### 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?

##### Age 11 to 14 Challenge Level:

Here is a 100 square board with a counter on 42:

Using either of the two operations $\times 2$ and $-5$, whereabouts on the 100 square is it possible to visit?

You might start like this: $$42, 37, 32, 27, 22, 17, 12, 7, 14, 9, 18, 13, 26, 52, 47, 42, 84 ...$$Notice that you are allowed to visit numbers more than once.

The board would look like this:

Is it possible to visit every number on the grid?

What if you start on a different number?