Here is an answer to “Test your intuition (8)”. (Essentially the answer posed by David Eppstein.)
(From Wolfram Mathworld)
Buffon’s needle problem asks to find the probability that a needle of length will land on a line, given a floor with equally spaced parallel lines at distance one apart. The problem was posed by Georges-Louis Leclerc, Comte de Buffon (1707 – 1788).
There is a familiar computation showing that when the probability is . A computation-free proof can be found in these pages on geometric probability written mostly by Molly McGinty. (They were pointed to me by Sergiu Hart.)
Briefly the computation-free argument goes as follows:
1) Consider the expected number of crossings of a needle with the lines rather than the probability of crossing.
2) Allow also polygonal needles.
3) Show, based on linearity of expectation, that for a polygonal needle the expected number of crossings is a linear function of the length of the needle.
4) Consider now a circular needle of radius 1/2. Note that with probability one it has two crossings with the lines. Deduce that .
This gives a proof for Buffon’s needle problem. But now consider any closed planar curve with constant width one. Again with probability one it will have two crossings with the parallel lines. Therefore, it has the same perimeter as the circle.
Update: The argument above (allowing broken needles) is known as the Buffon’s noodle problem. There are wikipidea articles about Buffon’s needle and about Buffon’s noodle and the later was started by Yuval Wigderson. A simulation where you can draw your own noodle can be found here. (While the expected number of crossings is the same for all noodles, the more entangled the noodle is, the more likely the simulation will lead to a large error.) The result about the perimeter of planar sets of constat width was proved by Joseph Emile Barbier.