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### Number and algebra

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### Advanced mathematics

# Where Can We Visit?

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Age 11 to 14

Challenge Level

*Where Can We Visit? printable sheet
100 squares printable sheet*

**Here is a 100 square board with a cou****nter 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 numb****er 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?