Given 9 balls, all of which weigh the same except for one, what is the minimum of weighings necessary to find the ball with the different weight?

1. Break Down the Problem

To determine the unique ball among 9 using a balance scale, we can utilize the properties of weighing. The goal is to find the ball that weighs differently (either heavier or lighter) with the least number of weighings.

2. Relevant Concepts

• In each weighing, a balance scale can group items and allow comparisons.
• By strategically dividing the balls into groups, we can eliminate possibilities with each weighing.

3. Analysis and Detail

• First Weighing: Divide the 9 balls into three groups of 3 each (let's call them Group A, Group B, and Group C). Weigh Group A against Group B.

  • If they balance, the different ball is in Group C.
  • If Group A is heavier or lighter, the different ball is in Group A or B.

• Second Weighing: Take the group that was determined to contain the different ball from the first weighing. Now we have 3 balls. Select 2 balls from this group and weigh them against each other.

  • If they balance, the different ball is the one not weighed.
  • If one is heavier or lighter, that ball is the different one.

4. Verify and Summarize

After two weighings, we have systematically reduced the number of possibilities and identified the ball that has a different weight.

Final Answer

The minimum number of weighings necessary to find the ball with a different weight is 2.