The wise men and the pearls (ancient Persia)

1. Think about information capacity: each weighing has 3 possible outcomes (heavier left, heavier right, balance).
2. First weighing: divide the 9 beads into three groups of 3 and compare (A, B, C) vs (D, E, F), leaving (G, H, I) out.
3. Second weighing: if the first balance is balanced, compare G vs H (if they tie, the weighing is I). If there is no balance, compare two of the heaviest trio to identify the pearl.

Answer: 2 heavy.
Explanation:
First weighing: Divide the 9 pearls into three groups of 3: $(A, B, C)$ vs $(D, E, F)$, leaving $(G, H, I)$ out.
If balanced:
The heavy pearl is in $(G, H, I)$.

• Second weighing: $G$ vs $H$
• If $G$ heavier $\to$ $G$ is false
• If $H$ heavier $\to$ $H$ is false
• If balanced $\to$ $I$ is false

If $(A, B, C)$ heavier:
The heavy pearl is at $(A, B, C)$.

• Second weighing: $A$ vs $B$
• If $A$ heavier $\to$ $A$ is false
• If $B$ heavier $\to$ $B$ is false
• If balanced $\to$ $C$ is false

Result: 2 weighings guarantee identifying the heaviest pearl.