Hello, everybody!
I have a puzzle for you,
and perhaps this is one you can try out on your friends and family.
Look at this picture.
Now the puzzle is this:
Can you draw
a continuous curve
without taking your pen off the paper,
that crosses each line once
and once only?
So,
it might look something like this--
OK, let's say I start over here,
I come into the box and I cross that line,
and I go up through the top,
and I go over,
I come back down,
cross that line,
go out of the picture,
come back in,
go through there,
which I'll cross this line,
through here,
down through this line.
Now, it's important to remember,
these two halves of this line
count as two different lines.
So I can go up
through here--that is allowed.
And now I come back down,
around,
into this box, and--ah!
Right.
Now I'll have to stop there,
so if I go down again, and I cross this line twice,
which is not allowed.
So I nearly did it,
but it didn't quite work.
Now, the question is
can you do it?
If you don't want to know the solution,
I suggest you pause the video now,
because I'm going to give you the solution in a second.
So if you want to try it,
do pause, stop, or do something like that,
because I can tell you
that this
is impossible.
But I want to know why it's impossible.
Let me show you how to prove it.
Here's that picture again.
Now what we want to do
is we want to cross
each line once, and once only.
So let's draw that in.
So I want to cross this line,
like that,
I want to cross this line,
and I want to cross all these lines,
and I want to cross them each once.
So this is exactly
what I want to do.
Now,
to finish this off,
really it should just be a case of joining up the dots.
So I might come into this box, and then I'll go up
through the top, like that.
Let me add just one more thing. Let's say
every time I enter a box,
I make a detour to the center.
So let me draw that in. I'm going to draw in the center points.
So there, there
there,
and there.
OK.
Let's join up the dots now.
If we do, this is what we'll get.
We'll get a picture
that looks
like this.
Now this
is a diagram
that tells us everything that we can do.
Now, I haven't connected the curve
outside the picture,
but certainly inside the picture,
this tells us everything
we can do.
So if I could
come into the box here,
and I go to the center,
I could go up through the top,
Or I could come in
and go to the center,
and I could go down this way.
Or I could come into the center
and I could keep going
and cross this line
like that, and then I could go up.
This really tells us everything
that we can do.
Now, everytime I enter the center
of a box,
I leave it again.
So lines into the center
come in pairs.
We should have an even number of lines
at each center point.
There are two exceptions to this,
and they are if the center point
is the beginning point,
or the end point.
In that case,
we may have an odd number of lines.
So we can have, at most, two points with an odd number of lines.
But here,
in this diagram
we can see that we have three center points
with an odd number of lines.
And that's why
it's impossible to draw.
This idea has a proper name,
and it's called
an Euler's Trail.
The idea of
drawing a curve
such that each point has an even number of lines--
perhaps--except at the beginning or at the end.
That's called an Euler's Trail.
And that is how you do it.
So, as usual,
if you have been,
thanks for watching.