It is possible to compute over encrypted data without access to the secret key: https://en.wikipedia.org/wiki/Homomorphic_encryption

It is possible to prove 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

It is possible to play poker by telephone in a trusted way which prevents cheating: http://math.stonybrook.edu/~scott/blair/How_play_poker.html

If customers take on average 10 minutes to serve and they arrive randomly at a rate of 5.8 per hour then the waiting time for one teller is five hours while the waiting time for two tellers is 3 minutes: https://www.johndcook.com/blog/2008/10/21/what-happens-when-you-add-a-new-teller/

There exists a set of three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time: https://en.wikipedia.org/wiki/Nontransitive_dice

Causation does not imply correlation: https://arxiv.org/abs/1505.03118

The Earth makes 366.25 rotations around its axis per year. (Related: 0% selected the right answer on this SAT question: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total? youtube.com/watch?v=kN3AOMrnEUs)

There is a surface that has only one side: https://en.wikipedia.org/wiki/Mobius_strip

It is possible to travel downwind faster than the wind: youtube.com/watch?v=jyQwgBAaBag (for a mechanical demonstration see: Under the ruler faster than the ruler youtube.com/watch?v=k-trDF8Yldc)

It is possible to read out the results of events that 'didn't happen' and whose 'probability of happening' can be driven arbitrarily low: https://fqxi.org/community/forum/topic/3345

Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it: https://en.wikipedia.org/wiki/The_Market_for_Lemons

Closing roads can improve everyone’s commute time: https://mindyourdecisions.com/blog/2009/01/06/why-the-secret-to-speedier-highways-might-be-closing-some-roads-the-braess-paradox/#.U4Ksl_ldUud

If you pay the value you think something is worth, you are going to end up with a negative net profit: http://en.wikipedia.org/wiki/Winner%27s_curse

Adding 3 feet to a tightly tied rope around the earth would allow you to raise it uniformly by almost 6 inches: http://puzzles.nigelcoldwell.co.uk/fortyone.htm

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

If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox

At any given moment on the earth's surface, there exist 2 antipodal points (on exactly opposite sides of the earth) with the same temperature and barometric pressure: youtube.com/watch?v=cchIr1OXc8E

100 prisoners problem -- The Riddle That Seems Impossible Even If You Know The Answer https://youtu.be/iSNsgj1OCLA

A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood

If you have galaxies of the same size at different distances, beyond a certain distance, the farther away the galaxy is the BIGGER it appears in the sky: https://twitter.com/AstroKatie/status/1516548836709343238

Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

A system cannot change while you are watching it: https://en.m.wikipedia.org/wiki/Quantum_Zeno_effect

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, Richard Elwes)

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

There is a shape with a finite volume but an infinite surface area (Gabriel’s Horn): https://en.wikipedia.org/wiki/Gabriel%27s_Horn

There are infinite sets that can be exhaustively searched over in finite time: http://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/

There are constant width curves other than a circle: https://en.wikipedia.org/wiki/Curve_of_constant_width

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)

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)

There are 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. https://en.m.wikipedia.org/wiki/Goodstein%27s_theorem

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

There exists a curve which fills an entire square: https://en.wikipedia.org/wiki/Space-filling_curve

There is a continuous and nowhere differentiable function: https://en.wikipedia.org/wiki/Weierstrass_function

At any given time there live at least two people in California with the same number of hairs on their heads: https://medium.com/cantors-paradise/the-pigeonhole-principle-e4c637940619

"...if you flip fair coins to generate n-dimensional vectors (heads => 1, tails => -1) then the probability they're linearly independent is at least 1-(1/2 + o(n))^n. I.e., they're very very likely independent!" twitter.com/michael_nielsen/status/1398408973657677825

An initial datapoint can be valuable, and the second worthless, but the third valuable again (due to discreteness of choice) twitter.com/ben_golub/status/1402780581029683203

If every truth is knowable, then every truth is known. https://en.wikipedia.org/wiki/Fitch%27s_paradox_of_knowability

Borromean rings: "No two of them link, but the three cannot be pulled apart. The three rings are trapped together until one of them leaves and sets the others free." https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-borromean-rings/

An example of a problem where the expected value and the long-run average are NOT the same! This is an example of a "non-ergodic" system: even as N→∞, individual runs DON'T converge to the expected value. Thread: https://threadreaderapp.com/thread/1592528685542047744.html | Video: https://www.youtube.com/watch?v=CCLtQHL-VUs

The end of a set of belts can be continuously rotated without becoming twisted or tangled: https://en.m.wikipedia.org/wiki/Anti-twister_mechanism

### Other classical and miscellaneous items:

Simple, yet counterintuitive mathematics | Why numbers don't always mean what you think youtu.be/xHjQhliXUB0

Truly brilliant examples from mathematics about why repeated confirmations don’t constitute proofs: The Most Misleading Patterns in Mathematics youtu.be/kp1C0E8Za7k

The Spring Paradox (watch the whole awesome video) youtube.com/watch?v=Cg73j3QYRJc

Rope, escape, topology, knots, creativity, geometry, mathematics, impossibility, access to higher dimensions of space-time. https://www.reddit.com/r/knots/comments/mhimtn/topology_demonstrations/

The Lifespan Dilemma http://lesswrong.com/lw/17h/the_lifespan_dilemma/

Bottema's theorem: Draw squares on AB and BC on two sides of the triangle ABC. Let R and S be the points on the squares opposite vertex B. Then the midpoint M of RS is independent of B. https://en.wikipedia.org/wiki/Bottema%27s_theorem

Monty Hall problem https://en.wikipedia.org/wiki/Monty_Hall_problem

Unexpected hanging paradox https://en.wikipedia.org/wiki/Unexpected_hanging_paradox

What the Tortoise Said to Achilles https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles

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

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

Stein's paradox twitter.com/johncarlosbaez/status/1298274201682325509

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

Gödel's incompleteness theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Hairy ball theorem https://en.m.wikipedia.org/wiki/Hairy_ball_theorem

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

Quantum Eraser Lottery Challenge youtube.com/watch?v=2Uzytrooz44

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

Simpson's Paradox https://en.wikipedia.org/wiki/Simpson%27s_paradox

Berkson's paradox https://en.wikipedia.org/wiki/Berkson%27s_paradox

Counterintuitive examples in probability https://math.stackexchange.com/questions/2140493/counterintuitive-examples-in-probability

What are some counter-intuitive results in mathematics that involve only finite objects? https://math.stackexchange.com/questions/2040811/what-are-some-counter-intuitive-results-in-mathematics-that-involve-only-finite

## The most counterintuitive facts in all of mathematics, computer science, and physics

hot water freezes faster than cold

https://en.wikipedia.org/wiki/Mpemba_effect

everything is subjective. especially laws.