### 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:

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?

Choose pairs of operations of your own and investigate what numbers can be visited.

You might like to use the interactive grid below, or print off some 100 squares.

Is there a way to predict which numbers it's possible to visit, for a given starting point and a pair of multiplication/subtraction operations?

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