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Famous Problems & Paradoxes: fun facts and quiz
The essentials in 7 ideas, with fun facts, an illustrated story and a quiz to see how much you know.
The essentials
Zeno’s Paradox
Zeno argued that to cross a room you must first cross half of it, then half of what’s left, forever, so you never arrive. Infinite sums solve it: the halves add up to 1.
Key fact
½ + ¼ + ⅛ + … adds up to exactly 1.
The Monty Hall Problem
A prize is behind one of three doors. You pick one; the host, who knows, opens another to show a goat. Switching wins two times in three.
Key fact
Many mathematicians got it wrong when it went viral in 1990.
The Liar Paradox
“This sentence is false.” If it’s true, it’s false; if it’s false, it’s true. Paradoxes like this have puzzled thinkers since ancient Greece.
Key fact
The liar paradox is over 2,000 years old.
Hilbert’s Hotel
A hotel with infinitely many rooms is full, yet it can fit a new guest: everyone moves to the next room up. Infinity doesn’t behave like ordinary numbers.
Key fact
David Hilbert described the hotel in 1924.
The Bridges of Königsberg
Could you walk through Königsberg crossing each of its seven bridges exactly once? In 1736, Leonhard Euler proved it’s impossible.
Key fact
Euler’s proof marked the birth of graph theory.
Fermat’s Last Theorem
In 1637, Pierre de Fermat scribbled that he had a proof too big to fit in the margin. It took until 1994 for Andrew Wiles to prove the theorem.
Key fact
The search took over 350 years.
Unsolved Problems
Some simple-sounding problems are still open. Goldbach’s conjecture says every even number above 2 is the sum of two primes; it’s been checked hugely far, but never proven.
Key fact
Seven Millennium Prize problems, each worth 1 million dollars, were set in 2000.
Fun facts
Zeno is said to have written about 40 paradoxes. Only a few survive, mostly because Aristotle wrote about them.
In a 2010 study, pigeons learned to always switch in the Monty Hall game, while most humans kept sticking with their first door.
An ancient cousin: Epimenides, a man from Crete, supposedly said “All Cretans are liars”.
Hilbert’s hotel can even fit infinitely many new guests: everyone moves to double their room number, freeing every odd room.
Some of Königsberg’s bridges were lost in the Second World War. With the ones left today, Euler’s walk is possible.
Wiles’s first proof, in 1993, had a gap. It took him over a year more to fix it.
Grigori Perelman, who solved one of the Millennium problems, turned down its 1 million dollar prize.
Story: Always Switch
Always Switch
1990. A reader asks columnist Marilyn vos Savant about the three doors: after the host shows a goat, should you switch?
She says yes. About 10,000 readers write in, many with PhDs, telling her she’s got it wrong.
She was right. Even the great mathematician Paul Erdős only believed it after seeing a computer simulation.
True or false?
Tap your answer to see if you got it, and why.
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Question 1 of 10
Infinitely many halves can add up to a finite total.
Show the answer
True. They add up to exactly 1.
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Question 2 of 10
In the Monty Hall problem, switching wins 2 times in 3.
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True. Your first pick was right only 1 time in 3.
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Question 3 of 10
“This sentence is false” can’t consistently be true or false.
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True. Either answer contradicts itself.
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Question 4 of 10
A full hotel with infinitely many rooms can still take a new guest.
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True. Everyone shifts up one room.
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Question 5 of 10
Euler proved the Königsberg bridge walk was impossible.
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True. In 1736, founding graph theory.
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Question 6 of 10
Andrew Wiles proved Fermat’s Last Theorem.
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True. In 1994.
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Question 7 of 10
Goldbach’s conjecture has not been proven.
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True. Nobody has proved it for every even number.
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Question 8 of 10
Zeno’s paradox proves that motion is impossible.
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False. Infinite sums show how you can finish.
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Question 9 of 10
Once a goat door is opened, sticking and switching are 50/50.
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False. The host’s choice gives away information.
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Question 10 of 10
The liar paradox was invented by computer scientists.
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False. Ancient Greek philosophers discussed it.
Keep going in the app
KnowSoda learns what you know and brings back what you’re about to forget. Famous Problems & Paradoxes has 6 more ideas at the Advanced and Expert levels.
Sources
- MacTutor History of Mathematics — Zeno, Euler, Fermat, Hilbert
- Clay Mathematics Institute — Millennium Prize Problems
- Stanford Encyclopedia of Philosophy — Zeno’s paradoxes, liar paradox
- Encyclopaedia Britannica — Monty Hall problem, Königsberg bridge problem
- Stanford Encyclopedia of Philosophy — sorites paradox, Russell’s paradox
- Clay Mathematics Institute — P vs NP
- Stanford Encyclopedia of Philosophy — sorites paradox, Russell’s paradox, the surprise exam paradox
