Problems

We start with the point \(S=(1,3)\) of the plane. We generate a sequence of points with coordinates \((x_n,y_n)\) with the following rule: \[x_0=1,y_0=3\qquad x_{n+1}=\frac{x_n+y_n}{2}\qquad y_{n+1}=\frac{2x_ny_n}{x_n+y_n}\] Is the point \((3,2)\) in the sequence?

Consider a graph with four vertices and where each vertex is connected to every other one (this is called the complete graph of four vertices, sometimes written as \(K_4\)). We write the numbers \(10,20,30,\) and \(40\) on the vertices. We play the following game: choose any vertex, and subtract three from that vertex, and add one to each of the three other vertices, so an example could be:

After playing this game for some number of steps, can we make the graph have the number \(25\) on each vertex?

Every year the citizens of the planet “Lotsofteeth" enter a contest to see who has the most teeth.
This year the judge notices:

1. Nobody has 0 teeth (everyone has at least 1).

2. There are more people in the contest than the most teeth that any one person has. (For example, if the most teeth anyone has is 27, then there are more than 27 people participating in the contest.)

Must there be two people who have exactly the same number of teeth? Explain why.

Fred and George had two square cakes. Each made two straight cuts on his cake from edge to edge. However, one ended up with three pieces, and the other with four. How could this be?

Remember that two shapes are congruent if they are the same in shape and size, even if one is flipped or turned around. For example, here are two congruent shapes:

Cut the following shape into four congruent figures:

A cube net is a 2D shape that can be folded into a cube. For example, in the following diagram we show a cube net and the steps that fold it into a cube:

Imagine that you want to cover an endless floor with this cube net, so there are no gaps or overlaps, how would you lay them out? This is called covering or tiling the plane.

Cut a square into three parts and use these pieces to form a triangle whose angles are all acute (i.e: less than \(90^\circ\))