Problems
Draw the plane tiling using trapeziums of the following shape:
Here the sides \(AB\) and \(CD\) are parallel.
For any triangle, prove you can tile the plane with that triangle.
Prove that one can not tile the whole plane with regular pentagons.
Draw the plane tiling using convex hexagons with parallel and equal opposite sides:
Draw how to tile the whole plane with figures, consisting of squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), \(5\times 5\), and \(6\times 6\), where each square appears equal amount of times in the design of the figure. Can you think of two essentially different ways to do this?
Find a non-regular octagon which you can use to tile the whole plane and show how to do that.
Observe that \(14\) isn’t a square number but \(144=12^2\) and \(1444=38^2\) are both square numbers. Let \(k_1^2=\overline{a_n...a_1a_0}\) the decimal representation of a square number.
Is it possible that \(\overline{a_n...a_1a_0a_0}\) and \(\overline{a_n...a_1a_0a_0a_0}\) are also both square numbers?
You may have seen the pigeonhole principle before, sometimes called Dirichlet’s box principle. It says that if you have more pigeons than pigeonholes, and you put all of the pigeons into some pigeonhole, then there exists at least one pigeonhole with at least two pigeons. While it sounds quite simple, it’s a powerful technique. The difficult thing is often choosing the appropriate pigeons and pigeonholes.
It has multiple applications in various situations.
Today we will see how to use it in geometric problems.
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.
Today we will solve some problems involving counting the same quantity twice (or more times) in different ways, which will let us learn about the components making it up.