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## Sealed Solution

A set of ten cards, each showing one of the digits from $0$ to $9$, is divided up between five envelopes so that there are two cards in each envelope. The sum of the cards inside it is written on each envelope:

What numbers could be inside the $8$ envelope?

Thank you to Alan Parr for allowing us to adapt one of his problems.

### 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 $0 - 5$, 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 but writing only 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.

Continue like this, gradually building up the level of complexity, and each time focusing on children's clear reasoning. You could put the cards in the envelopes without showing any to the children, only writing the totals. Invite the children to say which numbers could be in the envelopes. Next, using four envelopes and cards $0 - 7$, write $5$, $11$, $8$ and $4$ on the envelopes. How could
they put numbers in the envelopes for the totals to be right? The final challenge could then be to solve the problem as it stands, with five envelopes and ten digit cards.

You could leave this final problem up on the wall for children to contribute solutions to over a longer period of time.

### 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 extension

Children could make up their own problem along these lines.

### 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 out on paper.