Problems

I don’t know how the figure below can be made of several \(1\times5\) rectangles which do not overlap. I am willing to pay \(1\) pound if you show me a possible way of doing that which I have not seen before. What is the maximal amount of money a person can earn by solving this problem?

We wish to paint the \(15\) segments in the picture below in three colours. We want it such that no two segments of the same colour have a common end. For example, you cannot have both \(AB\) and \(BC\) blue since they share the end \(B\). Is such a painting possible?

There is a scout group where some of the members know each other. Amongst any four members there is at least one of them who knows the other three. Prove that there is at least one member who knows the entirety of the scout group.

You may remember the game Nim. We will now play a slightly modified version, called Thrim. In Thrim, there are two piles of stones (or any objects of your choosing), one of size \(1\) and the other of size \(5\).
Whoever takes the last stone wins. The players take it in turns to remove stones - they can only remove stones from one pile at a time, and they can remove at most \(3\) stones at a time.
Does the player going first or the player going second have a winning strategy?

We meet a group of people, all of whom are either knights or liars. Knights always tell the truth and liars always lie. Prove that it’s impossible for someone to say “I’m a liar".

We’re told that Leonhard and Carl are knights or liars (the two of them could be the same or one of each). They have the following conversation.

Leonhard says “If \(49\) is a prime number, then I am a knight."

Carl says “Leonhard is a liar".
Prove that Carl is a liar.

Prove that the set of all finite subsets of natural numbers \(\mathbb{N}\) is countable. Then prove that the set of all subsets of natural numbers is not countable.