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.
The simplest tree graph consists of one line with two vertices,
one at each end.
If a new line is added it must connect to one and only one of
the existing vertices.
If the new line connected to no vertices, the tree graph would
not be connected, as the new line's vertices could not be reached
from the existing vertices.
If each end of the new line connects to a vertex the graph will
have a circuit and will not be a tree graph.
So every new line added will join on to one existing vertex and
create a new vertex at its end. This adds one line and one vertex
to the tree graph making no change to the difference between the
numbers of edges and vertices. So the difference remains constant
at what it originally was.
Marcos's solution is a subtle variation
on the above method:
Proof . If there wasn't at least one such
vertex we could keep moving around the graph indefinitely and as
there is a finite number of edges it would mean that there is a
cycle, counter to the definition of a tree.
Take one such vertex as described in (2) and its respective
edge. So far we have 2 vertices and 1 edge. The difference is
(*) Add to this an adjacent edge (this is adding one edge and
one vertex. The difference is still one.
Generally, carrying out this step, (*), an arbitrary number of
times until we add the final edge will still result in a difference
of 1 (as the step was in no way linked to fact that the previous
edge was the starting one)
Hence, the number of vertices is one more than the number of