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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?

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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.

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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: Challenge Level:1

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 $C$.