Show that there are no rational numbers \(a,b\) such that \(a^2 + b^2 = 3\).
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.
Suppose \(x,y\) are real numbers such that \(x < y + \varepsilon\) for every \(\varepsilon > 0\). Show that \(x \leq y\).
Could you meet a person inhabiting this planet who asks you “Am I a Goop?"
On this planet you meet a couple called Tom and Betty. You hear Tom ask someone: “Are Betty and I both Goops?"
What kind is Betty?
You learn that one of the aliens living on this planet is a wizard. You learnt that by overhearing a certain question being asked on the planet. What question could that have been?
Suppose you meet a person inhabiting this planet and they ask you “Am I a Crick?" What would you conclude?
You meet two friends, Katja and Anja. Katja once asked Anja “Is at least one of us a Goop?"
What kinds are Katja and Anja?
You later learn that there is exactly one wizard on this planet of Cricks and Goops. You would like to find out who that is.
You meet an alien called Andrew. He asks you “Am I the kind that could ask whether I am not the wizard?"
Do you have enough information to tell for sure who the wizard is by now?