1. Pathways
  2. Freedom and Constraints
  3. A Trip to the Cinema
  4. Cinema Sweets

Cinema Sweets

Charlie has 3 coins in his pocket that add up to 32p - what could they be? Come and explore this and other exciting questions about the maths of partitioning.
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

Submitted by Abbey on Sun, 11/29/2015 - 12:14

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

Submitted by dp419 on Mon, 11/30/2015 - 09:41

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?

Submitted by Katie on Wed, 05/18/2016 - 14:17

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

9 - 5,5,5,2,2,1,10,1,1
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

Submitted by cfg21 on Thu, 06/02/2016 - 10:35

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?

Submitted by kriti on Wed, 06/22/2016 - 01:51

The answer for Abi's coins are
32 one penny coins

Submitted by jasmine on Tue, 02/21/2017 - 18:34

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