20-12-2012, 02:08 PM
Puzzles
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Here are some mathematical puzzles that I have enjoyed. Most of them are of the kind that you can discuss and solve at a dinner table, usually without pen and paper. So as not to spoil your fun, no solutions are given on this page, but for some problems I have provided some hints.
Planar configuration of straight connecting lines
Given an even number of points in general positions on the plane (that is, no three points co-linear), can you partition the points into pairs and connect the two points of each pair with a single straight line such that the straight lines do not overlap?
Reducing nearby enemies
You are given an irreflexive symmetric (but not necessarily transitive) "enemies" relation on a set of people. In other words, if person A is an enemy of a person B, then B is also an enemy of A. How can you divide up the people into two houses in such a way that every person has at least as many enemies in the other house as in their own house?
Transporting bananas
You have 3000 bananas that you want to transport a distance of 1000 km. The transport will be done by a monkey. The monkey can carry as many as 1000 bananas at any one time. With each kilometer traveled (forward or backward), the money consumes 1 banana. How many bananas can you get across to the other side?
Car and key hide-and-seek
In a room are three boxes that on the outside look identical. One of the boxes contains a car, one contains a key, and one contains nothing. You and a partner get to decide amongst yourselves to each point to two boxes. When you have made your decision, the boxes are opened and their contents revealed. If one of the boxes your partner is pointing to contains the car and one of the boxes you are pointing to contains the key, then you will both win. What strategy maximizes the probability of winning, and what is the probability that you will win?
The hidden card
In this problem, you and a partner are to come up with a scheme for communicating the value of a hidden card. The game is played as follows:
• Your partner is sent out of the room.
• A dealer hands you 5 cards from a standard 52 card deck.
• You look at the cards, and hand them back to the dealer, one by one, in whatever order you choose.
• The dealer takes the first card that you hand her and places it, face up, in a spot labeled "0"'. The next three cards that you hand her, she places, similarly, in spots labeled "1", "2", and "3". The last card that you hand her goes, face down, in a spot labeled "hidden". (While you control the order of the cards, you have no control over their orientations, sitting in their spots; so you can't use orientation to transmit information to your partner.)
Your partner enters the room, looks at the four face-up cards and the spots in which they lie and, from that information (and your previously-agreed-upon game plan)