Hilbert's Grand Hotel
A hotel with infinitely many rooms is completely full. At two in the morning, one more guest arrives. The manager finds them a room.
1924 (lectures), 1947 (popularised)13 min read
Kind
Paradox
Self-defeating arguments, impossible results, and situations where valid reasoning reaches an absurd conclusion.
69 entriesFilter these further →
Beginner
Ages 8+, no background needed.
Sunk Cost Fallacy
You paid for a concert ticket and now feel terrible. If you hadn't paid, you'd stay home. You paid, so you go. Why does a sunk cost change anything?
19857 min read
The Birthday Problem
How many people must be in a room before two of them probably share a birthday? The answer is far smaller than you expect.
1939 (first published), circulating from the 1920s14 min read
The Monty Hall Problem
Three doors, one car. The host opens a losing door and offers you the swap. Does it make any difference whether you take it?
1975 (posed), 1990 (made famous)9 min read
The Prisoner's Dilemma
Two people can each do better by betraying the other — and end up worse off than if neither of them had.
195018 min read
Buridan's Ass
A donkey stands exactly halfway between two identical bales of hay. With no reason on earth to prefer either, does it starve?
c. 1340 (the attribution; the example is far older)14 min read
The Barber Paradox
The barber shaves every man in town who does not shave himself, and no one else. So who shaves the barber?
1918 (Russell's lectures); the story circulated earlier11 min read
The Coastline Paradox
How long is the coastline of Britain? The answer depends on how you measure it. As the ruler gets shorter, the answer grows without bound.
1951 (Richardson), 1967 (Mandelbrot)7 min read
The Condorcet Paradox
A majority prefers A to B, and B to C, and C to A. Every voter was honest and nobody made a mistake.
178517 min read
The Friendship Paradox
Your friends have more friends than you do. So do mine. This is true for almost everybody, all at the same time.
199117 min read
Bertrand's Box Paradox
You reach into one of three boxes and pull out a gold coin. The chance the other coin in that box is gold is not one half.
188910 min read
The Abilene Paradox
Everyone at the table agreed to a trip that not one of them wanted to take. How does a group choose what every member is against?
197417 min read
Intermediate
Curious teenagers and general adult readers.
Achilles and the Tortoise
The fastest runner alive can never catch a tortoise with a head start. The argument is airtight. So where does it go wrong?
c. 450 BCE15 min read
Goodhart's Law
The moment you measure something and start rewarding the number, the number stops telling you what it used to.
1975 (Goodhart's formulation), 1997 (Strathern's phrasing)14 min read
Schrödinger's Cat
A cat sealed in a box, neither alive nor dead until somebody looks. The story was invented to show the idea was absurd.
193514 min read
Simpson's Paradox
A treatment does better for men, and better for women, and worse for people. None of the arithmetic is wrong.
1903 (Yule), 1951 (Simpson), 1972 (named)20 min read
The Fermi Paradox
The universe is ancient, enormous, and full of ordinary stars much like ours. So where is everybody?
1950 (Fermi's remark), 1975 (Hart's argument)22 min read
The Grandfather Paradox
Travel back in time and stop your own grandparents from ever meeting. Then who was it that went back?
1943 (Barjavel's novel), with antecedents from the 1920s18 min read
The Paradox of Choice
Put out twenty-four kinds of jam instead of six and fewer people buy any jam at all. More options, fewer decisions.
2004 (the book); 2000 (the jam study)19 min read
The Paradox of Tolerance
A society that tolerates absolutely everything, including those working to end it, does not remain tolerant for long.
19458 min read
The Ship of Theseus
Replace a ship's planks one at a time until not one original piece is left. Is it still the same ship?
c. 75 CE (Plutarch's report of an older dispute)20 min read
The Tragedy of the Commons
Every herder gains by putting one more animal on the shared field. The field is destroyed by people behaving reasonably.
196815 min read
The Twin Paradox
One twin flies away and comes home younger than the other. Yet each of them saw the other's clock running slow.
