Counter intuitive facts

Thanks to Alexander Kruel for this list:

1. It is possible to compute over encrypted data without access to the secret key Wikipedia.

2. 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.

3. It is possible to play poker by telephone in a trusted way which prevents cheating Wikipedia.

4. 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.

5. 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.

6. Causation does not imply correlation arxiv.

7. 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)

8. There is a surface that has only one side Wikipedia.

9. It is possible to travel downwind faster than the wind. YouTube (for a mechanical demonstration see "Under the ruler faster than the ruler" YouTube).

10. 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.

11. Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it. Wikipedia

12. Closing roads can improve everyone’s commute time. source

13. If you pay the value you think something is worth, you are going to end up with a negative net profit. Wikipedia

14. Adding 3 feet to a tightly tied rope around the earth would allow you to raise it uniformly by almost 6 inches. source

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

16. 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

17. 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

18. A one-in-billion event will happen 8 times a month. source

19. 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

20. A system cannot change while you are watching it: Wikipedia

21. 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)

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

23. There is a shape with a finite volume but an infinite surface area (Gabriel’s Horn). Wikipedia

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

25. There are constant width curves other than a circle. -Wikipedia

26. 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)

27. 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)

28. 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

29. The vast majority of real numbers can't be described. But it is impossible to give a single example. source

30. There exists a curve which fills an entire square Wikipedia

31. There is a continuous and nowhere differentiable function - Wikipedia

32. At any given time there live at least two people in California with the same number of hairs on their heads. source

33. "...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

34. An initial datapoint can be valuable, and the second worthless, but the third valuable again (due to discreteness of choice) Twitter

35. If every truth is knowable, then every truth is known. Wikipedia

## Other classical and miscellaneous items:

1. Simple, yet counterintuitive mathematics | Why numbers don't always mean what you think YouTube

2. Truly brilliant examples from mathematics about why repeated confirmations don’t constitute proofs: The Most Misleading Patterns in Mathematics YouTube

4. Rope, escape, topology, knots, creativity, geometry, mathematics, impossibility, access to higher dimensions of space-time. Source

5. The Lifespan Dilemma source

6. 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

7. Monty Hall problem source

10. Boy or Girl paradox source

11. Cheryl's Birthday source

14. German tank problem source

15. Two envelopes problem source

16. Sleeping Beauty problem Wikipedia

18. The ant on a rubber rope problem Wikipedia

20. 100 Prisoners Problem Wikipedia

21. Gödel's incompleteness theorems source

22. Hairy ball theorem source

23. Wheeler's delayed-choice experiment source

24. A Peculiar Connection Between the Axiom of Choice and Predicting the Future source

25. Quantum Eraser Lottery Challenge YouTube

26. Counterfactual mugging source

27. Vexing Expectations pdf

28. The Absent-Minded Driver source

29. The Hardest Logic Puzzle Ever Wikipedia

30. Seven Puzzles You Think You Must Not Have Heard Correctly pdf