Brain Teasers · Cars on a Single-Lane Road
Single-Lane Road Cars Puzzle - Expected Clusters
A hundred cars, random speeds, no overtaking: how many clusters form on average?
What is the single-lane road puzzle?
It is a records problem: a car leads a cluster exactly when it is a new record low looking back from the front of the road.
In a random order, the k-th item is a record with probability 1/k, so the expected number of records is the harmonic number H_n.
How it works
A hundred cars with different top speeds in random order on an endless road with no passing. Fast cars get stuck behind slow ones. The story asks the expected number of clusters. You type your answer and check it. Then you step through the answer and the breakdown.
How scoring works
This teaser checks your answer. Any correct form counts - a fraction, a decimal rounded correctly, a percent, or the number in words. You get two tries before the answer opens. A correct first try earns a ticket; a second try or opening the answer early is recorded but earns nothing. The story teasers share one lifetime unlock for free users.
The key insight
Car k leads a cluster with probability 1/k, the chance it is the slowest of the first k cars. Add the chances: 1 + 1/2 + ... + 1/100 ≈ 5.19.
The full step-by-step solution lives at /brain-teasers/single-lane-cars/solution.
A worked example
H_100 ≈ ln 100 + 0.577 ≈ 5.19. A thousand cars gives only about 7.49.
Common mistakes
• Answering 50 or 10 without setting up the indicator sum.
• Counting from the back of the road instead of the front.
• Thinking dependence between cars stops you adding the chances.
Why interviews test this
Spot the record-low pattern and write the count as a sum of indicators. The written solution is at /brain-teasers/single-lane-cars/solution.
Play Cars on a Single-Lane Road on Outcry
Cars on a Single-Lane Road is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play Cars on a Single-Lane Road and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.