Where can we visit?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Problem

Where Can We Visit? printable sheet

100 squares printable sheet

 

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

Image
The numbers 1 to 100 written in a 10 by 10 grid, with a counter on the number 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:
Image
The numbers 1 to 100 written in a 10 by 10 grid, with the numbers listed above that have said to have been visited in a different colour, and the number 84 in another colour to represent the final location of the counter.

Is it possible to visit every number on the grid?

What if you start on a different number?

Can you explain your results?

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

You might like to 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?



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