--I also make the trivial remark that, no matter how many poisoned bottles P there are among the W total bottles (and let us say we do not know the value of P ahead of time), you can always identify them by using R=W rats, 1 rat per bottle. Can we do better? No. Because there are 2^W possible bottle-subsets and 2^R possible dead-rat subsets, so you can't have enough info to identify the poisoned bottles unless you have at least R>=W rats.