A little mouse called Delia lives in a hole in the bottom of a
tree.....How many days will it be before Delia has to take the same
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
This drawing shows the train track joining the Train Yard to all
the stations labelled from A to S. Find a way for a train to call
at all the stations and return to the Train Yard.
You will have found that it is
impossible to trace over all of the diagonals of a hexagon without
going over the same line twice. Why is this? The trick is to look
at how many diagonals are connected to each vertex. Jenni offers us
a very clear explanation:
Each time you draw a line IN to a vertex, you also need to have
another line going OUT (unless you are on the end point). If there
is an even number of lines, then you will always be able to go in
and out of any vertex without using the same line twice. However,
if you have an odd number of diagonals connected to each vertex,
then sooner or later you will revisit a vertex and find that there
are no more available lines going out of it.
Thank you, Jenni.
Can you see that this will always be
possible for shapes with an odd number of sides, because they
always have an even number of diagonals connected to each vertex?
On the other hand it would never be possible for any shape with an
even number of sides, for the opposite reason.