What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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.
A graph is a network of points and lines. The points are called
vertices and the lines which connect them are called edges. Each
edge has a vertex at each end. The branch of mathematics called
graph theory studies the properties of various kinds of graphs.
A connected graph is a graph in which we can get from any vertex
to any other by travelling along the edges. A tree is a connected
graph with no closed circuits (or loops).
Prove that every tree has exactly one more vertex than it has