Simple Train Journeys
Problem
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!).
Your challenge is to find all the different journeys for visiting four stations.
Can you predict the number of different journeys for visiting seven stations? Were you right?
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.
Getting Started
Have you got a way of making sure you have got them all?
Perhaps there is a way you could use the three-station journeys to help you find all the four-station journeys?
When you're looking for a pattern in the numbers, it might help to go even simpler as well - how many journeys would there be for just two stations?
Can you see any way that the numbers of different journeys are connected to each other? It might help to write them in a list or table.
Student Solutions
We had a number of solutions come in with answers from Connor, Isobel, Brian, Josh, Stephanie and Ben.
H.C. from SIS School sent us this very good answer:
I started calculating the number of ways through 4, 5 and 6 stations first.I did it in an orderly manner.You must start with Dorby and then Ender and Dorby again so you will not miss any of the ways. Then you can start with Dorby, Ender and Floorin and so on ...
Here is the way I used from 4 stations to 6 stations:
4 stations:
1.Dorby-Ender-Dorby-Ender
2.Dorby-Ender-Floorin-Ender
3.dorby-Ender-Floorin-Gambolin
5 stations:
1.Dorby-Ender-Dorby-Ender-Dorby
2.Dorby-Ender-Dorby-Ender-Floorin
3.Dorby-Ender-Floorin-ender-Dorby
4.Dorby-Ender-Floorin-Ender-Floorin
5.Dorby-Ender-Floorin-Gambolin-Floorin
6 stations:
1.Dorby-Ender-Dorby-Ender-Dorby-Ender
2.Dorby-Ender-Dorby-Ender-Floorin-Ender
3.Dorby-Ender-Dorby-ender-Floorin-Gambolin
4.Dorby-Ender-Floorin-Ender-Dorby-Ender
5.Dorby-Ender-Floorin-Ender-Floorin-Ender
6.Dorby-Ender-Floorin-Ender-Floorin-Gambolin
7.Dorby-Ender-Floorin-Gambolin-Floorin-Ender
8.Dorby-Ender-Floorin-Gambolin-Floorin-Gambolin
3rd stat - 2 ways
4th stat - 3 ways (2+1)
5th stat - 5 ways (3+2)
6th stat - 8 ways (5+3)
7th stat - 12 ways (8+4)
This is great - you have really gone about it in a logical way.
So, I wonder if anyone can predict how many ways could be found for 8 stations? This is an example of the Fibonacci sequence. You can find out more about this special pattern of numbers in this article .
Teachers' Resources
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!).
Your challenge is to find all the different journeys for visiting four stations.
Can you predict the number of different journeys for visiting seven stations? Were you right?
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 |