Charlie's Delightful Machine has four coloured lights on it.

You can enter numbers into Charlie's machine.

Each light is controlled by a rule, and if your number satisfies the rule, the light will go on.

Some numbers may turn on more than one light!

Type in some numbers and see which lights you can switch on.

To start again with a new set of rules, click a Level button (each level is more difficult than the last).

**Can you develop a strategy to work out the rules controlling each light at Level 1?**

Once you have a strategy, challenge yourself to find four digit numbers that turn on each light.

Once you are confident that you can work out the rules for Level 1 lights, try finding rules for some Level 2 and 3 lights.

You may also want to try A Little Light Thinking.

## Comments

### LVL 1

Blue: 6n-4

Red: 11n-2

Yellow: 11n-4

Green:12n-2

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### LVL 1

Do you think it would be possible to find a number that would light up all four lights at the same time?

If not all four, what about three lights? Or two lights?

Can you explain why some pairs of lights will never light up at the same time?

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### Level 1 Reply

I'm not sure, but I did try to investigate. You can light up two lights at a time, but not three, and the Green and Yellow lights never lit up together in my problem because one was even numbers and one was odd. You could try looking for patterns like that.

In some cases, there is probably a number in millions (or minus millions) that lights up all the numbers but, unless you are willing to sit at a computer screen for the rest of your life, you'll probably never know!

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### Charlie's delightful machine

Yellow $= 2n^2 - 2n + 1$

Blue $= 2n^2 + 4n - 4 $

Red $= 2n^2 - 4n + 3$

Green $= \frac{1}{2}n^2 - \frac{1}{2}n + 10$

**Four Digit Numbers:**

Yellow = 7081

Blue = 7436

Red = 6963

Green = 1780

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### Charlie's Delightful Machine Level 3

For the blue light we have

5, 7, 9, 15, 19, 29, 35, 49 ,57

The only recurrence relation we have found is S(n+3) = 2( S(n+2) - S(n+1) ) + S(n) which only works for n = 2, 4, 6 !

Please help

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### Charlie's Delightful Machine Level 3

Can you find a rule for your first, third, fifth, seventh... values: 5, 9, 19, 35, 57?

What happens if you substitute negative numbers into your rule?

Does it help to draw a graph of your quadratic to see what is going on?

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### Charlie's Delightful machine

I generated the following results (Level 1):

Yellow: 6, 16, 26,.... 10n+6

Red: 2, 10, 18, 26, 34,.... 8n+2

Blue: 3, 7, 11, 15, 19, 23, 27, 31.... 4n + 3

Green: 5, 14, 23, 32,.... 9n + 5

So to light up all colours the number must be an element of each of the above sets, so can a single number be an element of each of the sets above?

Red and Blue lights will not light up together as for any n:

4n + 3 is a multiple of 4 + 3, and 8n+ 2 = 4(2n) + 2 is a multiple of 4 + 2

and we can't have a number that is even and odd at the same time.

Love this challenge.

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### Level 2

Yellow $= 5n - 1$

Red $= 10n - 9$

Blue $= n^2 + 5n$

Green $= n^2 + n$

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### It changes every time for me

It changes every time for me.

Eg: Level 1 - I put 123, it somehow lights up. I refresh the page, put in 123 and it doesn't light up.

I think, "That's weird." I refresh the page again, put in 123, and it lights up again...

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### It changes every time for me

Yes, every time you refresh, a new set of rules will switch on the lights. It makes it more interesting!

Sometimes you're able to switch on all the lights at the same time...

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### Strategy

This is a strategy.

Think about each light in turn. Start at 1, then keep going up by one number at a time noting down the numbers that turn it on.

When you think you have a pattern, move on to the next light.

I hope this helps.

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### Level 1

Red = 2n

Yellow =10n-6

Green = 3n-1

Blue = 9n-7

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### Level 1

Yellow: even numbers

Red: 7n-3

Blue: 8n+1

Green: odd numbers

I'm from Solihull School

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### Level One Awnsers

The yellow light is triggered by adding 5 each time starting from 2.

The red light is triggered by adding 7 each time starting on 3.

The blue one is triggered by adding 9 each time starting from 0.

The green one is triggered by adding 12 each time starting from 11.

A shorter way of writing these is:

yellow = 5n-3

red = 7n-4

blue = 9n

green = 12n-1

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### Answers for Charlie's Delightful Machine

Yellow = 5n-1

Red = 11n

Blue = 7n

Green = 3n

Solihull School

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### Level 1

Green: even numbers

Red: 6n-4

Blue: 11n-7

Yellow: 8n-1

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### Level 1

Do you think it would be possible to find a number that would light up all four lights at the same time?

If not all four, what about three lights? Or two lights?

Can you explain why some pairs of lights will never light up at the same time?

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### Level 3

Has anyone worked out a formula for green light at level 3?

Our numbers were 3, 5, 7, 13, 17, 27, 33, 47, 55, 73, 83, 105, 117, 143....

Any ideas?

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### Level 3

Can you find a rule for your first, third, fifth, seventh... values: 3, 7, 17, 33, 55, 83, 117?

What happens if you substitute negative numbers into your rule?

Does it help to draw a graph of your quadratic to see what is going on?

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