NIMCET 2013 — Reasoning PYQ

Solving

To find the minimum number of weightings for $n$ items where one is different, we use the base-3 logic because a balance scale has three possible outcomes (Left heavy, Right heavy, or Balanced). The formula is $3^k\ge n$, where $k$ is the number of weightings.

Step 1: First Weighting

• Divide the $8$ balls into three groups: Group A ($3$ balls), Group B ($3$ balls), and Group C ($2$ balls).

• Weigh Group A against Group B.

  • Outcome 1: If A = B, the heavy ball is in Group C ($2$ balls).

  • Outcome 2: If A > B, the heavy ball is in Group A ($3$ balls).

  • Outcome 3: If B > A, the heavy ball is in Group B ($3$ balls).

Step 2: Second Weighting

• If the ball is in Group C (2 balls): Weigh one ball against the other. The heavier one is identified.

• If the ball is in Group A or B (3 balls): Pick any two balls and weigh them against each other.

  • If they balance, the $3^{rd}$ ball (the one not weighed) is the heavy one.

  • If they don't balance, the heavier side contains the ball.

Conclusion:

In all scenarios, the heavy ball is identified in exactly 2 weightings.

Since $8\le 9$, only 2 weightings are required.