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Can You Traverse It?

Age 14 to 18
Challenge Level

You might like to try The Bridges of Konigsberg before exploring this problem.

A traversable network is one you can draw without taking your pen off the paper, and without going over any edge twice.

For each network below, decide whether or not it is traversable. It might be helpful to keep a track of where you started, the route you took, and where you finished.

You may find it useful to download a printable copy of the networks

What do you notice about traversable networks where you started and finished in the same place?
What about traversable networks where you started and finished in different places?
What do you notice about the number of times you visited each vertex (point)?
For the networks which are not transversable, what is the smallest number of edges that you need to add (or remove) so that the resulting network is traversable?

Can you find a condition that guarantees a network is not traversable?
Can you explain why?