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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 https://en.wikipedia.org/wiki/Monty_Hall_problem
Unexpected hanging paradox https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
A shape with a finite volume but an infinite surface area (Gabriel’s Horn) https://en.wikipedia.org/wiki/Gabriel%27s_Horn
Monty Hall problem https://en.wikipedia.org/wiki/Monty_Hall_problem
Unexpected hanging paradox https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
A shape with a finite volume but an infinite surface area (Gabriel’s Horn) https://en.wikipedia.org/wiki/Gabriel%27s_Horn
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? http://puzzles.nigelcoldwell.co.uk/fortyone.htm
As @QVagabond points out, Gödel's incompleteness theorems are a mandatory part of any list of counterintuitive results https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
"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
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
Maths 1001, @RichardElwes
Goodstein's Theorem https://en.m.wikipedia.org/wiki/Goodstein%27s_theorem
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.
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
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 https://en.wikipedia.org/wiki/Nontransitive_dice
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) https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester
See also: https://fqxi.org/community/forum/topic/3345
See also: https://fqxi.org/community/forum/topic/3345
Homomorphic encryption: Computing over encrypted data without access to the secret key. https://en.wikipedia.org/wiki/Homomorphic_encryption
Zero-knowledge proof: Proving that you know a value x, without conveying any information apart from the fact that you know the value x. https://en.wikipedia.org/wiki/Zero-knowledge_proof
Zero-knowledge proof: Proving that you know a value x, without conveying any information apart from the fact that you know the value x. https://en.wikipedia.org/wiki/Zero-knowledge_proof
Banach–Tarski paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
"The maths of queuing are absolutely brutal and counter-intuitive." https://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/
Many great examples from game theory https://williamspaniel.com/2014/05/25/game-theory-is-really-counterintuitive/
These will blow your mind: Simple, yet counterintuitive mathematics | Why numbers don't always mean what you think
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 https://en.wikipedia.org/wiki/Wheeler%27s_delayed-choice_experiment
A Peculiar Connection Between the Axiom of Choice and Predicting the Future https://web.archive.org/web/20100923004908/http://persweb.wabash.edu/facstaff/hardinc/pub/peculiar.pdf
More classics:
Zeno's paradoxes https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
Boy or Girl paradox https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
Cheryl's Birthday https://en.wikipedia.org/wiki/Cheryl%27s_Birthday
Zeno's paradoxes https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
Boy or Girl paradox https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
Cheryl's Birthday https://en.wikipedia.org/wiki/Cheryl%27s_Birthday
The Birthday Paradox http://en.wikipedia.org/wiki/Birthday_problem
Probability theory features a lot of counterintuitive results https://math.stackexchange.com/questions/2140493/counterintuitive-examples-in-probability
Probability theory features a lot of counterintuitive results https://math.stackexchange.com/questions/2140493/counterintuitive-examples-in-probability
Many more great examples can be found here https://math.stackexchange.com/questions/2040811/what-are-some-counter-intuitive-results-in-mathematics-that-involve-only-finite
Ross–Littlewood paradox https://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox
German tank problem https://en.wikipedia.org/wiki/German_tank_problem
Two envelopes problem https://en.wikipedia.org/wiki/Two_envelopes_problem
Sleeping Beauty problem https://en.wikipedia.org/wiki/Sleeping_Beauty_problem
The Lifespan Dilemma http://lesswrong.com/lw/17h/the_lifespan_dilemma/
German tank problem https://en.wikipedia.org/wiki/German_tank_problem
