OK, guys, so quickly go over this question, once posed the video, read the question, resolving
it and then we'll discuss it.
But there are two boxes in each one box contain some chips, other content and chips.
The deposition is generated by Ankomah envir and is greater than zero and is greater than zero.
The two players are alternate moving a move consist of empting one of the boxes and dividing the contents
of the other between the two boxes, with at least one chip in a two way, at least one chip.
Because these conditions should be.
It is.
But there is a unique position they believe in won the last Florida movement eight.
So let's discuss this problem initially was given is that one common one is a losing position, it's
a dominant position, right?
So dominant positions are losing position.
What can we do in one move we can to players alternate moving, one move consist of empting one of the
boxes and dividing the contents of the other between the two boxes with at least one job in each box.
So let's think it from where?
What are all those places from which I can come to one comma one, which is a language because we know
about this, right?
So in this question, we cannot take zero.
In other Emerin aid, because it has to be greater than the U.S. mentioning the great thing about all
those files or all those positions in which to come and or it is of the form government to.
But what can happen in this case, that if I pick up this file with Tacoma and I'm attempting this and
then this can be converted to.
One common one only because if I am 10, it becomes Tucumán zero, and then I have to divide these two
into equal parts aid and there's only one way of dividing.
Two, that is one comma one.
So all the positions of this type two comma and or M comma too, can be converted to one common.
Eight.
I hope are understanding this, but basically, if any of the.
But there it is at the end or end is, too, if any of them is to, then simply it's a winning position
by because I can directly give the other player a losing position.
Well.
I hope you are understanding despite that, how can I directly give my losing position because let's
say the first part is true, then I'll empty the second part.
Divide this two into one, comma one.
And vice versa, although the second part.
OK, so let me reiterate that.
All piles of type and and a more mature winning position.
So let me mention one to one is a losing position, but one common Toisa winning position and to no
one is assuming winning position right now.
Let's talk about one comma, three.
So from one government report, all options do I have I cannot empty this one by saying I cannot make
three because one can not be divided, it won't be a legal move.
Right.
I can only empty one eight.
So if I am the one, it is the right thing to do.
But then what can be the options?
One common to all to go over it again, guys, we can see that both of these are winning positions.
That means one commentary will be a.
Losing position will it's one comfort from one for I can only empty one because again, the same reason
I cannot empty for if I empty one now, I will divide four into different parts.
So what can be the parts one commentary or it can be Tacoma two.
It can also be Triggermen one.
But I'm not writing it again because one commentary is equally into Triggermen one.
Well, I wouldn't be writing the symmetric part again and again.
Eight.
OK, so to OK to commit to is a winning position because we saw that all all these types of numbers
which are of this form and contains to on the anagram are winning positions.
And it's a very important and one commentary's a losing position against which position is one of for
it has to be a winning position because I can give I'm going to a losing position.
Eight one for the.
Beating Bush and similarly, let's talk about one game of five and then we'll generalize for one game
of five, what we have, we have little empting one, then five can be divided into one comma for one
comma three.
Sorry to go three or three omertà.
It's eight or two.
Three is a winning position, but one Gomaa four is also a winning.
So what is one Gomo five days.
It has to be a losing position.
They similarly let's talk about the second row.
OK, let me write it.
One comma six is again a winning position, one comma seven is a winning position, losing position,
you can crosscheck and verify it yourself because it's easy to verify this one, comma nine is a losing
position and so on.
So it is basically losing, winning, losing, winning, losing, winning and so on.
Right now, let's talk about this column to column.
What we have to come three.
It's a winning position.
So as discussed earlier, that anything which contains two will be a winning position.
So quickly, let me write it.
Winning, winning, winning, winning, and.
Right now, let's talk about this trigonometry.
Will go from trigonometry, I can empty any one of the trees, no issues, and the second tree will
be divided into one, two or two gunmen when they are the same things, basically from trigonometry.
I can go to one comma to type situation.
What is one going to do?
It is the winning position.
So it has to be losing.
Well, again, from three four, let's talk about this in detail.
So we have three Gomo four, and that I can immediately or I can empty for, let's say, empty three
and I'm waiting for this four can be divided into one, comma three.
It can be divided into.
UGMA to.
Eight, so one three is a losing position and to come a Florida.
Oh, sorry to come out with a winning position so I can go to one of the losing positions, I do not
have to consider this case because I've already got the losing position.
If I got a losing position, this has to be a winning position.
Similarly, guys, for three five, it will be a losing position.
Three Gomo six becomes a winning position.
Triggermen seven is a losing position and so on.
And now we'll talk about this photo op ed column for Gummo four, four, five, four, six.
Think about this first yourself and then I'll discuss this.
Right, so there's a pattern now that we can observe in this whole column will be a winning position.
Eight days apart and then we can see over here that firstly alternate columns have evidence, so all
the columns with even number of even number is a winning position.
Right.
And then for all the or no columns, one, three, five and so on, we have losing, winning, losing,
winning, losing, winning into eight.
Basically, if to generalize it multiple, what we can say is if bought and comma M Artaud.
Then it's a.
Losing position.
Eight, but if any one of them is even at their em or and is even then it's a winning position.
Ladies days, this was about the cushion to we have finally, finally realized what the losing of an
implosion in this game that is empty in here right now for any given state and comma m, you can easily
tell which is a winning position and which is a losing position.
Right.
So what is the winning position if and when live and more to close to zero or and more to equals to
zero.
OK, or see in the next video in which we'll be discussing some other game then or we'll move on to
the next section eight.
Thank you.