Hello again. 
Games Without Chance, Tom Morley. 
Let's look at atomic weights. 
[SOUND] Now. 
Let's look at our, one of our favorite 
games. 
A nin heap of size one. 
Not terribly exciting. 
No one's really, oh let's go play in them 
heaps of size one. 
Nobody's really that excited about it. 
But, this is a new symbol. 
That's the integral sign, but this is the 
way the sign is used in theory. 
And has nothing to do really with 
integrals. 
This is the heating of star by three. 
So instead of zero. 
You say zero plus three over here. 
Instead of zero over here, you say zero 
minus three. 
And this is a heated game, and this is a 
hot game, because left is, wants to play 
this, because then left gets three 
points, or three, and right wants to play 
this because then right gets three 
points, or minus three, to the left. 
So this is a hot game. 
So we started off with a really kind of 
boring game and created a hot game. 
Okay, now I resolve to Simon-Norton that 
any game is actually a number plus the 
sum of the heated version of infetesimal. 
Okay. 
Infinitesimal would be a game that, 
that's less than one over two to the add 
for every add and greater than minus one 
over two to the add for any add. 
So, so you can't know anything, you can't 
know all about games unless you know all 
about infinitesimals. 
And infinitesimals can be really 
complicated. 
So here's one way of analyzing at least a 
very large class of infinitesimals. 
the game is called all small. 
If whenever left has a move so does 
right, and whenever right has a move so 
does left and this is true for both the 
game itself for any position possible in 
play. 
So for instance the game, hmm, zero this 
is the game one I believe. 
this right, left has a move but right 
doesn't have a move so this not all 
small. 
So an all small is your, you can prove 
are infinitesimal. 
Now hmm, here's the result and this is 
computable and this is probably actually 
the most complicated. 
Most intricate or long proof in winning 
weights by one. 
If g is all small then then there is 
again capital G, computable from g in the 
various ways and they go, go through the 
ways its computable such that g times, 
capital G times up, so this is a multiple 
of up. 
we have to eventually say what that means 
so that g minus this multiple of up is 
pretty close to zero. 
It's caught between up plus star plus an 
unspecified nim heap and greater than or 
equal to down plus star plus an 
unspecified nim heap and from this. 
this another approximation result that an 
all small game is very nearly subject to 
this error a multiple of up. 
And this is, this is, this can be used 
to, to analyze the play of all small 
games. 
But enough of this theory, let's look 
again at divided fair shares and varied 
pairs. 
If you remember what we have for fair 
shares and varied pairs is that you can, 
it's here somewhere. 
you can take, take, take a coin, take a 
take a stack of coins and, and divide it 
into any number of equal stacks. 
Or you can take two stacks that are not 
equal and combine them. 
So this is, this is a fun party game. 
Let's start with a stack of three over 
here, a stack of three over here. 
A stack of two and a stack of two. 
And let me remind you, you can take any 
stack, divide it into any number of equal 
stacks. 
You can take any two unequal stacks and 
combine them. 
So[SOUND] here we have a stack of three, 
stack of three, stack of two, stack of 
two. 
Ten coins in total. 
Go ahead, it's your first move.