Cheating Tax Collector 1 Weigh

This puzzle describes a cheating-detection scheme using a single measurement on a weighing device, in a setting with multiple identical-looking sources and one systematically defective one. The candidate must reason about how to design one carefully chosen sample from these sources so that the outcome of a single weigh reveals which one is bad. It is a classic brain teaser used in interviews for roles that value structured thinking and clever use of constraints, often appearing in tech and quantitative finance interviews as a warm-up logic problem rather than a domain-specific test.

Conceptually, the question leans on encoding information into the measurement by choosing different sampling patterns from each source. It relies on understanding linear relationships between the number of sampled items and the total deviation in the measurement. An interviewer is looking for the ability to move beyond brute force, to treat the weighing device as a one-shot information channel and design a scheme where each possible culprit yields a unique, decodable outcome. Clarity, systematic reasoning, and recognizing that the measurement can be "programmed" via sampling are what matter most.

What it tests

This problem class is governed by the principle of encoding information through controlled sampling and exploiting linearity. When only one measurement is allowed, you can design the sampling pattern so that each possible scenario produces a unique, decodable result. By assigning a distinct 'weight' (here, the number of coins) to each source, the effect of a single anomaly (the lighter coins) shifts the total by an amount that identifies its origin. The key is that the measurement outcome is a linear combination of the sources, so the deviation caused by any one source is proportional to the sampling weight you assigned it. This lets you invert the process: the observed deviation directly reveals the culprit because each possible outcome is mapped to exactly one source.