### 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?

# Attractive Tablecloths

##### Stage: 4 Challenge Level:

How many extra colours do you need to add to the $7$ by $7$ pattern to make the $9$ by $9$ pattern for each type of symmetry?

Where do those extra colours go?

How many extra colours would you need for the $11$ by $11$ design?

Where do those extra colours go?