Finding Fifteen

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Age

7 to 11

Challenge level

Primary

Number - Addition and Subtraction

Problem Solving

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

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.

Student Solutions

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:

We tried to be systematic by looking for other ways to make a number, for example 7. So we made 6+1, 3+4, 5+2 and 7 on its own and put 8 with it each time. It was good because then we could get more solutions to make 15. Here are our solutions to make 15 (reading across the table for the three piles):

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.

Then we asked you if these were all the combinations possible. Alicia and William of St Hugh's Woodhall Spa rose to the challenge, finding one more combination:

9, 6

8, 4, 3

7, 5, 2, 1

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

We worked systematically swapping combinations of the same value, ie 1 and 5 with 2 and 4. That way we took piles like (9, 1, 5), (8, 7), (6, 3, 2, 4). We did this for all the pairs, ones and threes that made up the same value, and then got rid of the ones we'd counted twice.

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 from 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.

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Finding Fifteen

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Finding Fifteen

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