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.