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Have you ever read a Winnie the Pooh poem called "Lines and Squares" by A.A. Milne? It tells us how Christopher Robin plays a game with himself as he walks along pavements, trying not to walk on the "lines" (the edges of the paving stones).

Have you ever done this? I know I have! This investigation is based on a similar idea - but here I want to walk on the lines!

This is a picture of the path leading up to my front door from the road:

102pic


I like to walk along the cracks of the paving stones, but not the outside edge of the path itself because I may tread on the grass.

The only way I can do it is by walking straight down the middle of the path like this:


If the path were three paving stones wide instead of just two it would look like this:


Remembering that I can only walk along the sides of the paving stones and I mustn't tread on the outside edge, how many different routes can you find for me to take? (By the way, you must not turn back on yourself, and you must head towards the door or sides - so you cannot walk towards the road on your journey.)

Do any of your routes have a repeating pattern?

Imagine now that the path is even wider, with four paving stones:


What different routes can you find now?

Perhaps you could group them into those with a repeating pattern and those without. Maybe you can find other ways to group the routes.

If I could also take steps diagonally across a paving stone, like this:


then there are even more possibilities. Try to find the new routes yourself, starting with 3 paving stones width. Remember all the same rules apply as before.