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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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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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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?

Square Pizza

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

You probably need to consider the two people having alternate segments of the pizza.

You could consider the areas of all the right angled isosceles triangles.

Alternatively, you could reduce the problem considerably by removing a symmetrial portion of the pizza.