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'Kay.
Welcome back.

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Still week six and we're doing some more
examples and talking a little about

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minimal excluded.
Okay, so let's, let's take a look at this

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quickly.
We have some examples of minimal excluded

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and excuse me while I look around the
camera.

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Okay.
So let's see.

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Zero is here, one is here, two is here,
three is here, four is here, five is here,

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six is here, seven is here, eight.
8 isn't.

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So the minimal excluded of this is 8.
And so that's the solution from last time.

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Alright, here's the general result.
If we have an impartial gain, all of whose

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moves are to nim heaps.
And this game is equivalent to a nim heap

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of size d, where d is the minimal excluded
natural number in the set consisting of

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this a, b, c, a, a, b, dot, dot, dot, up
to c.

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So the example from last time fits this
context here we have this.

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Here's a game, all of the options are nim
heaps and 3 is the first missing number.

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And so this game is equivalent to nim heap
of size 3.

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Okay.
Let's take a look at, and Any, any of

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these things we can think of as, as min
heaps, where we can add more coins, but

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you can reverse those moves by putting the
coins in your pocket.

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Alright.
Let's take a look at a, a, a, at a case

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yet another kind of case, another kind of
game, it's called a subtraction game.

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So we start with a heap of size n, n may
be 1, 2, 3 or 17 or 40 billion.

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A move for either player is to remove one,
two or five Coins, or beans or dice,

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whatever they are or bicycles, we could
have a heap of bicycles that would be fun

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to play.
Okay, so let's take a look at that.

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If we have, start off with a hepa size 0,
then there's no moves so that's equivalent

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to of size zero.
If we start off with an in heap of size 1,

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I'm sorry.
We start off with a heap of size 1 and

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play the subtraction game, 1, 2, or 5.
Then our only move is to a nim, to a heap

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of size 0.
That's 0.

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And the mex of zero all by itself is 1 so
that's a 1.

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Now, with a heap of size 2, we can remove
1 coin, giving us a heap of size 1.

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Equivalent to a hepa nim heap of size 1 or
we could remove 0 coins.

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Giving us equivalent to an in equal sign
0.

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The mex of 0 and 1 is 2.
For three coins, we can remove 1 or 2, so

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we can replace this by, a heap of size 1
or 2.

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Which is equivalent to a nim heap of size
1 or 2, and the mex of 1 and 2 is 0 for a

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heap of size 4.
We can remove 1 giving, an equivalent to a

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nim heap of size 0 or, 2, giving an
equivalent of nim heap of size 2.

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The max of 0 and 2 is 1.
For 5 we can, we move 0, 1 over 5, ooh,

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there we go.
Alright I feel like I'm playing the piano

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or something.
Alright and, and so the mex of zero, 0 and

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1 is 2.
For 6 here, this is for you to figure out.

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An we'll start next time with the, the
next module with this.

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So, I can, it goes back to removing one,
removing two or removing three so what do

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you think that should be?
Okay.

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Not too hard.
So, what we can do You say the file, in

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general, we can have subtraction gain
where we a, b, dot, dot, dot or c coins.

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First player with no moves loses.
And g of n is d this, the, the size of a

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nim heap equivalent to this game.
And this can be computed In exactly, this

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way and I think we're done.
It only works if you make a noise, take

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care.