# Delia's routes

A little mouse called Delia lives in a hole in the bottom of a
tree.....How many days will it be before Delia has to take the same
route again?

## Problem

A little grey mouse called Delia lives in a hole in the bottom of a tree in a small square garden (A).

Image

The garden is paved with 6 large square paving stones in each direction and has a circular pond right in the middle that has a diameter of 3 of the paving stones. Delia's tree is at the left hand corner at the bottom of the garden. At the top right hand corner of the paved area there is a bird table (B).

Image

How many days will it be before Delia has to take the same route again?

## Getting Started

Think about the number of choices there are at each point. Or simply count how many there are.

You could use different colours to find different routes on this sheet.## Student Solutions

David and Fiona told us how they started to tackle this problem:

First we tried different paths to find the shortest way for Delia to get to the bird table. We tried lots of paths and found that Delia must go along six tiles and up six tiles and so any path that does that and does not go down or left is a shortest path.

Then we worked out that to avoid the pond, Delia must only run along the edges of the outside tiles, because all the other edges run into the pond.

Lizzy explained how she continued from here:

First Delia can go up one square or right one square. That is two choices. Then she can go up one square or right one square. That is 2 x 2 = 4 choices, because she can do this for both of her first choices. Then she can go up one square or right one square. That is 4 x 2 = 8 choices, because she can do this for all four of her second choices. Then if she is on the side of the garden she can still go up one square or right one square, but if she is not then she might run into the pond if she makes the wrong move. I made a table of what she can do from here.

First move | Second move | Third move | Fourth move |

Along | Along | Along | Along |

Along | Along | Along | Up |

Along | Along | Up | Along |

Along | Up | Along | Along |

Along | Up | Up | Up |

Up | Along | Along | Along |

Up | Along | Up | Up |

Up | Up | Along | Up |

Up | Up | Up | Along |

Up | Up | Up | Up |

I saw that she cannot go along two and up two in any order because then she will be off the edges of outside squares and into the pond.

Then I looked at the rest of the path and saw that if she starts by going along two and up one or along one, up one and along one or up one, along two then she must continue going along until she gets to the edge of a tile that is on the righthand edge of the garden, and if she starts going up two then along one or up one, along one, up one or along up, up twothen she must go up until she gets to the edge of a tile at the top of the garden.

Then I worked out that if she starts going along, along, along then she must only go up once or not at all before getting to a tile on the righthand edge of the garden. If she goes up, up, up then she must only go along once or not at all before getting to a tile at the top of the garden.

This way I broke everything down so I could be sure I was counting all the paths she could take and none of them twice. I counted 74 different ways, so she must repeat every 75 days or less.

That's right,Lizzy! Thank you, Lizzy, David and Fiona.

## Teachers' Resources

### Why do this problem?

This problem has a story line that will encourage learners to become involved in Delia's nocturnal adventures. To answer the question How many days before Delia has to take the same route again? requires the children to find all of the possibilities. Given the number of pathways, this problem could be ongoing asking the children if they could find another three or four routes each day over a couple of weeks. This sheet gives 6 copies of Delia's garden plan.### Key questions

How many tiles along and how many tiles up must Delia go on
each journey?

Can you think of a way to record all the different pathways
Delia could take?

How many ways can you go when you get to this junction?

Which tiles can Delia not run along because they go into the
pond?