The thousand poisoned bottles

1. Strategy (binary encoding): For each bottle n, write n in 10-bit binary.
2. Example: If rats 0, 3, 4 and 9 die, the pattern is 1000011001_2, which corresponds to bottle 537.
3. With 10 rats and a binary outcome (live/die), there are 2^ 10 =1024 possible patterns, enough to identify 1000 bottles.

Answer: 24 hours (single test cycle).

Strategy (binary coding):
With 10 rats and binary outcome (live/die), there are $2^{10}=1024$ possible patterns, enough to identify 1000 bottles.

1. Number bottles from 0 to 999.
2. Number rats from 0 to 9 (each rat represents one bit).
3. For each bottle $n$, write $n$ in 10-bit binary.
4. If bit $i$ is set to 1, that bottle is given to rat $i$.

After 24 hours, the set of dead rats forms exactly the binary number of the poisoned bottle.
Example:
If rats 0, 3, 4, and 9 die, the pattern is $1000011001_2$, which corresponds to bottle 537.
Conclusion: the bottle is identified in a single round.