Efficiently Identifying the Heaviest Ball Among 500 Balls

In this article, we will explore a step-by-step method to efficiently identify the heaviest ball among a set of 500 balls, where 499 are 10 grams and 1 is 11 grams. This process uses a weighing machine to its maximum efficiency while minimizing the number of weighings required.

Introduction

The task at hand is to find the ball that weighs 11 grams among 500 balls, with the rest weighing 10 grams. To achieve this, we can employ a binary search strategy, which is a well-known algorithm for efficiently locating an element in a sorted list or, in this case, identifying a heavier ball.

Step-by-Step Approach

Divide the Balls

The first step is to divide the 500 balls into two groups of 250 balls each. This division helps in systematically narrowing down the search space by half with each weighing.

First Weighing

Weigh the first group of 250 balls against the second group of 250 balls. Based on the result, we can deduce the following:

If the scales balance, then the 11-gram ball is in the unweighed group (either the first 250 or the second 250 balls). If the scales tip, then the 11-gram ball is in the heavier group.

Narrow Down

Taking the group that contains the 11-gram ball (either the first 250 or the second 250), we can split it into two groups of 125 balls each and weigh them against each other. This process is repeated until we are left with just one ball.

Example of Continuing the Process

Let's walk through the example division and weighing process:

250 balls → 125 vs. 125 125 balls → 62 vs. 63 63 balls → 31 vs. 32 32 balls → 16 vs. 16 16 balls → 8 vs. 8 8 balls → 4 vs. 4 4 balls → 2 vs. 2 2 balls → 1 vs. 1

The heavier ball will be obvious at the final point.

Conclusion

By using this binary search method, we can identify the heaviest 11-gram ball in a maximum of log?(500) weighings, which is approximately 9 weighings. This method is highly efficient and provides a systematic approach to solving the problem.

Additional Tips and Guidelines

To further optimize the process, consider the following:

Accurate and consistent weighing is crucial. Make sure the weighing machine is calibrated properly. Ensure that the balls are placed equally on the weighing machine for each weighing to avoid skewed results. Organize the process to minimize errors and confusion, especially if dealing with a large number of balls.

Following these steps and tips will not only help in finding the heaviest ball efficiently but also in maintaining the integrity and accuracy of the process.

Frequently Asked Questions (FAQ)

How can I ensure the accuracy of my weighing machine?

To ensure the accuracy of your weighing machine, calibrate it regularly using known weights and follow the manufacturer's guidelines for maintenance and use.

What if there are more than one 11-gram balls?

In this scenario, the method would still work, but you will identify all the heavier balls. You can then follow the steps to isolate each one.

Is there a faster method?

Binary search is already the most efficient method for this problem. It is based on the principle of dividing the search space in half with each weighing, making it highly optimized.