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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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DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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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?

Triangles Within Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Image with six copies of the second triangular number and one of hte first triangular number added to make a square

The diagram above shows that: $$ 8 \times T_2 + 1 = 25 = 5^2$$

Use a similar method to help you verify that: $$ 8 \times T_3 + 1 = 49 = 7^2$$ Can you generalise this result?

Can you find a rule in terms of $ T_n $ and a related square number?

Can you find a similar rule involving square numbers for $T_{n}, T_{n+2}$ and several copies of $T_{n+1}$?