This is a quite interesting question that I have come across some earlier years in my Olympiad training. Due to my bad memories and notebook record, I failed to trace back where I found this question. I also lost the original solution to this problem. It was not until recently, I started reading about different number of systems rather than the decimal system that another solution crossed my mind.
I had solved the problem below using the binary system. But what troubles me is that, the number used was rather symbolic, which means, it has little to do with mathematics. So I added some extensions to the problem. The extension is an open question, and I didn't really know how to solve it.
Please give your solution to the original problem, it is okay if you don't use the binary system. And please help me solve my extension.
Any help is appreciated.
A king is throwing a party. During the party, a mandarin is reported to be ill due to wine poison. Therefore, the king ordered the soldiers to check the wines. There are $2019$ bottles in the cellar, and the guards decided to use rats to try and find the poisonous bottle. Assume that the attempts does not take time, and the rat also take one hour to die. There is also only one hour to do all the tasting. How many rats are there needed to find out the poisonous bottle?
Note that: a rat can taste many bottle (all $2019$ is okay, but of course this is not a solution) as long as it doesn't die. However, if a rat tasted both bottle A and bottle B, and it dies, we don't know whether A or B is poisonous.
So in the problem above, the number $2019$ and 1 hour has hardly any meaning. Now I am going to link all these numbers together.
A king is throwing a party. During the party, a mandarin is reported to be ill due to wine poison. Therefore, the king ordered the soldiers to check the wines. There are $x$ bottles in the cellar, and the guards decided to use rats to try and find the poisonous bottle. Assume that the attempts do not take time and the rats die after one hour. If the guards has 100 rats and one hour to find the bottle, what is the maximum value of $x$?