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>> Welcome back.
We have, what do we have?

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We have, we have a straight flush.
Small, small values.

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But nonetheless, straight flush.
Not quite in order, but you can check it.

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We're doing toads and frogs now.
You can, you can do this at home.

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With people and it's fun play with people
but you can also do it with, with coins.

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Here's toads and frogs.
Let's use nickels over here and Jefferson

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nickels.
Let's use Lincoln pennies over here.

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Here's toads and frogs.
The toads in this case are the nickels,

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they move right and are controlled by
left.

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The frogs are the Lincoln cents, they move
left and are controlled by right.

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The toads can move one space to the right
if it's empty.

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The frogs can move one space to the left
it it's empty.

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But they also can jump over each other.
A toad can jump over a frog, like this.

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Note that the, the coin when you jump over
isn't removed.

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'Kay?
But you can jump over and then they, then

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say the Lincoln cent can move over here
the, the nickel can do here.

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The Lincoln cent can move over here My
concern is the frog.

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And we see at this point that neither the
toads nor the frogs have a move so this

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Lincoln cent, which was a frog, was right.
Right made the last move, it's now left's

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turn, left doesn't have any moves, left
loses, Okay.

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That's toads and frogs.
And there's lots of examples involving up,

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down and integers and all that good stuff
involving with the small versions of toads

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and frogs.
So, so you can try this at home.

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You can either do this with, with, with
coins on a piece of paper or get your

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friends and, that way you can jump over
one another and you, you draw these on the

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floor, okay?
All right.

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So but be careful.
Don't hurt yourself and don't try that at

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home.
Probably do it with coins.

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All right.
So lets look at its, at a, toads and frogs

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position.
Now here we have toad, toad, they move

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right, they're controlled by left.
Blank means an empty space.

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Frog, frog, frogs are controlled by right.
They move left.

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So, the only possible opening move for
left is to move to toad, is to move this

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toad one to the right and that leaves toad
blank toad frog, frog.

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And the only possible opening move for
right is to move this frog one left and

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that's leaves the position, toad, toad,
frog, blank, frog.

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Now, let's go over here and look at this
position.

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If left moves in this position, then this
toad moves to the right.

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And we're left with toad, toad, frog,
frog.

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That's a zero position.
Okay?

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Up and if right moves then right can jump
out take this frog.

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The only possible case is for right to
move this frog jump over the toad and left

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with toad, frog, toad blank frog.
Now let me remind you that when we're

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analyzing games we have to consider like
two left moves in a row and right moves in

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a row, because we want to analyze this
game in the context of a much larger game.

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So we might have this toads and frogs over
here.

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We might have a nim game someplace else,
cut cake someplace else, a game of chess

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going on, a game of Go going on.
And then each player in turn chooses one

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of these, one of these games and makes a
move in it.

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This is how we add gain.
So and in doing so in adding this game to,

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to others, left might make two, two moves
in a row in this, okay?

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So, I'm going to leave out some, some
pretty long, I think, computation, but

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this game down here.
Toad, frog, toad, blank, frog is actually

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equal to star so we have this game up
here.

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Toad, blank, toad, frog, frog.
Its left option, only left option is to

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move to zero only left move is to move to
zero.

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Only right move is to move to star and
therefore this game toad, blank, toad,

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frog, frog is up.
Up, you remember, was left 0, right star.

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Now, if you look at this game over here,
it's the same as the game over here, with

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toads and frogs interchanged and left and
right interchanged.

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Just, Just take the mirror image and
change all the toads to frogs and all the

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togs, frogs to toads.
Therefore this gain on the right is the

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negative of the gain over here.
So this gain on the right is minus up

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which is sometimes written down.
So this whole gain is equal to up down.

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And that's what I want to look at in this
module.

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Okay.
So, there are no dominated moves.

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Because there's only 1 move possible for
each player.

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But down here.
If up is 0, star, then down is its

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negative which is star, 0.
Remember that the negative of star is star

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and the negative of 0 is 0.
And, so if right moves to down.

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Left can move,[SOUND].
Left can move to star.

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And my claim is, star is better.
For left than the original game.

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That is to say, g is less than or equal to
star.

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Which is the same thing as adding star to
both sides.

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G plus star is less than or equal to 0.
Which says, that right wins this game

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going second.
Okay, who will bit the[UNKNOWN] there?

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So, but what this means is that if right
moved down, left moves immediately to up.

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I'm sorry.
Right moves to down, left moves

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immediately to star.
And now since star is 0, 0, the right

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possibilities for to, to move From star
are only 0, so this reverses all the way

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down to just 0 which means that if right
plays down left and mutely[UNKNOWN] plays

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star and then rights only option is, is
down[SOUND].

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So this says that g is actually equal to
up zero.

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And this game simplifies a little bit.
But that's part of the exercises,

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homework, quiz.
Whatever you want to call it, for this

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week.
Alright.

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Now, let me make some general comments
about this.

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Simplifying games.
We, we came up with 2 ways of simplifying

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games.
Dominated moves.

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We can delete dominated moves.
If a move is better for left than another

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move, then you can, you never have to do
the poor move.

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If, if a move is better for right.
You never have to use the move that it's

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better than.
And reversible move and that's a little

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complicated but we have some examples and
that's probably technically the most

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complicated thing in the whole course.
Now, the question is, is this enough?

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That is are there other possible ways of
simplifying games?

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Of course there are.
But what one can show is that this is

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enough in the sense that if you keep doing
dominated moves and reversible moves and

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keep doing them over and over again,
you'll reach a point where no moves are

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dominated, no moves are reversible and
what's left is called the canonical form,

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and if, that's unique and there's so, and
so in some sense some technical sense, in

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fact that one can make very precise, the
simplest possible version of that game.

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So all that you really need is this.
Now in terms, and that's how some computer

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programs, and you can look up which ones
that are available open source up on the

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web actually simplified games And, they
keep doing dominated moves and reversible

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moves over and over again until there
aren't any.

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And then when its left, they say, this is
as simple as possible, and that's what it

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is.
Okay.

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Let me just leave you with this which will
be posted in, in the assignments, et

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cetera.
And the directions this week are simplify.

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So, write down the simplest form of, of
each of these games.

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Number 1 is numbers over here and numbers
over here.

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2 and 3 are turn out to be numbers, I
think.

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But, you try it yourself.
Number 4 was what we ended up with in the

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exact last answer.
We took up down and simplified to this.

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And now we want to simplify this further.
The last one is the toads and frogs

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position.
Remember, frogs move right.

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Toads move left.
Take this and see if you can come up with

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the simplest possible description of that
and we'll see you all in a week or two.

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Take care.
Next week I think is NIM.

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We finally will, will analyze NIM in
general and answer the question, with the

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arbitrary mid game how do you play and
when?

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Sp, we'll see that all next time.
Take care.