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### Number and algebra

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### Advanced mathematics

# A Little Light Thinking

Why do this problem?

### Possible approach

*This printable worksheet may be useful:* LightThinking.pdf

This problem follows on from Charlie's Delightful Machine, so we assume students have a strategy for determining the rules to switch individual**Level 1** lights on.

Here are some questions you may wish to ask, to prepare students for what comes next:

*What is special about a ***Level 1** rule where all the 'light on' numbers

"Did anyone find examples where two or more (level 1) lights switched on simultaneously?"

**"Our challenge now is to determine how to turn on two (level 1) lights (or more) simultaneously, if we know the rules for each individual light."**

If computers are available, students could work in pairs determining the rules for individual lights, then finding the rule for switching on pairs of lights simultaneously. Remind them of the importance of recording their findings to share with the class.

If computers are not available, students could explore pairs of linear sequences, searching for examples where two sequences have numbers in common.

### Key questions

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### Possible support

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### Possible extension

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## You may also like

### Euler's Squares

### There's a Limit

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Age 14 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Why do this problem?

In this problem, students are presented with a mystery to be solved (what are the rules which govern the lights, and can all the lights be switched on at once?).

To satisfy their curiosity and explain the mystery, they will need to go in search of the information they require, and work in a systematic way in order to make sense of the results they gather.

The questions encourage students to think about the properties of numbers, including divisibility and remainders. It may help students gain a deeper understanding of linear sequences (and perhaps quadratic sequences, if they explore Level 2 and 3).

This problem follows on from Charlie's Delightful Machine, so we assume students have a strategy for determining the rules to switch individual

Here are some questions you may wish to ask, to prepare students for what comes next:

*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?*

"Did anyone find examples where two or more (level 1) lights switched on simultaneously?"

If computers are available, students could work in pairs determining the rules for individual lights, then finding the rule for switching on pairs of lights simultaneously. Remind them of the importance of recording their findings to share with the class.

If computers are not available, students could explore pairs of linear sequences, searching for examples where two sequences have numbers in common.

One way they could record their work is by creating a table with sequence rules along the top and down the side, and indicating with a tick or a cross in each cell whether both lights could light up simultaneously. When appropriate, they could also indicate numbers which successfully switch on both lights.

Remind the class that the task is not simply to find examples, but to find a way of determining, by just looking at the sequence rules, whether or not a pair of lights will ever light up simultaneously.

When trying to produce convincing arguments, ideas from the problem Stars might be useful.

Leave some time for the class to come together to share examples of rules where more than one light could be switched on simultaneously, and examples where it was impossible, together with their reasoning.

What is true about any pair of rules where it is **not** possible to light up both lights?

If two sequences are described by the rules $an+b$ and $cn+d$, can you explain the conditions for determining whether the lights will ever switch on together?

The problem Shifting Times Tables offers an introductory challenge for exploring linear sequences.

The problem Remainders explores some properties of numbers which could be useful when thinking about this problem.

Students could use a 100 square as a visual way to record sequences and see where (if) they coincide.

Some students may wish to use ideas of modular arithmetic to prove their findings; reading this article first may help.

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?