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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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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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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?

A Little Light Thinking

Age 14 to 16 Challenge Level:

Can you figure out a method to make all four lights on the machine below switch on at once?

This problem follows on from Charlie's Delightful Machine, where you are invited to find efficient strategies for working out the rules controlling each light.

The rules for turning on the Level 1 lights are all given by linear sequences (like those found in Shifting Times Tables).


What is special about a Level 1 rule where all the 'light on' numbers
  • are odd?
  • are even?
  • are a mixture of odd and even?
  • are all multiples of 3? Or 4? Or...
  • have a last digit of 7?

Can you make two Level 1 lights light up together?




Once you have made two Level 1 lights light up together, can you find another number that will light them both up? And another? And another? ...

Can you find any connections between the rules that light up each individual Level 1 light and the rule that lights up the pair?
What about trying to light up three lights at once? Or all four?

Sometimes it's impossible to switch a pair of Level 1 lights on simultaneously.
How can you decide whether it is possible to switch a pair of lights on simultaneously?
Or a set of three lights? Or all four?