Circular Chase: Flee the Faster Dog

This puzzle describes a pursuit-and-escape scenario on a circular field: you begin at the center, while a faster pursuer is constrained to run along the boundary. The dog's leash to the perimeter fence means it can only move around the circle's edge, whereas you can move freely inside the disk and eventually must reach the boundary to escape. The question asks you to find a strategy that lets the slower runner get out without being intercepted, exploiting the fact that one agent is constrained to the circumference while the other is not. Variants of this style of continuous-time chase problem are popular in brainteaser interviews for quantitatively oriented roles.

The solution relies on geometric insight into circular motion, symmetry, and relative angular speed, plus careful reasoning about worst-case behavior by the pursuer. It implicitly uses ideas from kinematics and optimization: how to pick a path that minimizes your required travel while maximizing what the dog must cover to stay opposite you. Interviewers watch for candidates who set up relative motion cleanly, separate radial and angular components, recognize leverage from differing radii, and articulate a robust, stepwise strategy rather than guessing.

What it tests

When two agents move at different speeds on concentric paths (or with different constraints), the key is to exploit the geometry so that the slower agent can minimize the distance they must travel while maximizing the distance the faster agent must cover to intercept. This often involves using circular symmetry and relative angular speeds: by staying closer to the center, the slower agent can rotate around the center with less distance per revolution, while the faster agent, constrained to the perimeter, has to cover a much larger arc to keep up. The principle is that the ratio of angular velocities is inversely proportional to the radii of the paths, given equal linear speeds. This geometric leverage allows the slower agent to 'outrun' the faster one in angular terms, even if not in linear speed, by choosing the right radius and timing their escape.