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?

Where Can We Visit?

Stage: 3 Challenge Level:

Charlie and Lynne had this 100 square board and put a counter on 42.

They wondered if they could visit all the other numbers on the board moving the counter using just these two operations:

$\times 2$ and $-5$

This is how they started:

42, 37, 32, 27, 22, 17, 12, 7, 14, 9, 18, 13, 26, 52, 47, 42, 84 ...

(notice that they are allowed to visit numbers more than once)

and this is what their board looked like:

Will they be able to visit every number on the grid at least once?

What would have happened if they had started on a different number?

Can you explain your results?

They wondered if they would get the same sort of results with other pairs of operations.

This is what they tried next:

$\times 3$ and $-5$

$\times 4$ and $-5$

$\times 5$ and $-5\ldots$

And then they tried these:

$\times 5$ and $-2$

$\times 5$ and $-3$

$\times 5$ and $-4\ldots$

Find out what Lynne and Charlie discovered or choose pairs of operations of your own and investigate what numbers can be visited.

Can you explain your results?

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This problem is also available in French: Où irons-nous?

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