1905 (implicit in Einstein), 1911 (Langevin's traveller)20 min read
Gabriel's Horn
A shape you can fill with a finite bucket of paint but whose inside you could never finish painting. How?
16417 min read
Moravec's Paradox
A computer beat the world chess champion in 1997. Getting one to pick up a cup it has not seen before is still hard.
198811 min read
Olbers' Paradox
If the universe is infinite and full of stars, every line of sight should end on one. So why is the night sky dark?
1610 (Kepler), 1744 (de Chéseaux), 1823 (Olbers)17 min read
The Boy or Girl Paradox
A family has two children and one of them is a boy. What are the odds the other is a girl? It depends on how you found out.
1959 (posed), 1961 (retracted in part)13 min read
The Ellsberg Paradox
You know exactly how many red balls there are, and nothing about how many are black. Does knowing more make a bet more attractive?
19617 min read
The Jevons Paradox
Build an engine that burns less coal and the country ends up burning more coal. This was noticed in 1865 and has not stopped.
186518 min read
The Mpemba Effect
A pot of boiling water and a pot of cold water go into the same freezer. Which freezes first?
1963 (observed), 1969 (published)8 min read
The Paradox of the Court
A student agrees to pay for his lessons the day he wins his first case. He never takes a case. His teacher sues him for the money.
c. 420 BCE (traditional); earliest surviving account 2nd century CE20 min read
The Paradox of Thrift
Every family starts saving more. The economy ends up with less saving. How can everyone doing the right thing produce the wrong outcome?
19367 min read
The Two Generals' Problem
Two armies must attack at the same moment. Any message can be lost, including the one confirming the last one arrived.
1975 (proved), 1978 (named)18 min read
Berkson's Paradox
Two things with nothing whatever to do with each other can look strongly related — purely because of who ended up in the study.
194615 min read
Galileo's Paradox
There are exactly as many square numbers as there are numbers, even though almost every number is not a square.
163813 min read
Meno's Paradox
How can you search for something you don't know? And how would you recognise it if you found it?
c. 380 BCE9 min read
Peto's Paradox
A whale has a thousand times more cells than you and lives longer. It should be riddled with cancer. It is not.
1975 (observation), 1977 (Peto's fuller statement)17 min read
The C-value Paradox
An onion carries five times more DNA than you do. A single-celled amoeba carries a hundred times more. Nobody expected that.
1971 (named), 2001 (renamed the C-value enigma)17 min read
The Easterlin Paradox
Richer people are happier than poorer people. Yet a whole country getting richer does not seem to get happier.
197423 min read
The Grelling-Nelson Paradox
Is the word "heterological" heterological? Either answer generates a contradiction.
19087 min read
The Tristram Shandy Paradox
A man writes his autobiography but keeps falling behind. If he had infinite time, would he ever catch up?
1759 (novel), 1903 (paradox formulated)8 min read
Advanced
Undergraduate level in the relevant field.
Maxwell's Demon
A tiny creature sorting fast molecules from slow ones appears to make heat flow the wrong way, for free, forever.
1867 (letter to Tait), 1871 (published in Theory of Heat)18 min read
Russell's Paradox
Consider the set of all sets that do not contain themselves. Does that set contain itself? Either answer is wrong.
1901 (discovered), 1902 (letter to Frege), 1903 (published)19 min read
The Halting Problem
No program can ever be written that reliably tells you whether another program will finish running. This is not a limit of our cleverness.
193620 min read
The Liar Paradox
"This sentence is false." Decide whether you are looking at something true or something false, and see how far you get.
c. 600 BCE (the Epimenides line); c. 350 BCE (Eubulides' paradox)26 min read
Zeno's Dichotomy Paradox
To cross a room you must first cross half of it, then half of what is left, and so on without end. So how does anyone move?
c. 450 BCE18 min read
Braess's Paradox
Add a new road to a congested city and every journey gets slower. Close the road again and the traffic speeds up.
