Hello, everybody!
My name's James,
and welcome to Cambridge University,
where I work--
which could mean anything-
that could mean I'm the janitor.
But it's even better than that--
you see, I'm a mathematician.
And I know what you're thinking--
you're thinking
"Surely he's far too young and handsome
to be a mathematician," right? But--
OK--suit yourselves.
But no! And today,
I'm going to show you a problem,
and I thought I'd do it somewhere more academic.
So the problem I'm going to show you today
is called "The Utilities Problem,"
and it works like this:
There are three houses,
and I want to connect
the gas, the electricity, and the water
to these three houses.
I connect them by drawing a line.
I want to know is
can I do this without
crossing a line?
Now, if you haven't seen this problem before,
you might want to pause the video now
and have a go,
because I'm going to give the solution.
So
if you're ready,
I can tell you
that this is impossible,
and I can prove it's impossible, too.
Perhaps later I'll show you a way around it.
But to prove that it's impossible,
the first thing I'm going to do
is I'm going to reduce these houses
so that they're points--
so it looks something like this.
Now, I want to connect these three red dots
to these three blue dots
without crossing a line.
So it might look something like this.
Well, except you can see here
that I have crossed a line.
Now, this is called a graph,
feel free to move the points about,
to move the lines about, and the question still holds:
Can I do this without crossing a line?
Well, if I can do it without crossing a line,
then I can turn that flat, two-dimensional graph
into a three-dimensional solid,
just by inflating it.
So I can take something flat,
and I can go
[blows]
[popping noise]
and turn it into a three-dimensional solid.
Now, there's a rule that we know about
three-dimensional solids.
It's called "Euler's Formula,"
it was discovered by the Swiss mathematician Leonhard Euler
and it looks like this.
It says that the number of Faces [F]
plus the number of Vertices [V]--
that's the number of corners--
take away the number of Edges [E],
will always equal two.
This is a normal solid
with no holes in it,
or anything like that.
So, for example,
for this object here--
it's called an octahedron--
it has eight faces,
it has six corners,
and it has twelve edges.
If I put that into Euler's Formula,
it looks like this:
So it says 8+6-12=2,
which it should, and that's what I expect.
By the way, just as an aside,
I can use Euler's Formula
to work out that there are only five regular solids.
A regular solid
is like this.
It has the same number of faces at each corner.
The five solids are called
the tetrahedron,
the cube, which you probably know,
the octahedron, which we just saw,
the dodecahedron,
and the icosahedron.
There are only five regular solids.
But to get back to the problem--
now, this graph that we're going to inflate
and turn into a solid,
well, it has six corners,
and it has nine edges.
So if I put that into Euler's Formula,
I can say that the number of faces
is 2+9-6, which is 5.
There are five faces.
Now, there's one thing I know about this solid:
I know that I never connect
two houses together,
and I never connect two utilities companies together--
there's no point.
So I can say that each face has at least four edges.
So that means that if there's five faces,
there must be at least twenty edges.
But I've counted each edge twice there.
Each edge is connected to two faces,
so I can say I need at least ten edges.
But I only have nine.
And that's why it's impossible.
By the way, I said I was going to show you
a way around it.
Imagine this:
Imagine if you can go off this side here,
and reappear at the opposite side,
like an old computer game.
Then you can connect the houses.
Imagine this:
If I roll this up into a tube,
like that,
and then if I connected the two ends of the tube,
so it made a big, round donut-like shape,
then yes, I can connect the houses.
This is only excluded from
the Euler's Formula because
it has a hole in it,
and I said Euler's Formula
works for solids that don't have a hole in it.
And that's how you do it!
So, if you have been,
thanks for watching.