A team of N=100 contestants must choose a strategy for the following game. The members of the team are numbered from 1 to N. When the game begins, no further communications between team members are permitted. The game host creates a board with N little doors, numbered, on their fronts, from 1 to N. Behind the doors, he also writes the numbers from 1 to N, in a random permutation. (So behind door number 1, we might find the hidden number is 42; all permutations are equally probable.) Each hidden number is written either in red ink or in blue ink, chosen independently and randomly. Each contestant is allowed to open up to N/2 doors and look at the hidden numbers behind them. Then all the contestants must guess the colour of _their_ hidden number. For example, contestant number 42 must guess the colour of the hidden number 42, wherever it is located. The team wins a big prize if _all_ the contestants guess their hidden numbers' colours correctly. Find a strategy that gives the team a substantial probability of winning.
Hints and most of a Solution: cycles.ps | cycles.pdf
David MacKay; puzzle originally heard from MSRI, Berkeley.You arrive at a movie theater for a busy film, and there is a line about to form to buy tickets, but they have not yet started selling them. There is a large sign posted at the head of the line declaring that the first person to purchase a ticket who has the same birthday as someone who has already bought a ticket will win $100.
Since there is nobody in line yet, you have the option of joining the line at any time, and thus attaining any position you desire in line. Assuming that you don't know anyone else's birthday who plans to line up, and that the birthday of each person in line will be independently and uniformly distributed throughout the year, what position in line gives you the greatest chance of being the first duplicate birthday?
An insect with a very thin rope wants to hang from a sharpened pencil by putting a noose
_ ----<_)around the tip of the pencil. Assuming that the pencil is oriented vertically, that the tip is a perfect cone, and frictionless, and that the angle of the pencil's tip is alpha, for what range of alpha will the noose stay on the pencil?
Hint - there is a connection to the December lecture on general relativity.