### 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?

# Pick's Theorem

##### Stage: 4 Challenge Level:

* How many different figures can be described as $(4, 0)$?

* What do you notice about $(4,0)$ figures?

* Choose another particular value for $(p,i)$ and explore different shapes.

* Have you tried drawing shapes with the same area?

* What do you notice about those figures whose areas are the same?

* What ways are there of increasing the area by $1$ unit?

* Draw more figures; tabulate the information about their perimeter points ($p$), interior points ($i$) and their areas ($A$).