Paradoxes and puzzles

The friendship paradox is the finding that, on average, your friends have more friends than you do. It is a sampling effect of network structure, not a flaw in you.

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Braess's paradox is the finding that adding a road to a network of self-interested drivers can raise travel times for everyone, because the resulting equilibrium is worse than before.

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Benford's law says that in many real-world data sets, small leading digits like 1 and 2 appear more often than large ones. Auditors use departures from it as a flag, not proof, of fraud.

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Survivorship bias is reasoning from cases that passed a filter while ignoring those that did not. In WWII, Abraham Wald applied this insight to bomber damage, though the popular retelling embellishes the record.

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The gambler's fallacy is the false belief that past outcomes of independent random events change the odds of future ones, so a streak makes the opposite result "due."

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The Fermi paradox is the clash between the apparent likelihood of alien civilizations and the absence of any evidence for them. Proposed answers say life is rare, civilizations are short-lived or quiet, or we have not looked enough.

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It is a theorem that a solid ball can be split into finitely many pieces and reassembled into two balls of the same size. The pieces are non-measurable point sets, so it cannot be done physically.

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The sorites paradox shows that vague words like 'heap' seem to lead from true premises to an absurd conclusion. Philosophers disagree about where the reasoning fails.

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The Jevons paradox is the case where making a resource more efficient to use raises total consumption of it. How often it happens is debated, though researchers agree it can.

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Zeno argued that a faster runner can never catch a slower one with a head start. The standard modern answer is that infinitely many shrinking steps add up to a finite distance and time.

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A coin game has an infinite expected payoff, yet almost no one would pay much to play it. The puzzle challenges the idea that rational choice means maximizing expected value.

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Newcomb's problem is a thought experiment about choosing one box or two when a reliable predictor has already filled them. It divides people because two accepted decision principles give opposite answers.

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The liar paradox arises from a sentence like "This sentence is false," which seems to be true only if it is false and false only if it is true. Experts still disagree on how to resolve it.

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Hilbert's hotel is a thought experiment about a hotel with infinitely many rooms. Though full, it can still house one, infinitely many, or even infinitely many busloads of new guests.

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A time traveler who kills their own grandfather would prevent their own birth, and so the trip. Proposed resolutions range from self-consistent histories to ordinary mishaps and alternate timelines.

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A twin who travels at high speed and returns is younger than the twin who stayed home. The apparent paradox disappears because the twins' paths through spacetime are not symmetrical.

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Schrödinger devised the cat in 1935 as a reductio ad absurdum, to show that applying quantum superposition to large objects gives absurd results, not to claim cats can be both alive and dead.

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Kenneth Arrow proved that no method of turning voters' ranked preferences into one group ranking can satisfy a short list of reasonable fairness conditions once there are three or more options.

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The paradox of choice says more options can lower satisfaction and buying. Reviews find no average effect, but overload appears under certain conditions.

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Russell's paradox shows that the set of all sets that are not members of themselves cannot exist. It exposed a contradiction in naive set theory and prompted new foundations for mathematics.

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The base rate fallacy is ignoring how common a condition is when reading a test result. When a condition is rare, even accurate tests produce many false positives.

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