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# A Little Light Thinking

## You may also like

### Euler's Squares

### There's a Limit

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Links to the NRICH website Home page

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Age 14 to 16

Challenge Level

*A Little Light Thinking printable worksheet*

**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?**

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?