>> Welcome back. We have, what do we have?
We have, we have a straight flush. Small, small values.
But nonetheless, straight flush. Not quite in order, but you can check it.
We're doing toads and frogs now. You can, you can do this at home.
With people and it's fun play with people but you can also do it with, with coins.
Here's toads and frogs. Let's use nickels over here and Jefferson
nickels. Let's use Lincoln pennies over here.
Here's toads and frogs. The toads in this case are the nickels,
they move right and are controlled by left.
The frogs are the Lincoln cents, they move left and are controlled by right.
The toads can move one space to the right if it's empty.
The frogs can move one space to the left it it's empty.
But they also can jump over each other. A toad can jump over a frog, like this.
Note that the, the coin when you jump over isn't removed.
'Kay? But you can jump over and then they, then
say the Lincoln cent can move over here the, the nickel can do here.
The Lincoln cent can move over here My concern is the frog.
And we see at this point that neither the toads nor the frogs have a move so this
Lincoln cent, which was a frog, was right. Right made the last move, it's now left's
turn, left doesn't have any moves, left loses, Okay.
That's toads and frogs. And there's lots of examples involving up,
down and integers and all that good stuff involving with the small versions of toads
and frogs. So, so you can try this at home.
You can either do this with, with, with coins on a piece of paper or get your
friends and, that way you can jump over one another and you, you draw these on the
floor, okay? All right.
So but be careful. Don't hurt yourself and don't try that at
home. Probably do it with coins.
All right. So lets look at its, at a, toads and frogs
position. Now here we have toad, toad, they move
right, they're controlled by left. Blank means an empty space.
Frog, frog, frogs are controlled by right. They move left.
So, the only possible opening move for left is to move to toad, is to move this
toad one to the right and that leaves toad blank toad frog, frog.
And the only possible opening move for right is to move this frog one left and
that's leaves the position, toad, toad, frog, blank, frog.
Now, let's go over here and look at this position.
If left moves in this position, then this toad moves to the right.
And we're left with toad, toad, frog, frog.
That's a zero position. Okay?
Up and if right moves then right can jump out take this frog.
The only possible case is for right to move this frog jump over the toad and left
with toad, frog, toad blank frog. Now let me remind you that when we're
analyzing games we have to consider like two left moves in a row and right moves in
a row, because we want to analyze this game in the context of a much larger game.
So we might have this toads and frogs over here.
We might have a nim game someplace else, cut cake someplace else, a game of chess
going on, a game of Go going on. And then each player in turn chooses one
of these, one of these games and makes a move in it.
This is how we add gain. So and in doing so in adding this game to,
to others, left might make two, two moves in a row in this, okay?
So, I'm going to leave out some, some pretty long, I think, computation, but
this game down here. Toad, frog, toad, blank, frog is actually
equal to star so we have this game up here.
Toad, blank, toad, frog, frog. Its left option, only left option is to
move to zero only left move is to move to zero.
Only right move is to move to star and therefore this game toad, blank, toad,
frog, frog is up. Up, you remember, was left 0, right star.
Now, if you look at this game over here, it's the same as the game over here, with
toads and frogs interchanged and left and right interchanged.
Just, Just take the mirror image and change all the toads to frogs and all the
togs, frogs to toads. Therefore this gain on the right is the
negative of the gain over here. So this gain on the right is minus up
which is sometimes written down. So this whole gain is equal to up down.