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Welcome back.
Tom Morley, Games without Chance.

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Actually that's not so bad.
There's some game you play with these.

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I, I, it, it has chance in it, so we won't
use it.

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We're looking now at examples from Nim and
let's look at it.

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Now, Nim is an example of an impartial
game, so what we're going to do is

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abbreviate impartial games like this.
So what this means is the left player has

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the option of moving to a Nim heap size 0,
1, 3, or 7 and the right player has

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exactly the same options.
The right player can, can move to a Nim

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heap size 0, size 1, size 3, or size 7.
Now, my claim is, so we neither leave 0 on

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the table, leave 1 on the table, leave 3
on the table, or leave, let's see how many

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more do we need.
Leave 7 on the table and then the other

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player can do so why is the, okay oh, all
at once kind of proof like there in the

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camera.
So, so that's the game.

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Now this is not by itself, not an
interesting game, but you can be part of a

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larger, a larger Nim game with other
stacks of coins on the table.

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Now, let's analyze this, in terms of
analyzing this, my claim is, this is a

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disguised version of a Nim heap of size 2.
Why?

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Let's take a look what you can do with a
nim heap of size 2.

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With a nim heap of size 2, you can leave
no coins on the table.

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That's star 0 or you can remove one coin,
and leave one coin on the table and that's

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star 1.
So that's, that's star 2.

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Now, this up here, these two moves don't
do you any good, because they reverse.

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If, if you put down three coins, another
player removes one, we're left back with

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star 2.
If you put down 7 coins instead, one, two,

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three, four, five, six, seven, then the
other player removes 5 is that right?

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Is that how you do arithmetic?
Check it with your calculators.

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Okay, and then you're left with two.
So these two moves are reversible.

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If you ever make those two moves, the
other player can reverse, strike back to

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star 2, so this impartial game is actually
star 2.

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It's actually the same as an n heap of
size 2.

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Try one for yourself.
How, what about this one, star 1, star 0,

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star 2, star 101.
Try that, see what you can get.

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You should conclude that this is the same
as an n heap of size three.

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Short module this time.
You all come back now you hear.