*In this problem, you can choose to work with the UK currency (1p, 2p, 5p, 10p, 20p, 50p, £1 and £2) or you could use your own currency if you prefer.*

Charlie, Alison, and Abi want to buy some sweets at the cinema. Sweets cost 32p, and they all have exactly the right amount.

Charlie has 3 coins:

*20p, 10p, 2p.*

Alison has 7 coins:

*10p, 5p, 5p, 5p, 5p, 1p, 1p.*

Abi has 32 coins. We're not going to list them, we expect you'll be able to work out what she has!

**Is there a way of making 32p with 4 coins? ****How about 5 coins?**

**Or ANY number of coins between 3 and 32?**

Explore ways of making other totals using different numbers of coins.

**Is it always possible to make a total using every number of coins between the minimum and the maximum?**

## Comments

### A trip to the Cinema

10p 10p 10p 2p

10p 10p 10p 1p 1p

10p 10p 5p 5p 1p 1p

10p 5p 5p 5p 5p 1p 1p

5p 5p 5p 5p 5p 5p 1p 1p

10p 5p 5p 5p 2p 2p 1p 1p 1p

5p 5p 5p 5p 5p 2p 2p 1p 1p 1p

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### Re: A trip to the Cinema

Well done Abbey, this is a good start!

Can you describe the changes that you've made at each stage to increase the number of coins by 1?

Can you use these same manoeuvres to keep increasing the number of coins by 1 each time? Will you be able to keep going until you reach the combination that uses 32 coins?

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### Maths

Following on from Abbey, here are solutions for 3 coins to 32 coins:

3 - 20,10,2

4 - 10,10,10,2

5 - 10,10,10,1,1

6 - 10,5,5,5,5,2

7 - 5,5,5,5,1,1,10

8 - 5,5,5,5,5,5,1,1

10 - 5,5,10,2,2,2,2,2,1,1

11 - 2,2,1,5,5,5,5,2,2,1,2

12 - 2,2,2,2,2,5,5,5,2,2,2,1

13 - 1,1,2,2,2,2,5,5,5,2,2,2,1

14 - 1,1,1,1,10,1,1,1,1,1,1,1,1,10

15 - 1,1,1,1,5,5,1,1,1,1,1,1,1,1,10

16 - 16 x 2

17 - 15 x 2, 2 x 1

18 - 14 x 2, 4 x 1

19 - 13 x 2, 6 x 1

20 - 12 x 2, 8 x 1

21 - 11 x 2, 10 x 1

22 - 10 x 2, 12 x 1

23 - 9 x 2, 14 x 1

24 - 8 x 2, 16 x 1

25 - 7 x 2, 18 x 1

26 - 6 x 2, 20 x 1

27 - 5 x 2, 22 x 1

28 - 4 x 2, 24 x 1

29 - 3 x 2, 26 x 1

30 - 2 x 2, 28 x 1

31 - 1 x 2, 30 x 1

32 - 32 x 1

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### Maths

Well done, Katie!

Can you use these manoeuvres to make a total other than 32 with different numbers of coins? Are there totals for which some numbers of coins don't work?

If the total gets very large, it can take a long time to list the solutions. Could you describe an algorithm that would generate a solution for each number of coins for a given total?

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### maths

The answer for Abi's coins are

32 one penny coins

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### a trip to the cinema

5, 5, 5, 5, 5, 2, 2, 2, 1

5, 5, 5, 5, 5, 5, 2

20, 5, 2, 1, 2, 1, 1

10, 10, 10, 2

10, 2, 5, 10, 5

20, 5, 5, 1, 1

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