196819 min read
Cantor's Diagonal Argument
There are more numbers between 0 and 1 than there are whole numbers in all of eternity. Both lists are infinite.
189116 min read
Newcomb's Problem
The boxes are already filled. Does choosing one still change what is in them?
c. 1960 (posed), 1969 (published)10 min read
The Allais Paradox
Offered the same improvement in odds twice, most people take it once and refuse it the second time. Both choices feel obviously right.
1952 (posed at the Paris colloquium), 1953 (published)22 min read
The Byzantine Generals Problem
Several generals must attack together or be destroyed. They can only send messengers, and some of the generals are traitors.
198222 min read
The CAP Theorem
A system spread across several machines can give you consistent answers, or always answer at all. The network makes you choose.
2000 (conjectured), 2002 (proved)23 min read
The Sleeping Beauty Problem
Woken with no memory of having been woken before, how confident should you be that the coin came up heads?
1997 (posed), 2000 (named and popularised)24 min read
The Sorites Paradox
One grain of sand is not a heap. Adding a single grain never turns a non-heap into a heap. So heaps do not exist.
c. 350 BCE20 min read
The St. Petersburg Paradox
A coin-flipping game whose average winnings are infinite. How much of your own money would you actually pay to play?
1713 (posed by Nicolaus Bernoulli), 1738 (published by Daniel Bernoulli)19 min read
The Two Envelopes Problem
One envelope holds twice as much money as the other. The arithmetic says swap — then says swap back, forever.
1953 (Littlewood, crediting Schrödinger), 1989 (envelope form)19 min read
The Unexpected Hanging Paradox
A prisoner is told he will hang next week, on a day he cannot possibly predict. He proves it cannot happen. It happens.
1943 (in circulation), 1948 (first in print)21 min read
Berry's Paradox
"The smallest number that cannot be described in fewer than twelve words." That description is eleven words long.
1906 (published by Russell, crediting Berry); communicated c. 190419 min read
Levinthal's Paradox
A protein has more possible shapes than the universe has atoms, yet it finds the right one in less than a thousandth of a second.
1968–196917 min read
The Giffen Paradox
Some things sell more as the price goes up, and the people buying them are behaving perfectly sensibly.
1895 (Marshall's third edition); 2008 (first credible field evidence)18 min read
The Ladder Paradox
A ladder too long to fit inside a barn fits inside it — and also does not — depending entirely on who is watching.
1905 (the physics), 1961 (first prominent published version)19 min read
The Productivity Paradox
Computers arrived in every office through the 1970s and 80s. Measured productivity did not move for two decades.
1987 (Solow's remark), debate ongoing17 min read
The Ross-Littlewood Paradox
You add ten balls and remove one, infinitely many times. At the end, the vase is empty. What happened to all the balls?
19887 min read
Expert
Graduate level or specialist readers.
Gödel's Incompleteness Theorems
In any system of arithmetic strong enough to be useful, there are true statements it can never prove. Including this one.
193125 min read
The Banach–Tarski Paradox
Cut a solid ball into five pieces, move them around, and you have two balls the same size as the first. Nothing was stretched.
192423 min read
Arrow's Impossibility Theorem
There is no fair way to turn what a group of people each want into what the group wants. This is not a complaint about politics. It is a theorem.
1950 (paper), 1951 (book), 1963 (corrected second edition)25 min read
The Black Hole Information Paradox
Physics insists information can never be destroyed. Black holes appear to destroy it. Both of those cannot be true.
1974/75 (Hawking radiation), 1976 (the paradox stated)24 min read
The EPR Paradox
Measure one particle here and you instantly know something about its partner light-years away. Einstein thought this proved physics was unfinished.
1935 (argument), 1964 (Bell's theorem), 2015 (loophole-free tests)21 min read
The FLP Impossibility Result
A handful of computers cannot be guaranteed to agree on a single value if even one of them might be slow. Not merely hard — impossible.
198523 min read