Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Marbles

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

More Marbles

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

How Much Can We Spend?

Stage: 3 Challenge Level:

Why do this problem?

This problem offers students opportunities to explore fundamental ideas about number theory in a simple context. They are encouraged to explore, conjecture, generalise and justify.

There are opportunities for older students who are familiar with algebraic manipulation or modulo arithmetic to produce rigorous proofs.

Possible approach

Introduce the context, just two coins 3z and 5z.

Ask students to combine these to make totals up to 20z.

Which totals can they make? Ask them to keep a record of their working.

Bring the class together to share results. Did anyone have strategies for working these out?

Ask for strategies for making totals greater than 20z.

It can be hoped that some students will have noticed that once they have found three consecutive totals they can make all successive numbers by adding on three.

There should now be agreement that:

7 is the largest total that cannot be made
8 and all numbers above can be made

'Could we have predicted this?'

Are there any conjectures?
'What do you think would happen if the two coins were 3z and 6z?'
'What about 3z and 7z? 3z and 8z? 3z and 9z?......'

Allow some time for the students to test and refine their conjectures.

Bring the students together to share their findings and display their results, focussing on the largest total that cannot be made and the number that follows.

Were any of the original conjectures correct? Which?
Can they now use the combined results to make some generalisations?
Can they use their generalisations to predict what will happen for any pairing of 3z with another coin?

With older students who are familiar with algebraic manipulation, there is an opportunity here to express the findings algebraically in different ways and convince themselves that they are equivalent.

Ask students to consider now what would happen when the pairings contain:

5z and another coin
7z and another coin...
any prime number value and another coin.

Working in twos or threes, ask students to make a display of their results.

Key questions

How do you know you can make all the totals after a certain total?

What happens when the other coin is a multiple of 3z? (or 5z or 7z...)

Does it make a difference if the other coin is 1 more or 2 more than a multiple of 3z? (or 5z or 7z...)

Possible extension

Can students explain/justify/prove their findings?

Ask students to consider what happens if neither coin is a prime number.

Possible support

The whole class introductory activity as described above should provide the necessary support for all students to access this problem.

Creating expressions/formulae. Visualising. Modulus arithmetic. smartphone. Making and testing hypotheses. Number theory. Working systematically. Generalising. Interactivities. Mathematical reasoning & proof.

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