>> Alright, welcome back.
None of this, none of these, no dice, no
cards, games without random moves.
Now, now that we know from last time how
to add things up, add up games, maybe
we'll play them simultaneously in the ways
we described last time.
We want to talk about what it means for
two games to be equal, which is going to
mean something like, whenever we play
these games together with some other game,
the same thing happens.
Now, I want to be much more precise about
that in a minute.
But that's esesentially what's giong on.
And so, to start this, I want to start
with a simple, a very simple case.
And our definition of win a game is 0.
This is a definition and mathematics
definitions are prescriptive, that means
they're, what is the case.
Okay.
So what does it mean for game to be zero
and this means in best play, the first
player to move, loses.
So, just like the zero game, whoever moves
first in the zero game, loses, because
there's no moves.
Let me, let me show you.
The zero game happened to bush is this.
Nothing to cut.
The zero game in, in a heap, is a heap of
zero coins, which you can see over here to
your left.
And so that's the zero game.
And then, we'll say the game is equal to
0, if in best play the first player to
move loses.
Okay, let's look at an example.
Nim-heap of size 2 over here, nim-heap of
size 2 over here.
It just matters that there's two dice over
here, two dice over here, the numbers on
the dice don't change anything.
So, so if, if left, whatever left does
over here, right does over here.
Whatever left does over here, right does
over here, and left loses.
So, if left goes first, left loses.
Similarly, if right goes first, right
loses.
So, this game 2 nim-heaps, a nim-heap of
size 2, which we now take this way, plus a
nim-heap of size 2, is first player lose,
so it's equal to 0.
Okay.
So, we know when a game is equal to 0.
So now, let's look, let's try to see what
a negative of a game is.
We have a game, G.
We can compute, have an associated game
called minus G.
Now, let me show you this for hackenbush.
Say, this hackenbush game is G.
Then, minus G is the same thing with left
and right interchanged, okay?
Now, let's look at other possibilities.
In, in cutcake, remembering that right,
cuts left to right, and left cuts up and
down.
Then, if that's a game of cutcake, if we'd
simply interchange rows and columns, then
whatever became a, a, a possibility for
left is now a possibility for right, and
whatever became a possibility for right is
now a possibility for left, okay?
So, the negative of this cutcake game is
this game.
The, the negative of, say nim-heap of size
3, is now interchanged the versions of
left and right, but since left and right
are, are, have the same moves in them, the
negative of this is the same, same as, as
that.
So, the negative, let's do it, of nim-heap
of size 2, is a nim-heap of size 2.
Interchanging left and right, a nim
doesn't do anything.
What is it in chess?
Well, the negative of, of, the negative
chess game is a game, a chess game where
black goes first, I guess.
Or maybe it's the negative of a chess game
is where instead of you having white, you
have black.
So it's the interchange of the, of the two
sides.
Left becomes right, right becomes left,
black becomes white, white becomes black,
whatever the, the moves in the game are.
In Go the negative of the game, which is
corresponding to a different player going
first.
So instead of white stones, you have black
stones etc., etc.
So, that's, that's how a negative of game
is.
And now, we're ready to, to define G
equals H, means G minus H is 0, or G minus
H is just an abbreviation for this, okay?
We'll look at some examples.
For next time, this is a short one.
Let's just take a look at, I'll give you
an example to, to work on yourself.
Let's look at a cutcake and a hackenbush,
and my claim is, are these equal?
Try it out and we'll see you next time.
Take care.