A 'doodle' is a closed intersecting curve drawn without taking
pencil from paper. Only two lines cross at each intersection or
vertex (never 3), that is the vertex points must be 'double points'
not 'triple points'. Number the vertex points in any order.
Starting at any point on the doodle, trace it until you get back to
where you started. Write down the numbers of the vertices as you
pass through them. So you have a [not necessarily unique] list of
numbers for each doodle. Prove that 1)each vertex number in a list
occurs twice. [easy!] 2)between each pair of vertex numbers in a
list there are an even number of other numbers [hard!]

How many different cubes can be painted with three blue faces and
three red faces? A boy (using blue) and a girl (using red) paint
the faces of a cube in turn so that the six faces are painted in
order 'blue then red then blue then red then blue then red'. Having
finished one cube, they begin to paint the next one. Prove that the
girl can choose the faces she paints so as to make the second cube
the same as the first.

Tree Graphs

Stage: 4 Challenge Level:

What is the very simplest graph you can think of? How many edges
and vertices does it have?