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