Thanks to Alexander Kruel for this list:
It is possible to compute over encrypted data without access to the secret key Wikipedia.
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 Wikipedia.
It is possible to play poker by telephone in a trusted way which prevents cheating Wikipedia.
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 source.
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 Wikipedia.
Causation does not imply correlation arxiv.
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? source)
There is a surface that has only one side Wikipedia.
It is possible to travel downwind faster than the wind. YouTube (for a mechanical demonstration see "Under the ruler faster than the ruler" YouTube).
It is possible to read out the results of events that 'didn't happen' and whose 'probability of happening' can be driven arbitrarily low. source.
Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it. Wikipedia
Closing roads can improve everyone’s commute time. source
If you pay the value you think something is worth, you are going to end up with a negative net profit. Wikipedia
Adding 3 feet to a tightly tied rope around the earth would allow you to raise it uniformly by almost 6 inches. source
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. Wikipedia
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
A one-in-billion event will happen 8 times a month. source
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: Wikipedia
A system cannot change while you are watching it: Wikipedia
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). Wikipedia
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. -Wikipedia
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. Wikipedia
The vast majority of real numbers can't be described. But it is impossible to give a single example. source
There exists a curve which fills an entire square Wikipedia
There is a continuous and nowhere differentiable function - Wikipedia
At any given time there live at least two people in California with the same number of hairs on their heads. source
"...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
An initial datapoint can be valuable, and the second worthless, but the third valuable again (due to discreteness of choice) Twitter
If every truth is knowable, then every truth is known. Wikipedia
Other classical and miscellaneous items:
Simple, yet counterintuitive mathematics | Why numbers don't always mean what you think YouTube
Truly brilliant examples from mathematics about why repeated confirmations don’t constitute proofs: The Most Misleading Patterns in Mathematics YouTube
The Spring Paradox (watch the whole awesome video) YouTube
Rope, escape, topology, knots, creativity, geometry, mathematics, impossibility, access to higher dimensions of space-time. Source
The Lifespan Dilemma source
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. Wikipedia
Monty Hall problem source
Unexpected hanging paradox source
Zeno's paradoxes source
Boy or Girl paradox source
Cheryl's Birthday source
The Birthday Paradox source
Ross–Littlewood paradox source
German tank problem source
Two envelopes problem source
Sleeping Beauty problem Wikipedia
Stein's paradox source
The ant on a rubber rope problem Wikipedia
Infinite offset paradox Wikipedia
100 Prisoners Problem Wikipedia
Gödel's incompleteness theorems source
Hairy ball theorem source
Wheeler's delayed-choice experiment source
A Peculiar Connection Between the Axiom of Choice and Predicting the Future source
Quantum Eraser Lottery Challenge YouTube
Counterfactual mugging source
Vexing Expectations pdf
The Absent-Minded Driver source
The Hardest Logic Puzzle Ever Wikipedia
Seven Puzzles You Think You Must Not Have Heard Correctly pdf
Simpson's Paradox wikipedia
Berkson's paradox wikipedia
Counterintuitive examples in probability stack exchange
What are some counter-intuitive results in mathematics that involve only finite objects? stack exchange