. The $2^{nd}$ prisoner will be forced to taste all the bottles whose binary labels have digit $1$ at the $2^{nd}$ position, and so on. After $24$ hours, line up those prisoners both alive and dead according to their initial order. An alive one represents $0$ and a dead one represents $1$. The result number, in binary format, identifies the poisonous wine.
$\textbf{Note}$
This brain teaser is related to the information encoding and error detection. It also illustrates the use of the binary method which has wide applications in solving math and science problems.