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?