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>> Welcome to Week 2, games without, oops,
what have we got here?

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Dice.
No dice.

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Cards.
No cards.

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No cards.
No cards.

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No dice.
No win moves at all.

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Games without dice or cards.
Combinatorial game theory, I'm Tom Morley

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again.
Welcome.

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So, in the study of these simple games,
what we need in order to look at examples

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by hand, is a whole bunch of games.
We have a couple that we've look at

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already, and we'll look at again this
week.

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But I want to introduce you to a new game,
new game.

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It's called Cutcake.
Okay, in this game, like all the other

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games that we've looked at, the players
are left and right, they move alternately

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left moves, then right moves, then left
moves, then right moves.

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And the first player, who has no move at
all, loses.

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So, cutcake is played on squares like
this, that's 2 by 2.

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You could start off with a 3 by 3, a 10 by
10, a 6 by 4, whatever you like.

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And let me show you the various moves that
are possible just in this example, on a 2

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by 2.
Left always cuts up and down, right always

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cuts left to right.
We're actually looking at a couple other

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cut games before we finish the course and
we'll always use this convention, left

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cuts, cuts up and down, right cuts left to
right, okay?

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So, if left goes for first here, left is,
only move, possible move, is left to cut

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the center seam here.
Oops, left, left cuts up and down, I've

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got that backwards.
Okay, so left goes up and down, like this.

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That leaves two pieces like this.
Now, what is right's movement here?

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Right doesn't get to cut in both of them,
right has to pick one or the other and cut

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in it.
We see that they're approximately the

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same, no matter which piece right picks.
If right picks, say, the first one, right

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cuts across like that, that's the way
right cuts.

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That leaves two pieces here.
In both of these pieces, no cuts left are

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possible.
They've been cut into small squares.

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There's this piece left here, which right
has a cut in.

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But remember, it's left's turn next, and
left has no move.

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Left loses.
So in, in this cutcake, that's 2 by 2, if

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left goes first, left loses.
You might want to look and see what

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happens if right goes first.
Try it out.

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Try it out for yourself.
Let's remember a little bit about

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Hackenbush.
And let me actually extend the game

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slightly.
Hackenbush you played on we have this

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ground here which is green.
The green grass on the ground.

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And growing out of the ground is red,
blue, green plant of some sort, okay?

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The, the edges are red and right can cut
the red edges, and the edges are, some

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edges are blue, and blue edges can be cut
by left.

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So, right for red, right has an r in it.
Blue for left.

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Blue has an l in it.
And the green edges can be cut by both

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players.
Now, this is, these green edges are new

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from last time.
And I'll try to point this out.

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On some monitors, this, this looks very
similar.

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The blue and the green look very similar.
I'll try to point when there's some

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confusion.
Okay, let's look at this, this example

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down here we have this whole plant here.
Suppose it's, rights move and right cuts,

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by pencil mark, this right edge, this
right edge then disappears and then,

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everything that's not hooked up to ground
also disappears.

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So, once this, this, this red edge here is
disconnected, this whole piece of the

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plant here will die because it's not
connected to the ground.

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So, we erase all of that.
So, players alternate right cuts right

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edges or green edges.
Left cuts blue edges or green edges.

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Each time you cut an edge, that edge
disappears together with everything above

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it.
And the first player, who can't move,

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loses.
Alright.

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Now, it's fun to play one game at a time,
but it's even more fun to game, play five

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or six games all at once.
So, I don't know let, let's play over

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here, let's play a game of chess.
Over here, let's play a, a game of go.

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Over here, let's play I don't know, a
cutcake of some sort.

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Let's, let's, let's, let's play a
hackenbush.

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Suppose some of these edges are green and
some are red, I'm not going to draw it.

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I just mean this to be a schematic.
Over here, we have a nim-heap of some

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size.
And let's play all of these at once.

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Now, what do I mean by playing all of
these at once?

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Well, the players now alternate.
But instead of playing one game or the

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other, you pick one game out of the five,
and they can move in that game, and that's

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your, that's your turn.
The other player, if you're left, say you

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do that first, the other player, say,
right, picks one game out of the five,

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makes a move or not, okay?
We know the following, even though the,

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the players go left, right, left, right,
left, right, left, right, in each of the

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individual games, the players may not, the
plays in that, inside that individual game

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may not be left, right, left, right,
because left might make a move in here and

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then right moves over here and then left
moves again over here.

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So, in the individual games, we may not
have left, right, left, right, left, right

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alternation but in this, this combination
game of playing these games all at once,

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we do have left, right, left, right
alternation.

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Now, if these individual games are called
G, H, K, L, and M, then, then playing

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these games at once simultaneously, like
we just went through is called the sum of

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the games and is written just like
ordinary sums.

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So, G plus H plus K plus L plus M,
represents the game.

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We have these, these 5 games, I guess.
Let's see, 1, 2, 3, 4, 5, I'm not very

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good at numbers.
But I think there's five games there.

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We have these five games and then each
player, left and right, in alternation

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picks one of these games and makes a move
in that one game.

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Now, the first player who can't move at
all, loses, just like we have in all our

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other games.
So you can't move after you're checkmated.

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Go, you can't move after you lose.
Cutcake, we know the rules for cutcake.

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Hackenbush, we know the rules for
hackenbush.

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We have a pile of, of 4 coins here, so
whoever move, once the coins are gone,

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then there's no moves there.
So, once there's no moves left in any of

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these games, then there's no moves left
for that player, and that player loses.

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So, let's, let's maybe look at a an
example, and try to play through part of

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the game.
So here, we have a cutcake and two

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hackenbush games.
And we've been looking at left first for a

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while.
So, let's do right first.

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So, suppose right moves first by cutting
this in 2.

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Therefore, what's left there on the table
is, is one of these and another one of

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these.
And now, it's, in some sense this is now 4

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games.
This game, this game, this game, and this

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game.
Now, it's left's move.

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Left I don't know, what does, what does
left want to do?

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Let's say, cuts one of these off.
And so, this now becomes, and now, we have

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this game, this game, this game, this
game, and this game.

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Right has a move.
Right, maybe, cuts this off that gets rid

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of the blue also.
Every, this, this all goes up in the air.

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There's no moves left in that game.
Left's move again.

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Oh, I don't know, left might cut off
another square down here.

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And we can continue on left, right, left,
right, left, right.

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And I think in a move or 2, you'll see
that left can win this game.

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So, so, try that out for yourself.