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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?

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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?

Squaring the Circle and Circling the Square

Age 14 to 16 Challenge Level:

Hint for part 2

You might need to ask your teacher to explain to you how to sum a Geometric Progression.

Alternatively there are some ideas on the NRICH site- but this diagram might give you some ideas: