Bulbs Left On After 100 Passes

This classic brain teaser is about uncovering a hidden pattern in a simple, repetitive process. It looks like a toy problem, but underneath the story sits a clean mathematical structure involving how many times each position is touched. It's a perfect example of how quant interview questions wrap discrete math and number patterns inside an easy-to-state puzzle.

It trains comfort with divisors, parity, and pattern recognition under constraints, all of which are central in quant prep. You practice mapping a verbal description to an indexed system, tracking state changes, and spotting which indices behave differently from the rest. It also develops the habit of abstracting away the story into a more general mathematical object.

This matters for quant interviews because it tests fast abstraction, rigorous counting, and clear reasoning. Interviewers see whether you reduce noisy descriptions to concise math, reason about structure instead of brute force, and communicate your thinking cleanly. These are exactly the skills needed for trading problems, algorithm design, and probability puzzles that dominate high-end quant interviews.

What it tests

The core structure here is that the number of times an object is affected in a sequence of actions often corresponds to the number of divisors of its label or index. In such problems, the parity (odd or even) of the number of divisors determines the final state. Most numbers have an even number of divisors because divisors come in pairs $(d, n/d)$, but perfect squares have a repeated divisor, resulting in an odd count. This principle holds because only perfect squares have a factorization where a divisor is paired with itself, breaking the even pairing and leaving one unpaired toggle.