This problem, which could be introduced using a 'real' toy train and tracks, is the classic network problem redesigned for young mathematicians. It is based on Euler's Bridges of Konigsberg puzzle.

If you have train sets available, building the track arrangements will be of enormous value to children who are less able to transfer their thinking into the more abstract diagrammed representation.

After the children have worked either as a group or in pairs on the first track in the problem, they should share their findings, discussing if there is more than one way of travelling around the tracks and tell of any other observations they have about the number of pieces the track that were used and how many pieces of track feed in at each junction or node. Focussing their attention around these features will assist them in their observations and predictions and are also a prelude to future investigations of Euler's theory about networks and nodes.

This information might help you:

1. If the number of lines at each junction is even then a network can be drawn that starts from any point and finishes there.
2. If just 2 (and only 2) junctions have an odd number of lines meeting at them, a network can be drawn that starts at one of the 'odd junctions' and finishes at the other.
3. However, if there are more than 2 'odd junctions' then a network can't be drawn.

Before actually setting about the next task of drawing the route and making their discoveries, the children should look carefully and visualise the route the train might take. Each child should then be encouraged to predict if the train can travel each set of tracks in the way described - without moving along the same piece of track more than once.

As an extension or follow up activity, children should be asked to design a track that can be travelled by the train (using the same condition) and a track that can't be travelled. You might want to set a limit on the maximum number of pieces of track that can be used.