Problems

Today we will be solving problems using the pigeonhole principle. What is it? Simply put, we are asked to place pigeons in pigeonholes, but the number of pigeons is larger than the number of pigeonholes. No matter how we try to do that, at least one pigeonhole will have to contain at least 2 pigeons. By “pigeonholes" we can mean any containers and by “pigeons" we mean any items which are placed in these containers. This is a simple observation, but it is helpful in solving some very difficult problems. Some of these problems might seem obvious or intuitively true. Pigeonhole principle is a useful way of formalising things that seem intuitive but can be difficult to describe mathematically.

There is also a more general version of the pigeonhole principle, where the number of pigeons is more than \(k\) times larger than the number of pigeonholes. Then, by the same logic, there will be one pigeonhole containing \(k+1\) pigeons or more.

A formal way to prove the pigeonhole principle is by contradiction - imagine what would happen if each pigeonhole contained only one pigeon. Well, the total number of pigeons could not be larger than the number of pigeonholes! What if each pigeonhole had \(k\) pigeons or fewer? The total number of pigeons could be \(k\) times larger than the number of pigeonholes, but not greater than that.

A natural number \(p\) is called prime if the only natural divisors of \(p\) are \(1\) and \(p\). Prime numbers are building blocks of all the natural numbers in the sense of the The Fundamental Theorem of Arithmetic: for a positive integer \(n\) there exists a unique prime factorization (or prime decomposition) \[n = p_1^{a_1}p_2^{a_2}...p_r^{a_r}.\] Today we will explore how unusual prime numbers are.
Essentially there is only one way to write an integer number as a product of prime numbers, where some of the prime numbers in the product can appear multiple times.

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 this problem set we only consider graphs in which every pair of vertices is connected with one or zero edges.

In a mathematical problem, one may use vertices of a graph to represent objects in the problem, i.e. people, cities, airports, and edges of the graph represent relations between the objects such as mutual friendship, railways between cities, plane routes. As you will see in the examples below, representing the initial problem as a graph can considerably simplify the solution.

Imagine you are a manager of a very special hotel, a hotel with an infinite number of rooms, where each room has a natural number on the door \(1,2,3,4,...\). Only one guest can stay in each room and in most cases the hotel will be initially full with no vacant rooms left.
You will have to deal with unusual situations that may occur.

Today we will solve some geometric problems using the triangle inequality. This is an inequality between the lengths of the sides of any triangle, or between the distances of any three points.

The shortest path between any two points \(A\) and \(B\) is a straight segment - every other path is longer. In particular, a path through another point, \(C\), is equal or longer. \[AC + BC \ge AB\] The triangle inequality says that the sum of lengths of any two sides of a triangle is always larger than the length of the third side. The inequality only becomes an equality if \(ABC\) is not actually a triangle and the point \(C\) lies on the segment from \(A\) to \(B\).

Even though it is a simple idea, it can be a really helpful tool in problem solving.

A set is a collection of objects of any specified kind, the objects are called elements or members, the objects in one set cannot repeat, namely \(\{1,2,3\}\) and \(\{1,2,2,2,3\}\) are identical sets. We denote a set by a capital letter \(A\), or \(B\) and write \(x\in A\) if \(x\) is an element of \(A\), and \(x\notin A\) if it is not. The notation \(A=\{a,b,c,...\}\) means that the set \(A\) consists of the elements \(a,b,c,...\). The empty or void set, \(\emptyset\), has no elements. If all elements of \(A\) are also in \(B\), then we call \(A\) a subset of \(B\) and we write \(A\subseteq B\). It is an axiom that the sets \(A\) and \(B\) are equal \(A=B\) if they have the same elements. Namely, \(A\) is a subset of \(B\) and \(B\) is a subset of \(A\) at the same time.
For any sets \(A\) and \(B\), we define their union \(A\cup B\), intersection \(A\cap B\), and the difference \(A-B\) as follows:

• the union \(A\cup B\) is the set of all elements that belong to \(A\) or \(B\);

• the intersection \(A\cap B\) is the set of elements that belong to both \(A\) and \(B\);

• the difference \(A-B\) consists of those \(x \in A\) that are do not belong to \(B\).

Sometimes it is useful to draw sets as Venn diagrams, on the diagram below the pink circle represents the set \(A\), the yellow circle represents the set \(B\), the orange part is the intersection \(A\cap B\), the pink part is \(A-B\), the yellow part is \(B-A\), and the whole picture is the union \(A\cup B\).

A useful common problem-solving strategy is to divide a problem into cases. We can divide the problem into familiar and unfamiliar cases; easy and difficult cases; typical and extreme cases etc. The division is sometimes suggested by the problem, but oftentimes requires a bit of work first.

If you are stuck on a problem or you are not sure where to begin, gathering data by trying out easy or typical cases first might help you with the following (this list is not exhaustive):

1. Gaining intuition of the problem

2. Isolating the difficulties

3. Quantifying progress on the problem

4. Setting up or completing inductive arguments

Let us take a look at this strategy in action.

Let’s play some games today! We will play a classic game known as nim, which is thought to be one of the oldest games.

Typically people play nim using matchsticks, though stones and coins are popular too. There are a few heaps of matchsticks in nim. Players take turns to remove matchsticks from a heap of their choosing. The player can remove any number of matchsticks they wish from that heap. Whoever has no matchsticks left to take loses.

This following position will be written as \(\text{Nim}(3,3,3)\):

As another example, this is \(\text{Nim}(1,2,3,4)\):

We will omit heaps of size zero, so \(\text{Nim}(3,0,3,0,3)\) is the same as \(\text{Nim}(3,3,3)\).

Nim is important because a large class of games are equivalent to it despite its simple appearance. The interested reader should look up “Sprague-Grundy Theorem".

Let us introduce a few terms that will be helpful for analyzing games. A game \(G\) consists of some positions and a set of rules. A position \(g\) in the game \(G\) is called a winning position if the player starting this turn has a winning strategy. This means as long as the player starting this turn continues to play optimally, the second player has to lose. Conversely, a position \(g\) is a losing position if the player starting this turn has no winning strategy.

Today we will draw lots of pictures.

The subject is Topology. It is often called “rubber-sheet geometry" because while it is the study of shapes, topologists typically do not pay too much attention to rigid notions like angle and lengths. We have much more flexibility in topology. Some common words describing the operations here might include “gluing", “stretching", “twisting" and “inflating".

Although we will not define continuity, it is a more or less intuitive idea. Topological operations should be continuous. If you have a line segment, no amount of stretching, twisting or bending can make it into two disconnected segments.

There are various ways to prove mathematical statements. One of the possible methods which might come in handy in certain situations is called proof by contradiction. To prove a statement we first assume that the statement is false and then deduce something that contradicts either the condition, or the assumption itself, or just common sense. Due to the contradiction, we have to conclude that the first assumption must have been wrong, so the statement is actually true.

A closely related method is called contrapositive proof. An example should make the idea quite clear. Consider the statement “if the joke is funny, then I will be laughing". Another completely equivalent way of saying it would be “if I am not laughing, then the joke is not funny". The second statement is known as the contrapositive of the first statement.

We can often prove a statement by proving its contrapositive. Many statements are proven by deriving a contradiction. However, one can often rewrite them as either a direct proof or a contrapositive proof.

Let’s take a look at both of these techniques.