There are infinitely many couples at a party. Each pair is separated to form two queues of people, where each person is standing next to their partner. Suppose the queue on the left has the property that every nonempty collection of people has a person (from the collection) standing in front of everyone else from that collection. A jester comes into the room and joins the right queue at the back after the two queues are formed.
Each person in the right queue would like to shake hand with a person in the left queue. However, no two of them would like to shake hand with the same person in the left queue. If \(p\) is standing behind \(q\) in the right queue, \(p\) will only shake hand with someone standing behind \(q\)’s handshake partner. Show that it is impossible to shake hands without leaving out someone from the left queue.
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.