Heavier ball | Puzzles | Math for fun and glory | Khan Academy

TL;DR

It is possible to find the heaviest ball out of nine identical balls in only two weighings.

Transcript

• [Voiceover] So we have these nine balls right over here. We're going to assume that they are completely identical. At least they are identical in appearance. But one of the nine balls is heavier, just a little bit heavier, is heavier than the other eight balls. And my question to you is: What's the minimum number of times that we can use this sca... Read More

Key Insights

• 💬 Finding the heaviest ball out of nine identical balls can be done with two weighings of the scale.
• 💬 Each weighing allows you to rule out 2/3 of the balls as candidates for the heaviest ball.
• 💬 The first weighing involves dividing the balls into three groups of three.
• 💬 The second weighing narrows down the possibilities to one out of three balls.
• 👥 The strategy can be applied to larger groups of balls by dividing them into smaller groups and repeating the two-step weighing process.
• 🧠 This puzzle is commonly used in brain teasers and sometimes appears in job interviews to test problem-solving skills.
• 💬 Variations of this puzzle exist with different numbers of balls, but the same principle of ruling out candidates applies.

Questions & Answers

Q: What is the objective of finding the heaviest ball out of the nine balls?

The objective is to determine the heaviest ball out of nine identical balls by using the scale to weigh them.

Q: How many weighings are required to find the heaviest ball?

It is possible to find the heaviest ball in just two weighings of the scale.

Q: How is the first weighing divided into groups of three?

The first weighing involves dividing the nine balls into three groups of three, with two groups placed on each side of the scale.

Q: What do the outcomes of the first weighing indicate?

If the scale balances, indicating that the left and right sides are equal, the heavy ball is in the third group. If the left side is heavier, the heavy ball is in the first group. If the right side is heavier, the heavy ball is in the second group.

Q: How is the second weighing conducted?

In the second weighing, one ball from the group that contains the heavy ball is placed on one side of the scale, and one ball from any other group is placed on the other side.

Q: How do the outcomes of the second weighing determine the heaviest ball?

If the scale balances, the heavy ball is the one that was not weighed. If the left side is heavier, the heavy ball is the one that was placed on the left side of the scale. If the right side is heavier, the heavy ball is the one that was placed on the right side.

Q: Can this strategy be applied to larger numbers of balls?

Yes, the same strategy can be applied to larger groups of balls by dividing them into smaller groups and using the two-step weighing process repeatedly.

Q: Are there variations of this puzzle with different numbers of balls?

Yes, variations of this puzzle exist with different numbers of balls, such as eight balls. However, the same principle of dividing the balls into groups and using repeated weighings applies.

Summary & Key Takeaways

• The goal is to find the heaviest ball out of nine balls, one of which is slightly heavier than the rest.

• Two weighings of the scale can definitively identify the heaviest ball.

• In the first weighing, three balls are placed on each side of the scale, narrowing down the possibilities to one of three groups. Then, in the second weighing, one ball is placed on each side of the scale, determining the heaviest ball.