Two envelopes problem https://en.wikipedia.org/wiki/Two_envelopes_problem
Sleeping Beauty problem https://en.wikipedia.org/wiki/Sleeping_Beauty_problem
The Lifespan Dilemma http://lesswrong.com/lw/17h/the_lifespan_dilemma/
Counterfactual mugging https://wiki.lesswrong.com/wiki/Counterfactual_mugging
Vexing Expectations https://authors.library.caltech.edu/7496/1/NOVmind04.pdf
The Absent-Minded Driver http://lesswrong.com/lw/182/the_absentminded_driver/
The Hardest Logic Puzzle Ever https://en.wikipedia.org/wiki/The_Hardest_Logic_Puzzle_Ever
Seven Puzzles You Think You Must Not Have Heard Correctly https://math.dartmouth.edu/~pw/solutions.pdf
Vexing Expectations https://authors.library.caltech.edu/7496/1/NOVmind04.pdf
The Absent-Minded Driver http://lesswrong.com/lw/182/the_absentminded_driver/
The Hardest Logic Puzzle Ever https://en.wikipedia.org/wiki/The_Hardest_Logic_Puzzle_Ever
Seven Puzzles You Think You Must Not Have Heard Correctly https://math.dartmouth.edu/~pw/solutions.pdf
Simpson's Paradox https://en.wikipedia.org/wiki/Simpson%27s_paradox
Berkson's paradox (via @naive_skeptic) https://en.wikipedia.org/wiki/Berkson%27s_paradox
Berkson's paradox (via @naive_skeptic) https://en.wikipedia.org/wiki/Berkson%27s_paradox
The long line is longer than the real line https://en.wikipedia.org/wiki/Long_line_(topology)
The vast majority of real numbers can't be described. But it is impossible to give a single example. https://blog.ram.rachum.com/post/54747783932/indescribable-numbers-the-theorem-that-made-me#:~:text=The%20answer%20to%20the%20question,of%20the%20familiar%20describable%20variety.
The ant on a rubber rope problem
https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope
Infinite offset paradox
https://en.wikipedia.org/wiki/Block-stacking_problem
100 Prisoners Problem
https://en.wikipedia.org/wiki/100_prisoners_problem
via @DavidB52s
https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope
Infinite offset paradox
https://en.wikipedia.org/wiki/Block-stacking_problem
100 Prisoners Problem
https://en.wikipedia.org/wiki/100_prisoners_problem
via @DavidB52s
There are infinite sets that can be exhaustively searched over in finite time http://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/
A curve may fill an entire square. https://en.wikipedia.org/wiki/Space-filling_curve
There is a surface which has only one side. https://en.wikipedia.org/wiki/Mobius_strip
There are constant width curves other than a circle. https://en.wikipedia.org/wiki/Curve_of_constant_width
There is a surface which has only one side. https://en.wikipedia.org/wiki/Mobius_strip
There are constant width curves other than a circle. https://en.wikipedia.org/wiki/Curve_of_constant_width
There is a continuous and nowhere differentiable function. https://en.wikipedia.org/wiki/Weierstrass_function
It is possible to play poker by telephone in a trusted way which prevents cheating. http://www.math.stonybrook.edu/~scott/blair/How_play_poker.html
It is possible to play poker by telephone in a trusted way which prevents cheating. http://www.math.stonybrook.edu/~scott/blair/How_play_poker.html
"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→∞."
Littlewood’s Law of Miracles: a one-in-billion event will happen 8 times a month. https://www.gwern.net/Littlewood
Why Unlikely Events Are Not Unlikely http://daviddfriedman.blogspot.com/2015/09/why-unlikely-events-are-not-unlikely.html
Why a Polish Village Hasn’t Seen a Baby Boy Born for Almost 10 Years http://blogs.discovermagazine.com/crux/2019/08/21/why-a-polish-village-hasnt-seen-a-baby-boy-born-for-almost-10-years/#.XV7o5egzabg
Why Unlikely Events Are Not Unlikely http://daviddfriedman.blogspot.com/2015/09/why-unlikely-events-are-not-unlikely.html
Why a Polish Village Hasn’t Seen a Baby Boy Born for Almost 10 Years http://blogs.discovermagazine.com/crux/2019/08/21/why-a-polish-village-hasnt-seen-a-baby-boy-born-for-almost-10-years/#.XV7o5egzabg
Quantum Zeno effect: "a system cannot change while you are watching it" https://en.m.wikipedia.org/wiki/Quantum_Zeno_effect
Truly brilliant examples from mathematics about why repeated confirmations don’t constitute proofs: The Most Misleading Patterns in Mathematics
