Simple Train Journeys
Well, here's a train route. The train starts at the top and makes a number of visits to the stations.
Now let's suppose that the train is going to make visits to three stations (they do not have to be different stations  each station can be visited several times!).
So the first station would be Dorby  this will always be the case! (Why?)
Then the train can go on to Ender. When at Ender it could return and visit Dorby again OR it could go on to Floorin.
So two different journeys:
Dorby  Ender  Floorin
Dorby  Ender  Dorby
Your challenge is to find all the different journeys for visiting four stations.
You could then go on to find all the different journeys for visiting more stations  try five.
How about six stations?
Can you predict the number of different journeys for visiting seven stations? Were you right?
How would you predict the number of different journeys for visiting eight stations?
You might like to then invent your own routes that may go further than this one and then answer similar questions that you can think up.
Initially, this problem could be introduced in a similar way as is suggested in the notes for Train Routes, but it would be good to focus more on looking for patterns and generalising in this case. You might like to work on the different routes for four stations as a whole class then ask small groups to look at
five and six stations so that you can pool results. Ask the children how they are recording the different routes  using initial capitals to stand for the stations is a great help, but share any good ways the pupils have found.
In order to look for a pattern in the numbers of routes, it might be helpful to make a table, something like this:
Number of station visits 
Number of different journeys 
3 
2 
4 

5 

and so on ...
Encourage the class to look carefully at how the number of different journeys in each case is related to the number of different journeys for smaller numbers of station visits. Once they have identified the pattern, ask them to think about why the pattern occurs.
Making up their own rail networks and investigating them with similar questions would be a good next step. Alternatively, you could challenge them to devise networks with certain criteria.