# Sealed Solution

## Problem

*Sealed Solution printable sheet*

Here is a set of ten cards, each showing one of the digits from 0 to 9:

The ten cards are divided up between five envelopes so that there are two cards in each envelope.

The sum of the two numbers inside it is written on each envelope:

What numbers could be inside the "8" envelope?

*Thank you to Alan Parr who inspired this task.*

## Getting Started

What are the possible ways of making the numbers on the envelopes?

Which number has the fewest possible combinations? It might be worth starting from this envelope and looking at what could be in the others.

## Student Solutions

We had lots of solutions sent in for this activity, so thank you to everybody who shared their ideas with us.

Anika from the National Academy for Learning in Bangalore, India sent in a table of possibilities for the numbers in each envelope, with this explanation:

I made five columns, one each for the envelopes. For each number, I wrote down the possible pairs that could be in the envelope. I found that the envelope 8 can contain the numbers (0,8),(5,3),(1,7),(6,2).

Well done for finding all these possibilities, Anika. In Anika's table we can see all the pairs of numbers that make the total on each envelope.

Leo from Kings' School in Dubai, UAE used the same idea as Anika to draw a table of all the different possibilities. He then worked out which of these possibilities would work with each other, as the same number can't be in more than one envelope:

I started off by creating a table of all of the possible number bonds to each of the targets on the envelopes.

I then worked systematically so that I tried every potential pathway across the table. Some of them proved to be impossible but I was able to find all 3 solutions. I have also included an example of a pathway that was impossible; it was where I started the process.

I know that there are only 3 possible solutions as I worked through every possibility in a systematic fashion.

Well done for working systematically through all of the different possibilities, Leo. Leo's solution shows that the "8" envelope can contain 0 and 8, 5 and 3, or 1 and 7, but it looks like the cards 6 and 2 can't be in the "8" envelope. I wonder why?

James and Oais from the UK found the same solutions as Leo, and they explained why 6 and 2 can't be in the "8" envelope:

We made a chart with all the possibilities for making the numbers on each envelope, then we tested the different ways it could go.

If you start by looking at how to make 14, there are two: 5/9 or 6/8. Then, if you try to make 13, you can see that there is only one way that works for each. When you choose 5/9 for 14, you have to choose 6/7 for 13. When you choose 6/8 to make 14, you have to choose 4/9 to make 13 because you've used up the other numbers.

The numbers you have to make 8 are 0/8, 1/7, 2/6 or 3/5 but you'll never be able to use 2/6 because you always have to use the 6 to make 13 or 14. So the answer is 0,8,1,7,3 or 5!

Good explanation! I wonder if it's more helpful to start with the smallest numbered envelopes like Leo did, or to start with the largest numbered envelopes like James and Oais did?

Thank you as well to these children who sent in excellent solutions: James from Hamstel Junior School in England; Mia and Hari from Kings' School in Dubai; Anna from Sahuaro Elementary in the USA; Lauren from Australia; K from Crossflatts in the UK; Saanvi from Newcastle-under-Lyme in the UK; Isla from Walton and Holymoorside Primary in the UK; and Ci Hui Minh Ngoc from Kong Hwa School in Singapore.

## Teachers' Resources

**Why do this problem?**

Sealed Solution offers the chance for children to work in a systematic way and is a great context in which to encourage them to explain and justify their reasoning.

### Possible approach

Begin by familiarising children with the context: Using digit cards 0 to 5, invite the class to watch as you put 0 and 1 in one envelope and write their total on the outside (or on a 'post-it' note stuck to the envelope). Put 3 and 5 in another envelope, again writing their total on the envelope. Explain that the other two cards will go in the last envelope. What will the total be? How do they know?

Try this again, this time putting 0 and 5 in one envelope and recording the total. But then put two cards in another envelope without showing them to the children. Write the total on the outside of the envelope. Repeat this for the third envelope. (For example you could have 1 and 3 in the first and 2 and 4 in the second.) What numbers are in the two envelopes? How do they know?

Try again, this time keeping 0 and 5 in the first envelope but suggest that you want to put the other cards in pairs into the envelopes, so that the totals on the other two are the same. What could you do? How do they know? At each stage, children can be working in pairs, perhaps using mini-whiteboards and digit cards to try out their ideas.

*could*be inside each envelope. In this case, there are two possible solutions, which will prepare them well for the challenge as written in the problem.

### Key questions

Which envelope shall we try first? Why?

What could be in this envelope?

Are there any numbers which you know definitely *aren't* in this envelope? Why?

Are there any other solutions?

### Possible support

Having digit cards available for children to use will free up their thinking and will make it easier to try out different ideas without worrying about crossing ideas out on paper.

### Possible extension

Children could make up their own problem along these lines.

Alan Parr, the creator of this task, wrote to tell us:

'I've recently returned to this for the first time in ages, working with some Year 6s [10 and 11 year olds]. They found it so accessible and involving that we took it to places I'd never previously dreamt of.' You can read what they did in the first April 2015 post on Alan's blog, and he writes about the task again in
two March 2017 posts.