Ants on a circle
You have a circle with a number of ants scattered around it at distinct points. Each ant starts walking at the same speed but in possibly different directions, either clockwise or anticlockwise. When two ants meet they immediately change directions, and then continue with the same speed as before. Will the ants ever, simultaneously, be in the same positions as when they started out?
(Thanks to OMD.)
Four switches and a lightbulb
Outside a room there are four switches, and in the room there is a lightbulb. One of the switches controls the light. Your task is to find out which one. You cannot see the bulb or whether it is on or off from outside the room. You may turn any number of switches on or off, any number of times you want. But you may only enter the room once. (Thanks to Tomfr.)
In a dark room there is a table, and on this table there are 52 cards, 19 face up, 33 face down. Your task is to divide the cards into two groups, such that in each group there must be the same number of face up cards. You can't switch on a light, ask a friend for help, all the usual disalloweds. Is this even possible?
(Thanks to golftango and Bruno Dupire.)
A group of children are playing and some of them get mud on their foreheads. A child cannot tell if he has mud on his own forehead, although he can see the mud on the foreheads of any other muddy children. An adult comes to collect the children and announces that at least one of the children has a dirty forehead, and then asks the group to put up their hand if they know that they have mud on their forehead. How can each child determine whether or not their forehead is muddy without communicating with anyone else? (Thanks to weaves.)
The Oracle at Delphi
On January 1st you go to the Oracle at Delphi who tells you the opening and closing prices of a small non-dividend-paying stock every trading day for the rest of the year. Every opening price is the same as the closing price the day before. You have a 0.5% one-way transaction cost in buying or selling the stock, and can buy every day at the opening price and sell every day at the closing price... if you choose. On the last day of the year you must not own the stock. What is the best you can do, having this perfect foresight? Every day you can buy stock at the opening price if you don t own it, and sell stock at the closing price if you do own it. Keep the problem simple, no leveraging, no short selling, no options or futures, etc. (Thanks to cdmurray80.)
You need to hire a secretary. There are n possible candidates to interview and you want to find the best, the most talented. The problem is that there is great demand for secretaries, so if you want to make sure that you get one you ll have to offer her the job on the spot. Once she walks out of the door she s gone. You start interviewing candidates one after the other, they can all be ranked, so this one is better than that, or that one is worse than another, etc. There are no ties. But the order in which they are interviewed is random. What is the best strategy for maximizing the probability of getting the best secretary?
There are 10 pirates in a rowing boat. Their ship has just sunk but they managed to save 1,000 gold doubloons. Being greedy bastards they each want all the loot for themselves but they are also democratic and want to make the allocation of gold as fair as possible. But how?
They each pick a number, from 1 to 10, out of a hat. Each person in turn starting with number 1, decides how to divvy up the loot among the pirates in the boat. They then vote. If the majority of pirates approve of the allocation then the loot is divided accordingly, otherwise that particular pirate is thrown overboard into the shark-infested sea. In the latter case, the next pirate in line gets his chance at divvying up the loot. The same rules apply, and either the division of the filthy lucre gets the majority vote or the unfortunate soul ends up in Davy Jones's locker.
Question, how should the first pirate share out the spoils so as to both guarantee his survival and get a decent piece of the action?