# Finding Fifteen

*Finding Fifteen printable sheet*

Tim had nine cards, each with a different number from 1 to 9 on it.

He put the cards into three piles so that the total in each pile was 15.

How could he have done this?

Can you find *all* the different ways Tim could have done this?

You may like to print off and cut out some digit cards to help you.

Try starting with one number, for example 1. Which other two numbers can you add to 1 to make 15? Is there more than one way of doing it?

How many ways can you make 15?

Thank you for the many solutions to this problem. It was interesting to see that some of you presumed there had to be three cards in each pile which totalled 15. In fact, the question simply said there had to be three PILES, which makes the problem a little trickier than it looks at first.

Jessica and Ruby from Aldermaston C of E Primary School told us how they went about tackling the problem:

2, 3, 9, 1 | 6, 5, 4 | 7, 8 |

3, 8, 4 | 6, 7, 2 | 9, 1, 5 |

5, 2, 8 | 1, 3, 4, 7 | 9, 6 |

6, 1, 8 | 5, 7, 3 | 2, 4, 9 |

Wilbury Primary School Mathletics Club also got the idea. Some of the solutions they found were the same as Jessica's and Ruby's, but here are their different solutions:

1, 2, 3, 4, 5 |
9, 6 | 7, 8 |

9, 1,5 | 7, 8 | 4, 3, 6, 2 |

9, 6 | 3, 5, 7 | 1, 8, 2, 4 |

So, in total Jessica, Ruby and the Mathletics Club at Wilbury have found seven different ways of putting the cards into three piles.

9, 6 | 8, 4, 3 | 7, 5, 2, 1 |

and telling us why they thought they'd found them all:

That makes eight ways altogether. Well done, Alicia and William! I think there might be one more to find ...

Then, early in 2015 we had a solution fromĀ the year five pupils at Applegarth Academy in Croydon. Also at the end of 2015 from Wool Primary School, they both found 8,7 with 4,9,2 with 6,5,3,1, which we think is the remaining one. Well done those year 5 pupils to find that one that previous pupils did not manage to find.### Why do this problem?

This task will encourage children to develop a systematic approach. It will also give them opportunities to practise simple addition.

### Possible approach

Introduce the problem without saying too much more and then give children chance to have a go in pairs. Having digit cards will help them try out their ideas without feeling inhibited. Suggest that they record each solution on a different piece of paper, large enough so that it could be seen from some distance away.

After a while, bring the whole group together and invite several pupils to come up holding one of their solutions. Keep adding to those standing at the front until the group doesn't have any more different solutions. How do we know that there aren't any other solutions? If no-one offers an idea, suggest to the children that they arrange the solutions in some kind of order or
pattern which will then reveal any that are missing. In this way, a system is imposed afterwards. This will help them to see the value of working systematically on this kind of problem.

### Key questions

How do you know you haven't got that solution already?

How will you know when you have found them all?

Can you convince me that you haven't left any out?

### Possible extension

Investigating magic squares is a nice follow-on activity. The game Fifteen also links well.

### Possible support

Having digit cards available will make this activity accessible for most children.