Take Three from Five

Problem | Teachers' Notes | Hint | Solution | Printable page |
Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem looks like a number task, possibly revision about multiples, but it becomes a question about establishing why something can never happen, and creating a real proof of this. When it comes, the proof often feels powerful, satisfying and complete, and students leave feeling they have achieved something.

Possible approach

This problem could follow on from What Numbers Can We Make?
Ask students for a set of five numbers and then choose three that add up to a multiple of $3$. Make a note of what they add up to. Repeat several times. Write all the sets of five numbers on the board clearly, with the three (that make the multiple of three) circled, or marked clearly.

Don't say anything - let students work out what is special about the sum of the numbers you select. Suggest that if they know what is going on they may like to choose $5$ numbers that stop you achieving your aim. At some stage check that they all know what is going on.

Challenge them to offer five numbers that make it impossible for you to choose three that add up to a multiple of $3$. Allow them time to work on the problem, and suggest that they write any sets they find up on the board. Students may enjoy spotting errors among the solutions on the board.

Allow negative numbers, as long as they will allow you negative multiples of $3$ (and zero).

At some stage there may be mutterings that it's impossible. A possible response might be:
"Well if you think it's impossible, there must be a reason. If you can find a reason then we'll be sure."

Once they have had sufficient thinking time, bring the class together to share ideas.

You may need to prompt them by talking about multiples:

All numbers fall into one of these 3 categories:

type A (ie multiple of $3$)
XX...XXXXXXXX
XX...XXXXXXXX
XX...XXXXXXXX

type B (ie of the form $3n+1$)
XX...XXX
XX...XX
XX...XX

type C (ie of the form $3n+2$)
XX...XXXXX
XX...XXXXX
XX...XXXX

We have found that trying to use algebraic expressions as above, is tricky, students often end up with n having two or more values at once. Students are unlikely to know the notation of modular arithmetic, but the crosses notation above is sufficient for the context, and it suggests a geometrical image that students can use in explaining their ideas.

Ask students which combinations of As Bs and Cs would add to a multiple of three. For each suggestion,(e.g. A+A+A, A+B+C, B+B+B...) ask students to return to the list of numbers that were offered earlier and find all the examples of each case.

Ask students to continue until they have found all the possible combinations of A, B and C that sum to a multiple of $3$. Ask them to find five numbers that don't satisfy any of these combinations."
Soon they should realise that this will be impossible!

Ask them to set down the logic of the argument in order, like a mathematical proof, trying to be as clear as possible each time, and to state clearly what they have proved.

Key questions

Which types of numbers always add up to a multiple of three?

Possible extension

You can guarantee being able to get an even number when you select $2$ from $3$ (check).
You can guarantee being able to get a multiple of $3$ when you select $3$ from $5$.
Can you guarantee being able to get a multiple of $4$ when you select $4$ from $7$?

Possible support

Perhaps a good idea to start by having a go at Make 37

Multiples of $2$ added together make other multiples of $2$.
What other numbers can we add to make multiples of $2$? Why?

Multiples of $3$ added together make other multiples of $3$.
What other numbers can we add to make multiples of $3$? Why?



Published October 2003.