### 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?

### 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?

### Equilateral Areas

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

# Nim

##### Age 14 to 16 Challenge Level:

Describe the strategy for winning the game of Nim. The rules are simple. Start with any number of counters in any number of piles. Two players take turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

Try out your strategy by playing against the computer (although in this version, the winner takes the last counter).