\(n\) is an integer (positive or negative). For which \(n\) is \(4^{\frac{n-1}{n+1}}\) an integer?
Klein tosses \(3\) fair coins and Möbius tosses \(4\) fair coins. What’s the probability that Möbius gets more heads than Klein? (Note that a fair coin is one that comes up heads half the time, and comes up tails the other half of the time).
The letters \(A\), \(E\) and \(T\) each represent different digits from \(0\) to \(9\) inclusive. We are told that \[ATE\times EAT\times TEA=36239651.\] What is \(A\times E\times T\)?
The kingdom of Rabbitland consists of a finite number of cities. No matter how you split the kingdom into two, there is always a train connection from a city in one part of the divide to a city in the other part. Show that one can in fact travel from any city to any other, possibly changing trains.
A poetry society has 33 members, and each person knows at least 16 people from the society. Show that you can get to know everyone in society by a series of introductions if you already know someone from the society.
Some Star Trek fans and some Doctor Who fans met at a science fiction convention. It turned out that everyone knew exactly three people at the convention. However, none of the Star Trek fans knew each other and none of the Doctor Who fans knew each other. Show that there are the same number of Star Trek fans as the number of Doctor Who fans at the convention.
Sometimes it is hard to rigorously formulate an intuitively correct reasoning. We might not know the proper words, the proper language, we might not have the right tools. Maths problems are not an exception. When we start learning to solve them, we know nothing about possible mathematical approaches and mathematical models. Today you will learn a very useful way to visualise information: you will learn how to represent information as a graph.
A graph is a finite set of points, some of which are connected with line segments. The points of a graph are called vertices. The line segments are called edges.
In a mathematical problem, one may use vertices of a graph to represent objects in the problem, i.e. people, cities, airports, etc, and edges of the graph represent relations between the objects such as mutual friendship, railways between cities, plane routes, etc. As you will see in the examples below, representing the initial problem as a graph can considerably simplify the solution.