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
Type
Infinity
Unbounded quantities, limits, and completed infinities.
15 entriesFilter these further →
Beginner
Ages 8+, no background needed.
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
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
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
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
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
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.
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 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 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.
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
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