>> 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.