Heavier Ball Weighing Puzzle Solution

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

A two-pan balance allows us to compare the weight of items placed on its pans. There are three possible outcomes when weighing: the left pan is heavier, the right pan is heavier, or both pans are of equal weight.

Weighing Strategy

To find the single heavier ball efficiently, we need a strategy that eliminates as many possibilities as possible with each weighing. The optimal approach involves dividing the balls into groups.

First Weighing

Divide the nine balls into three equal groups:

• Group 1: Balls 1, 2, 3
• Group 2: Balls 4, 5, 6
• Group 3: Balls 7, 8, 9

Place Group 1 (3 balls) on the left pan of the balance and Group 2 (3 balls) on the right pan.

There are three possible outcomes from this first weighing:

• Outcome A: The balance stays level. This means Group 1 and Group 2 have equal weight. The heavier ball must be in the group not weighed, which is Group 3.
• Outcome B: The left pan goes down. This indicates that Group 1 is heavier than Group 2. The heavier ball must be within Group 1.
• Outcome C: The right pan goes down. This indicates that Group 2 is heavier than Group 1. The heavier ball must be within Group 2.

After this first weighing, you have successfully narrowed down the location of the heavier ball to a specific group of 3 balls.

Second Weighing

Take the group of 3 balls that was identified in the first weighing as containing the heavier ball. Let's call these three balls A, B, and C.

Place Ball A on the left pan and Ball B on the right pan.

There are three possible outcomes from this second weighing:

• Outcome X: The balance stays level. This means Ball A and Ball B are of equal weight. The heavier ball must be the one not weighed, which is Ball C.
• Outcome Y: The left pan goes down. This indicates that Ball A is heavier than Ball B. The heavier ball is Ball A.
• Outcome Z: The right pan goes down. This indicates that Ball B is heavier than Ball A. The heavier ball is Ball B.

After this second weighing, you have definitively identified the single heavier ball among the original nine.

Minimum Weighings Required

Using this strategy, we were able to isolate the heavier ball in all possible scenarios with just two weighings. Therefore, the least number of weighings needed to definitely identify the heavier ball among nine is two.