Given 12 balls with one of them either heavier or lighter than the rest, what is the minimum number of weighings needed on a balance scale to identify the unique ball?

Official Answer

Certainly, navigating through probability and logic-based questions, especially ones like identifying the unique ball among 12, with a balance scale, illuminates not just the analytical prowess but also the critical thinking and problem-solving methodology a candidate brings to the table. Drawing from my extensive experience as a Data Scientist, where dissecting complex problems to arrive at data-driven insights is part of my daily repertoire, I would approach this intriguing puzzle with a structured and strategic framework, much like how I tackle data anomalies or irregularities in large datasets.

The essence of solving this problem lies in efficiently partitioning the set of balls to minimize the number of weighings required to pinpoint the unique ball and determine whether it's heavier or lighter. The minimum number of weighings needed is three, and here’s a strategic approach to achieve that.

In the first weighing, we divide the 12 balls into three groups of four and weigh two groups against each other. This initial step is pivotal as it branches into two scenarios: either the scales balance, indicating the unique ball is in the unweighed group, or they don’t, revealing it's in one of the weighed groups. This is akin to initial data exploration where we segment data to identify areas that warrant a deeper dive.

If the scales balance, indicating the unique ball is in the unweighed group of four, we proceed to the second weighing with a refined strategy. Here, we take three balls from the unweighed group and weigh them against three balls we know are of standard weight. If the scales don’t balance, it’s clear the unique ball is one of the three being weighed, and its deviation in weight will also be apparent – whether it’s heavier or lighter. However, if the scales balance, it signifies the unique ball is the one not being weighed in this round, and its subsequent weighing will reveal if it's heavier or lighter. This mirrors the analytical process of isolating variables and conducting controlled experiments to validate hypotheses.

On the other hand, if the first weighing doesn't balance, we know the unique ball is among the eight we initially weighed. For the second weighing, we take three balls from the heavier side and three from the lighter side and swap them, also including two balls of known standard weight for balance. The outcome of this weighing, whether it tips the same way, the opposite way, or balances, will not only identify the unique ball as one of the three switched but also its nature – heavier or lighter. This step is reminiscent of advanced analytical techniques where we adjust models based on new data or findings to refine predictions or insights.

This methodology not only showcases the analytical and problem-solving skills inherent to a Data Scientist but also demonstrates a logical, step-by-step approach to dissecting and solving complex problems. It’s a testament to the ability to navigate through ambiguity, apply logical deductions, and utilize a structured approach to arrive at a solution – qualities that are indispensable in the realm of data science and analytics. Leveraging this framework, candidates can effectively tailor their responses, drawing parallels to their experiences and showcasing their problem-solving acumen in a compelling narrative that resonates with the hiring managers.