### 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 Abi 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?

#### 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 Abi and Charlie discovered or choose pairs of operations of your own and investigate what numbers can be visited.