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
Status
Resolved
Settled. There is a correct answer and experts agree on it.
30 entriesFilter these further →
Beginner
Ages 8+, no background needed.
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
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 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
Intermediate
Curious teenagers and general adult readers.
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 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
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 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
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
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
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
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
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 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
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 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