### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Nim

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

### Loopy

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

# Beelines

##### Age 14 to 16Challenge Level

Start by drawing some lines and recording how many squares they cross.

What do you notice about the coordinates of lines that pass through the corners of grid squares?

Alternatively, you can install GeoGebra on your computer and download and run the GeoGebra file yourself.