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Take Three from Five

Age 14 to 16 Challenge Level:


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

It would be very useful for students to work on What Numbers Can We Make? before attempting this problem.
Introduce the problem the way Charlie does in the video, by inviting students to suggest sets of five whole numbers, circle three of them that add up to a multiple of three, and write down the total.
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 don't include three that add up to a multiple of $3$. Allow them time to work on the problem in pairs or small groups, and suggest that they write any sets they find up on the board. Students may enjoy spotting errors among the suggestions 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.

If it hasn't emerged, share with students Charlie's representation from What Numbers Can We Make?

All numbers fall into one of these 3 categories:

Type A (multiple of $3$)

Type B (of the form $3n+1$)

Type C (of the form $3n+2$)

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.

"Which combinations of A, B and C give a multiple of three?"
"Can you find examples in our list on the board where you gave me one of those combinations?"

A few minutes later...

"Great, then all you have to do is find a combination of As, Bs and Cs that doesn't include AAA, BBB, CCC or ABC!"

Later still...

"It's impossible! All the combinations will include AAA, BBB, CCC or ABC!"

"OK, but can you prove it? Can you convince me that it's impossible?"


Possible support

Select sets of $3$ numbers. Your sets will always include two numbers that add up to an even number. Why?

Possible extension

What Numbers Can We Make Now? is a suitable follow-up task.
A challenging extension:
You can guarantee being able to get a multiple of $2$ when you select $2$ from $3$.
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 $7$?