Weighing Nine Balls

The question asks for the minimum number of weighings required to definitely identify a single heavier ball among nine identical balls using a two-pan balance.

A two-pan balance allows us to compare the weight of items placed on each pan. There are three possible outcomes for each weighing:

• The left pan goes down (left side is heavier).
• The right pan goes down (right side is heavier).
• The pans remain level (both sides have equal weight).

Identifying the Heavier Ball in One Weighing

Let's consider a specific weighing strategy using a two-pan balance with the nine balls.

1. Divide the nine balls into three groups: Group A (4 balls), Group B (4 balls), and Group C (1 ball).
2. Place the balls from Group A on the left pan of the balance and the balls from Group B on the right pan. The ball from Group C is left off the balance.

Possible Outcomes and Identification

When weighing Group A against Group B, there are three possible outcomes:

• Outcome 1: The balance remains level. This means the total weight of balls in Group A is equal to the total weight of balls in Group B. Since one ball is heavier than the others, and it's not in Group A or Group B (as they balanced), the heavier ball must be the one left aside in Group C.

In this specific outcome where the pans balance, the heavier ball is identified directly in this first and only weighing.

While other outcomes (left side heavier or right side heavier) would require further weighings to pinpoint the heavier ball within the set of 4 on the heavier side, the question asks for the least number of weighings needed for definitely identifying the heavier ball. By demonstrating a scenario where the ball is identified in just one weighing (when the scale balances), it shows that finding the heavier ball can be achieved in a single weighing using a two-pan balance in certain circumstances.