Distance Traveled by the Fly Before Collision

This brain teaser describes two vehicles starting some distance apart and driving directly toward each other at fixed but different speeds, while a fast-moving third agent shuttles continuously between them. The setup sounds like it should require tracking an ever-growing number of back-and-forth legs as the gap shrinks, which makes it a classic puzzle in mental misdirection. Variants of this question are common in general brainteaser interviews and casual quantitative screening, because the story feels intricate while the underlying structure is quite simple once recognized.

The problem leans on understanding relative motion in one dimension and separating the "timeline" of an event from the detailed path of any one participant. It tests whether the candidate can strip away the distracting iterative description and instead identify the key invariant quantities: initial separation, constant speeds, and the effective closing speed. Interviewers watch for comfort with basic rate-time-distance reasoning, the ability to model the situation with a single clean calculation instead of summing infinitely many segments, and clear explanation that shows conceptual grasp rather than rote puzzle familiarity.

What it tests

When two objects move toward each other at constant speeds, the time until they meet depends only on their initial separation and the sum of their speeds, not on the details of their individual motions. Any third object moving continuously during this interval, regardless of its own path or how many times it changes direction, will cover a distance equal to its speed multiplied by the meeting time. This principle arises because the meeting event is determined solely by the relative motion of the two converging objects, and any continuous process running in parallel is unaffected by the complexity of its own trajectory. The key is that the system's timeline is set by the closing speed and initial gap, so all other continuous motions are simply scaled by this fixed duration. This approach sidesteps the need to analyze or sum up infinitely many sub-intervals or path segments.