Problems

A surface P is created by gluing every pair of antipodal points of a disc (a circle with inside filled in). We represent P on the plane by a disc in the following picture and bear in mind that the antipodal points are glued.

Explain why the two diameters in the pictures are in fact two circles on P and how to stretch it so that it becomes a single loop not touching any of the glued points.

It is possible to play tic-tac-toe on a torus: gluing the sides means that the bottom row is above the top row and the right most column is also to the left of the left most column. Is one of the players guaranteed to win if they play all the right moves?

Describe the surface we get if we start with a rectangular sheet of paper and then glue the opposite sides of the paper band in the same direction as in the picture.

Show that \(\sqrt[3]{3}\) is irrational.

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 \(x \leq y\).

What is logically the opposite of the statement “every \(n\) is odd or \(p<q\)"?

The picture below shows a closed disc, which is just a circle with the inside filled. The grey interior represents the interior of the disc. Describe the resulting shape when you glue the circular boundary to one point.

Label the vertices of a cube with the numbers \(1,2,3,\dots,8\) so that the sum of the labels of the four vertices of each of the six faces is the same.