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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?

Converging Means

Age 14 to 16 Challenge Level:
Take any two positive numbers and call the larger one $a_1$ and smaller $b_1$. Calculate the arithmetic mean of the two numbers and call this $a_2$, where: $$a_2 = (a_1+ b_1)/2.$$Calculate the geometric mean of $a_1$ and $b_1$ and call this $ b_2$ so that: $$b_2 = \sqrt{(a_1b_1)}.$$ Suppose you start with 3 and 12, then the arithmetic mean is 7.5 and the geometric mean is 6.

Repeat the calculations to generate a sequence of arithmetic means $a_1$, $a_2$, $a_3$, ... and a sequence of geometric means $b_1$, $b_2$, $b_3$, ... where $$a_{n+1} = (a_n+ b_n)/2,$$ $$b_{n+1} = \sqrt{(a_nb_n)}.$$In the example given $$a_2 = 6.75,$$ $$b_2 = \sqrt{(45)}= 6.708\; \mbox{to 3 decimal places}.$$Calculate the first 5 terms of each sequence and mark them on a number line. Calculate a few more terms and make a note of what happens to the two sequences.

Now repeat the same calculations starting with different choices of positive values for $a_1$ and $b_1$. You should notice the same behaviour of the two sequences whatever starting values you choose. Describe and explain this behaviour.

You may like to write a short program for a calculator or computer to calculate the sequences and if so you should send in your program with your solution.