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
Domain
Mathematics
Where rigorous proof leads somewhere your gut refuses to follow.
28 entriesFilter these further →
Beginner
Ages 8+, no background needed.
Occam's Razor
When two explanations fit the evidence equally, which should you believe? The one that assumes less.
c. 13206 min read
Pareto Principle
80% of results come from 20% of effort. 80% of bugs from 20% of code. Is this a law of nature, or a coincidence with a catchy ratio?
18966 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 Butterfly Effect
Can a butterfly in Brazil cause a tornado in Texas? Not by force — but by changing everything that follows.
1963 (discovery), 1972 (butterfly metaphor coined)7 min read
Hofstadter's Law
It always takes longer than you think — and that's still true even after you account for the fact that it always takes longer than you think.
19796 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 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.
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
Benford's Law
Look at the first digit of every number in a newspaper. You'll see 1 far more often than 9 — and it is a mathematical law, not a coincidence.
19388 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
Zipf's Law
The most common English word is twice as frequent as the second. Why does language obey the same mathematical law as city populations?
19357 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 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.
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
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
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
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 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