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Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Cuisenaire Squares

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?


We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Simple Train Journeys

Age 5 to 11
Challenge Level

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
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.