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Hello again. 
Games Without Chance, Tom Morley. 

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Let's look at atomic weights. 
[SOUND] Now. 

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Let's look at our, one of our favorite 
games. 

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A nin heap of size one. 
Not terribly exciting. 

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No one's really, oh let's go play in them 
heaps of size one. 

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Nobody's really that excited about it. 
But, this is a new symbol. 

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That's the integral sign, but this is the 
way the sign is used in theory. 

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And has nothing to do really with 
integrals. 

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This is the heating of star by three. 
So instead of zero. 

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You say zero plus three over here. 
Instead of zero over here, you say zero 

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minus three. 
And this is a heated game, and this is a 

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hot game, because left is, wants to play 
this, because then left gets three 

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points, or three, and right wants to play 
this because then right gets three 

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points, or minus three, to the left. 
So this is a hot game. 

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So we started off with a really kind of 
boring game and created a hot game. 

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Okay, now I resolve to Simon-Norton that 
any game is actually a number plus the 

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sum of the heated version of infetesimal. 
Okay. 

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Infinitesimal would be a game that, 
that's less than one over two to the add 

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for every add and greater than minus one 
over two to the add for any add. 

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So, so you can't know anything, you can't 
know all about games unless you know all 

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about infinitesimals. 
And infinitesimals can be really 

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complicated. 
So here's one way of analyzing at least a 

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very large class of infinitesimals. 
the game is called all small. 

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If whenever left has a move so does 
right, and whenever right has a move so 

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does left and this is true for both the 
game itself for any position possible in 

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play. 
So for instance the game, hmm, zero this 

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is the game one I believe. 
this right, left has a move but right 

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doesn't have a move so this not all 
small. 

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So an all small is your, you can prove 
are infinitesimal. 

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Now hmm, here's the result and this is 
computable and this is probably actually 

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the most complicated. 
Most intricate or long proof in winning 

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weights by one. 
If g is all small then then there is 

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again capital G, computable from g in the 
various ways and they go, go through the 

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ways its computable such that g times, 
capital G times up, so this is a multiple 

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of up. 
we have to eventually say what that means 

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so that g minus this multiple of up is 
pretty close to zero. 

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It's caught between up plus star plus an 
unspecified nim heap and greater than or 

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equal to down plus star plus an 
unspecified nim heap and from this. 

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this another approximation result that an 
all small game is very nearly subject to 

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this error a multiple of up. 
And this is, this is, this can be used 

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to, to analyze the play of all small 
games. 

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But enough of this theory, let's look 
again at divided fair shares and varied 

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pairs. 
If you remember what we have for fair 

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shares and varied pairs is that you can, 
it's here somewhere. 

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you can take, take, take a coin, take a 
take a stack of coins and, and divide it 

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into any number of equal stacks. 
Or you can take two stacks that are not 

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equal and combine them. 
So this is, this is a fun party game. 

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Let's start with a stack of three over 
here, a stack of three over here. 

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A stack of two and a stack of two. 
And let me remind you, you can take any 

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stack, divide it into any number of equal 
stacks. 

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You can take any two unequal stacks and 
combine them. 

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So[SOUND] here we have a stack of three, 
stack of three, stack of two, stack of 

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two. 
Ten coins in total. 

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Go ahead, it's your first move.