Weigh One Bag to Find the Fake

This brain teaser presents a collection of identical-looking groups of objects, all but one of which are genuinely identical in weight. One group is counterfeit, with every object in that group uniformly lighter or heavier than the rest. You have access to a perfectly accurate digital scale, but are allowed to use it only once. The challenge is to plan a single weighing that reveals exactly which group is fake, without any trial and error or additional information gathering.

The puzzle leans on ideas from information encoding, number patterns, and basic algebraic reasoning. A strong answer uses a systematic scheme for choosing how many items from each group to weigh so that every possible outcome maps cleanly to one specific group. Interviewers watch whether the candidate recognizes that the weighing is not just measurement but also a communication channel, and whether they can design a one-shot strategy instead of relying on sequential testing. Clarity of reasoning, ability to generalize the method, and avoidance of ad hoc guesswork are key.

What it tests

This problem class is governed by the principle of encoding information through uniquely weighted combinations. By assigning a distinct number of items (or weights) to each category, you ensure that any deviation in the aggregate measurement can be traced back to a single source. The key is that the mapping from possible deviations to sources is injective: each possible outcome corresponds to exactly one scenario. This works because the sum of distinct positive integers (like 1 through n) produces a set of possible deviations that are all different, so the measurement uniquely identifies the source. The underlying structure is using the linearity of measurement to turn a single aggregate observation into a code that reveals which category is anomalous.