1
00:00:00,012 --> 00:00:06,392
So, here's an outline of, of how to show
that 1 over 2 to the n, minus up is

2
00:00:06,392 --> 00:00:10,971
positive.
Let's just look at the left going first

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00:00:10,971 --> 00:00:15,037
move.
If left moves first, I'd claim that the

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00:00:15,037 --> 00:00:18,832
best move for left is to play to this
star.

5
00:00:18,832 --> 00:00:22,554
What's left is then 1 over 2 to the n plus
star.

6
00:00:22,554 --> 00:00:25,856
Now, now it's right's turn, what can right
do?

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00:00:25,856 --> 00:00:31,485
Right can, can move from star to zero.
Remember that the two, the only option in

8
00:00:31,485 --> 00:00:35,658
star is for left to move to zero or for
right to move to zero.

9
00:00:35,658 --> 00:00:41,449
In which case, what's left is 1 over 2 to
the n, left just chops the blue edge, and

10
00:00:41,449 --> 00:00:44,688
now there's nothing left, and so left
wins.

11
00:00:44,688 --> 00:00:50,389
The other possibility is for right to cut
off one of these red edges, and that just

12
00:00:50,389 --> 00:00:54,263
delays things.
So, right cuts off one of those edge,

13
00:00:54,263 --> 00:01:00,906
edges the left then moves from star to 0,
and what's left is right cuts off another

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00:01:00,906 --> 00:01:05,444
red edge, blue cuts off the blue edge, and
now left wins.

15
00:01:05,444 --> 00:01:09,174
So, so that's the argument when left goes
first.

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00:01:09,174 --> 00:01:15,525
When left goes second, there's a little
bit more stuff to do, but the argument is

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00:01:15,525 --> 00:01:19,826
very similar.
And again, if left goes second, left wins.

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00:01:19,826 --> 00:01:26,408
Put this all together and that says that 1
over 2 to the n minus star is positive.

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00:01:26,408 --> 00:01:33,181
Which says, with what we did previously,
that up is bigger than zero and less than

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00:01:33,181 --> 00:01:38,281
1 over 2 to the n for any n.
And there's no number that does that.

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00:01:38,281 --> 00:01:41,226
So, up is not a number.
Okay.

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00:01:41,226 --> 00:01:49,134
That ends week 3.
I'll post a little quiz and I'll, I'll

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00:01:49,134 --> 00:01:55,291
talk through it for the last part of week
3.

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00:01:55,291 --> 00:02:00,027
Okay.
So, the quiz for week 3 is, I have a whole

25
00:02:00,027 --> 00:02:05,671
bunch of games.
You have to decide in each game whether

26
00:02:05,671 --> 00:02:11,437
it's positive, negative, equal to zero or
fuzzy with zero.

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00:02:11,437 --> 00:02:15,603
It can't be two of these, it has to be
exactly one of these.

28
00:02:15,603 --> 00:02:21,290
The first game is these hack and bush with
three stems like this.

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00:02:21,290 --> 00:02:27,469
And second one, hack and bush with, with
one green stem and one blue stem.

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00:02:27,469 --> 00:02:32,653
Game number three is 2 copies of one green
stem and one red stem.

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00:02:32,653 --> 00:02:39,391
And number four is a little more abstract,
go back and look at the definitions of up

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00:02:39,391 --> 00:02:46,024
and star and figure out whether up plus
star is positive, negative, zero or fuzzy

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00:02:46,024 --> 00:02:53,304
with zero.
Work this out and we'll post the solutions

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00:02:53,305 --> 00:02:57,853
by the time you see this.
Take care.