Sixteen Bottles, One Round

Each mouse gives one bit: dies or survives.

Label bottles $0$ to $15$ in binary.

1. 1. Encoding

   Mouse $i$ drinks from every bottle whose binary label has a $1$ in bit $i$.

2. 2. Decode

   The pattern of deaths spells the bottle's label. $2^4=16$, so $4$ mice suffice and $3$ cannot.

3. 3. Key idea

   One round of binary outcomes is a binary code: you need $\lceil {\log }_{2}n\rceil$ testers.

Answer: 4