Thread by @XiXiDu

What's the most counterintuitive fact of all of mathematics, computer science, and physics?

Here are some suggestions I received. Let's start with some of the classics:

Monty Hall problem en.wikipedia.org/wiki/Monty_Hal…

Unexpected hanging paradox en.wikipedia.org/wiki/Unexpecte…

A shape with a finite volume but an infinite surface area (Gabriel’s Horn) en.wikipedia.org/wiki/Gabriel%2…

Suppose you tie a rope tightly around the Earth's equator. You add an extra 3 feet to the length. All around the Earth the rope is raised up uniformly as high as is possible to make it tight again. How high is that? puzzles.nigelcoldwell.co.uk/fortyone.htm

As @QVagabond points out, Gödel's incompleteness theorems are a mandatory part of any list of counterintuitive results en.wikipedia.org/wiki/G%C3%B6de…

"Any positive rational number x can be written as a finite sum of distinct numbers of the form 1/n."
Calculus, 4th edition by Michael Spivak

"In two dimensions, there are infinitely many regular polygons. In three dimensions, there are five Platonic solids. In four dimensions, there are six platonic polychora. In all higher dimensions than four, there are only ever three regular polytopes."
Maths 1001, @RichardElwes

Goodstein's Theorem en.m.wikipedia.org/wiki/Goodstein…

Sequences of numbers which grow unimaginably enormous and continue for an unimaginably long number of terms... but which always eventually get back down to zero.

"Let

alpha = 0.110001000000000000000001000...,

where the 1's occur in the n! place, for each n.

Then alpha is transcendental."
Calculus, 4th edition by Michael Spivak

Stein's paradox

https://twitter.com/johncarlosbaez/status/1298274201682325509

There are as many whole positive numbers as all fractions (including the whole negative and whole positive numbers).

The existence of non-transitive dice en.wikipedia.org/wiki/Nontransi…

Elitzur–Vaidman bomb tester: "we can read out the results of events that 'didn't happen' and whose 'probability of happening' can be driven arbitrarily low." (via @ESYudkowsky) en.wikipedia.org/wiki/Elitzur%E…

See also: fqxi.org/community/foru…

Homomorphic encryption: Computing over encrypted data without access to the secret key. en.wikipedia.org/wiki/Homomorph…

Zero-knowledge proof: Proving that you know a value x, without conveying any information apart from the fact that you know the value x. en.wikipedia.org/wiki/Zero-know…

Banach–Tarski paradox en.wikipedia.org/wiki/Banach%E2…

"The maths of queuing are absolutely brutal and counter-intuitive." johndcook.com/blog/2008/10/2…

Many great examples from game theory williamspaniel.com/2014/05/25/gam…

These will blow your mind: Simple, yet counterintuitive mathematics | Why numbers don't always mean what you think

Hairy ball theorem en.m.wikipedia.org/wiki/Hairy_bal…

via @btcvol

Two 12 Inch Pizzas have less Pizza then one 18 inch pizza.

https://twitter.com/fermatslibrary/status/1082273172114862083

Quantum Eraser Lottery Challenge

Wheeler's delayed-choice experiment en.wikipedia.org/wiki/Wheeler%2…

A Peculiar Connection Between the Axiom of Choice and Predicting the Future web.archive.org/web/2010092300…

More classics:

Zeno's paradoxes en.wikipedia.org/wiki/Zeno%27s_…

Boy or Girl paradox en.wikipedia.org/wiki/Boy_or_Gi…

Cheryl's Birthday en.wikipedia.org/wiki/Cheryl%27…

The Birthday Paradox en.wikipedia.org/wiki/Birthday_…

Probability theory features a lot of counterintuitive results math.stackexchange.com/questions/2140…

Many more great examples can be found here math.stackexchange.com/questions/2040…

Ross–Littlewood paradox en.wikipedia.org/wiki/Ross%E2%8…

German tank problem en.wikipedia.org/wiki/German_ta…

Two envelopes problem en.wikipedia.org/wiki/Two_envel…

Sleeping Beauty problem en.wikipedia.org/wiki/Sleeping_…

The Lifespan Dilemma lesswrong.com/lw/17h/the_lif…

Counterfactual mugging wiki.lesswrong.com/wiki/Counterfa…

Vexing Expectations authors.library.caltech.edu/7496/1/NOVmind…

The Absent-Minded Driver lesswrong.com/lw/182/the_abs…

The Hardest Logic Puzzle Ever en.wikipedia.org/wiki/The_Harde…

Seven Puzzles You Think You Must Not Have Heard Correctly math.dartmouth.edu/~pw/solutions.…

Simpson's Paradox en.wikipedia.org/wiki/Simpson%2…

Berkson's paradox (via @naive_skeptic) en.wikipedia.org/wiki/Berkson%2…

The long line is longer than the real line en.wikipedia.org/wiki/Long_line…

The vast majority of real numbers can't be described. But it is impossible to give a single example. blog.ram.rachum.com/post/547477839….

The ant on a rubber rope problem
en.wikipedia.org/wiki/Ant_on_a_…

Infinite offset paradox
en.wikipedia.org/wiki/Block-sta…

100 Prisoners Problem
en.wikipedia.org/wiki/100_priso…

via @DavidB52s

There are infinite sets that can be exhaustively searched over in finite time math.andrej.com/2007/09/28/see…

A curve may fill an entire square. en.wikipedia.org/wiki/Space-fil…

There is a surface which has only one side. en.wikipedia.org/wiki/Mobius_st…

There are constant width curves other than a circle. en.wikipedia.org/wiki/Curve_of_…

There is a continuous and nowhere differentiable function. en.wikipedia.org/wiki/Weierstra…

It is possible to play poker by telephone in a trusted way which prevents cheating. math.stonybrook.edu/~scott/blair/H…

"The volume of a unit sphere of dimension n first grows as n grows (2,π,4π/3,…) but starts decreasing for n=6 and eventually converges to 0 as n→∞."