A lone rover, dispatched from Earth to explore a distant exoplanet, faces a peculiar dilemma: it wants to return to its starting point after following a precise, day-by-day travel schedule. The rover must move forward a growing distance each day (1 km, 2 km, 3 km, and so on, for eight days) and turn 90 degrees at the end of each segment. The question is, can it choose left or right turns strategically to end up exactly where it began?
The Solution
The rover can return to its starting point. The key lies in recognizing a simple pattern: the rover needs to make an equal number of left and right turns. Since the mission lasts eight days, it must turn four times left and four times right.
To achieve this, the rover should alternate directions. For example, turning right on day one, left on day two, right on day three, left on day four, and so on. This ensures that after eight movements, the rover has completed a full square loop, effectively canceling out its movements and bringing it back to the original landing site.
The Larger Puzzle: Rover Missions Across Planets
The bonus question extends this concept to a fleet of 100 rovers, each assigned missions of varying lengths from one to 100 days. Which rovers can successfully return home?
The answer is that any rover with an even-numbered mission length can return to its starting point. This is because an even number of days allows for an equal split between left and right turns. The rover simply needs to alternate directions consistently.
Conversely, rovers on odd-length missions (1, 3, 5, etc.) cannot return to their starting point. An odd number of movements will always result in an imbalance between left and right turns, leaving the rover stranded.
In conclusion, the homesick rover’s journey highlights a simple yet elegant mathematical principle. Rovers can only return home if their missions are structured to allow for balanced movements, proving that even in the vastness of space, a little planning can bring you back to where you started.
