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?

Multiplication Arithmagons

Stage: 4 Challenge Level:

How does the product of the numbers on the edges relate to the product of the numbers at the vertices?
It may help to label the numbers at the vertices $A$, $B$ and $C$ and then express the edge numbers in terms of $A$, $B$ and $C$.