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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:

This is how they started:

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

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

This is what they tried next:

And then they tried these:

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

Can you explain your results?

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

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:

(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?

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.

Can you explain your results?

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