### Redblue

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

### Diagonal Trace

You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?

### Rail Network

This drawing shows the train track joining the Train Yard to all the stations labelled from A to S. Find a way for a train to call at all the stations and return to the Train Yard.

# Delia's Routes

##### Stage: 2 Challenge Level:

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.