There are 30 cans, one of which is poisoned. A person who tastes even a small…

Concept: To identify one poisoned item among N possibilities using test subjects whose only observable outcome is “dies” or “survives” within the time limit, use binary encoding: assign each possibility a unique binary code, and let each test subject represent one bit. With k test subjects there are 2k distinct die/survive patterns, so k subjects are enough to uniquely identify one item among N possibilities exactly when 2k ≥ N.

Application:

1. There are 30 cans, and exactly one is poisoned; each can must get a unique die/survive signature across the mice used.

2. With k mice there are 2k possible signatures, so we need 2k ≥ 30.

3. Check k = 4: 24 = 16, which is less than 30 — not enough signatures to cover every can.

4. Check k = 5: 25 = 32, which is at least 30 — enough signatures.

5. Assign each can a distinct 5-bit binary code between 1 and 30. For a given can, feed its liquid to every mouse whose bit is 1 in that can's code.

6. Within the 24-hour window, observe which mice die; reading the die/survive pattern as a binary number identifies exactly which can was poisoned.

7. So the minimum number of mice required is 5.

Cross-check: The number of cans that can be distinguished grows as a power of two with the number of mice, not linearly.

This confirms that 5 mice are both necessary and sufficient, while 3 or 4 mice fall short.