Rats and wine bottles’ puzzle

I have tried to solve the puzzle starting with smaller numbers of wine bottles.

Let’s take 2 bottles and 1 rat.

Feed 1 bottle to the rat. This leads to 2 possible outcomes: either the rat dies or the rat doesn't die. This will tell us which of the 2 bottles is poisoned.

Depending on the rat’s life/death outcome, it would be easy to figure out which bottle was poisoned.

I have tried to show this using in table 1. Where 0 means the bottle was not fed to that rat 1 and means the bottle was fed to that rat. See table 1 below

Let’s say the rat dies that means the bottle 2 with the ‘1’ was poisoned.

Let’s say the rat survives that means the bottle 1 with the ‘0’ was not poisoned.

Let’s take 4 bottles and 2 rats.

There is one of four possibilities for the rats: See table 2 below

if neither dies, bottle 1 poisoned, if only rat 2 dies that means bottle 2 was poisoned, if only rat 1 dies that means bottle 3 was poisoned, and if both die that means bottle 4 was poisoned

Table 2 built on the same logic as table 1. Depending on which rats die and live, we can figure out which bottle was poisoned.

That means: 2² = 4.

Carrying on, with 3 rats we can actually find the poisoned bottle out of 8 bottles.

Table 3 built on the same logic as table 2. Depending on which rats die and live, we can figure out which bottle was poisoned.

That means: 2³ = 8. In this manner, we can find the poisoned bottle out of 2ⁿ bottles with n rats.

We can test 16 bottles by taking 4 rats; we can test 32 bottles by taking 5 rats for 32 bottles, and so on. 2¹⁰ = 1024 fortunately, we have 1000 bottles only.

There is only 1 rat died after testing 999 wine bottles.