What is the fewest number of testers needed to identify the poisoned wine? The King has 1000 bottles of wine, exactly one of which is poisoned. Your
job is to identify and throw out the poisoned bottle as quickly as possible by having the royal
taste-testers drink the wines. Since the poison takes a little while to take effect, quickly means
that the taste-testers only have the time to take a single drink—so you can’t have a single
taste-tester go through the wines one by one. On the other hand, you can mix the wines in
arbitrary ways before handing them out. What is the fewest number of taste-testers needed to identify the poisoned wine, and why?
I cannot figure out where to start

As mentioned in the comments, you should use binary representation of numbers. Let's start with fewer bottles, say $6$. We can write the binary numbers for the labels as $001$, $010$, $011$, $100$, $101$, $110$. Now let's assign a tester for each bit. For example, tester 1 will check the last bit, tester 2 will check the next to last and so on. What does it mean to check the bit? In this case, prepare a drink for tester 1 that contains a mixture of all the drinks where the last digit in the label is $1$. For tester 2, the mixture contains all the wines where the digit next to last is $1$. And so on.
$$\begin{array}{c|c|c|c|c}
Nr&Label& Tester\ 3 & Tester\ 2 & Tester\ 1 \\ \hline
1& 001& 0&0 &1\\  \hline
2& 010& 0 &1 &0\\ \hline
3& 011& 0 &1 &1\\\hline
4& 100&1&0&0\\\hline
5& 101&1&0&1\\\hline
6& 110&1&1&0
\end{array}$$
You can see now that if a tester detects poison, the label must have the corresponding digit equal to $1$. If a tester does not detect poison, the corresponding digit on the label is $0$. Then you just need to see which tester detects the poison. For example, if tester 1 and 3 detect poison, the bottle with the problem is number 5.
To get more bottles just extend the numbers of testers. Show that you need 10 for 1000 bottles.
