So when mathematicians solve a problem
they don't really stop. They try to
generalise it, they try to create new
games, new problems and see if they're
solvable or not. So in this case we can
play beyond the game in two different
directions. Well, we had initially three
clones correct in this configuration, we
can change the configurations. We can for
instance do the simplest thing possible,
start with one clone in the first cell.
So that's one way change the initial
clones configuration. Another way to
change the game is change the initial
barbed fence, change what the prison is.
So let's try the simplest possible game
which is not trivial.
So what would that game be? Same prison
as before, three cells, but only one clone.
Is this game possible or impossible? Now
you can start calculating invariant sums,
this that, but really why don't you try
to play the game? Play it, maybe it will be
possible. Because if you can show me one
way to do it then you don't need to show
me anything else, I will be convinced.
All right? So, play play play - and in fact
just about anything you try here in this
game should work, again you have to play
by the rules, alright. So we have five
clones out and the prison is empty. So
this game is solvable; and we move on to
the next one, but now we enlarge the prison.
We make it a nice bigger shape with six
cells - can you escape this prison? Again
you'll start playing - play play play - but
after a while- and you stop the video and
you played for ten minutes right? Okay,
you may conjecture that it is impossible
to free the clones. So we're back to
square zero trying to figure out why
this is impossible. However we have some
progress made from the previous game,
because the rules of the game have not
changed. Therefore you can use exactly
the same invariant as before, we just
have to make different calculations with
it. We again want to free the clones
into Greenland, let's see if we have
enough gasoline, enough weight there.
The initial prison sum, if you calculate
it, will be 2 and 3/4. The Greenland sum
will be 1 and a quarter. But let's not
forget that the initial sum is just 1
because we have only 1 clone in one cell,
in the first cell. So is this a
contradiction?
The initial sum is equal to the final
sum, that's one, and that's less than what
we have in Greenland. - (Brady: Whoo! That's good.)
Really? What did it prove? It proved that
we have enough weight in Greenland to
free the clones - but did we really free
the clones? Did we show a way to do it? We
didn't reach a contradiction and we
didn't find a way to free the clones; so
we really are still back to the very
beginning. We have to do something more
subtle if we still believe it's
impossible to free the clones. And the
subtler argument comes from a very
simple observation. So as we are playing
the game let's watch what happens on Row
1, just on Row 1. I started with 1 clone
there, after one move I have another
clone, just one. I can make another move
and I will again have one clone there. I
can make lots of other moves here that
don't change anything, but the moment I
try to change something on the first row
I'll be replacing one clone there with
one clone. So we are talking about the
first row, and at any point during the
game there will be exactly one clone
there - and you can check this with
experimenting. And symmetrically there
would be exactly one clone in the first
column. So really there are restrictions
in Greenland, you cannot have for example
two clones like this or perhaps three
clones like that, this is impossible, you
must have exactly
one clone in the first row and in the
first column. Let's see how this changes
our calculations. So if you free the
prison, the largest contribution in
Greenland from the first row will be 1/8;
because this is the first cell after the
prison with the largest weight. You
cannot have anything else here or else
you will have a smaller contribution. So
on the table this is precisely position
4. And from the first column the
largest contribution will be made by
having the clone right here, also in
position 4. Let's see what the
calculations show us now: we have the
prison, and we disappear all of the
clones beyond the fourth cell in the
first row and in the first column,
because their contribution is less than
1/8 and we cannot have two clones there.
So let's start adding up. How much did we
subtract? In other words how much are
those tails that we took away? Well they
are another geometric series starting
with 1/16, also with ratio 1/2. And if you
use our formulas from before, a similar
reasoning, you will end up with 1/8. So
those green tails that we took out are
precisely 1/8 each. What does this do to
Greenland's sum? The total sum of the whole
table minus the prison sum minus the tails. Total
sum 4, prison sum two and three
quarters, and 1/8 for each of the cells,
this is 1. Now, is this a contradiction?
It looks very similar to the situation
from the previous problem where we just
squeezed in a contradiction.
So the initial sum is 1, we have one
clone in the first cell, like that. The
final sum has to be 1. But how am I
going to get 1 out of this whole table?
Once again I have to populate every
single cell in Greenland except for the
two tails. But that means again
I need to have infinitely many clones
in the end. And this is a finite game,
therefore again we reach a contradiction,
we cannot free the prison. - (Pretty cool.)
Is that the end? Well you can play the game
with larger and larger prisons; or perhaps
with more clones in different
configurations. But as you can see the
problem becomes harder and harder once you
start changing the shape of the prison
or changing the clones configuration.
(Why are we doing this?) - Because it's fun! Once
you solve a particular problem, the
original one, which in itself was so much
fun you don't stop there - you don't want
to stop the fun. Mathematicians have
found the whole time, especially when
they're doing research, that's why
they're into this profession, they love
it. - (But you're get- you get paid. Like,)
(people are paying you money.)
(You're just being paid to have fun?)