Brain Teasers · The Amoeba Population
Amoeba Extinction Puzzle - Play It (Root 2 Minus 1)
Each minute an amoeba dies, stays, splits in two or splits in three. Will the family die out?
What is a branching process?
A branching process tracks a population where each member independently leaves a random number of offspring. The question is usually whether it eventually dies out.
The extinction probability p satisfies p = E[p^offspring], and it is the smallest root of that equation between 0 and 1.
How it works
A single amoeba, and each minute every amoeba independently dies, stays, becomes two or becomes three, each with probability 1/4. The story asks the probability the population eventually dies out. 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
Look at the first minute: p = (1 + p + p² + p³) / 4. That rearranges to (p - 1)(p² + 2p - 1) = 0, and the smallest root in [0, 1] is √2 - 1 ≈ 0.414.
The full step-by-step solution lives at /brain-teasers/amoeba-population/solution.
A worked example
Average offspring is (0 + 1 + 2 + 3) / 4 = 1.5, above 1, so extinction is not certain. The chance of dying out by minute n climbs from 0 and settles at √2 - 1.
Common mistakes
• Answering 1/4, the chance of dying in the first minute.
• Writing 2p instead of p² for two amoebas, which must both die out.
• Picking the root p = 1.
Why interviews test this
This tests first-step analysis on a process that renews itself, and knowing why p = 1 is the wrong root. The written solution is at /brain-teasers/amoeba-population/solution.
Play The Amoeba Population on Outcry
The Amoeba Population is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play The Amoeba Population 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.