We're gonna talk about the game of Amazons. It was invented in Argentina
by a chap named Walter Zamkauskas, I believe in 1980's. So it's a fairly new game.
For purposes of this video, I think we should use
only a 6x6 board because this makes the game faster. This is a good game for kids at that size.
The official size preferred by Zamkauskas is 10x10,
which is also the size of a checkerboard in Argentina. And you can also play an 8x8 or, you could play any size
or could even be a rectangular board.
Well, there's two black pieces and two white pieces and there are various possible starting configurations.
This is one that I like.
A legal move is - let's just take one piece here.
A legal move is - by an amazon, if you choose to move this amazon,
it can go as far as it wants in any direction either diagonal or horizontal or vertical,
like a chess queen.
Now after it moves it must shoot an arrow and the arrow also goes like a chess queen.
The same Amazon that moved must also shoot so maybe it shoots like that, okay.
Now the square in which it lands, this is like a flaming arrow,
it is burned off the board and that square cannot be occupied nor can any other Amazon jump over it.
Or...
capture it. There's no capturing, all the Amazons are immortal.
They run about shooting flaming arrows, and eventually the board gets blocked up with flaming arrows, and
somebody can't move.
And when that happens, because they're unable to move then the game's over. And the player unable to move is the player who loses.
(Brady: Or the player who did the last legal move is the winner?)
That's right.
(Brady: Okay)
Yes
(Brady: Sounds fun)
Okay, well, it is. Brady's gonna play black and I'm gonna play white and we're gonna play a game.
Well, let's say you can go first. It is believed there is an advantage to going first but no one really knows.
(Brady: I'm gonna go, so I'll go... there.)
(And then so I could shoot my arrow, for example, to there?)
Sure.
That's okay. All right I'll go here. Aaand shoot there.
(Brady: I'm gonna go to...)
(here.)
(Shoot my arrow to there.)
Okay
So I'll go here and shoot there.
(Brady: I'm feeling that black player getting a bit penned in there. So for example, I might, I might move to here.)
(And then try and start penning in your white, your white amazon there)
Yep, that's highly plausible. So I'll go here and shoot...
there, okay.
(Brady: I'll go here)
(And shoot to there. Was that a good or a bad move?)
Uhh...hard for me to tell.
I...think I better - Let's go here and shoot here.
(Brady: Okay, my black player is getting quite pinned in there, isn't he?)
Yeah
(I feel like I need to get that guy out of trouble.)
(Because if - because I could get boxed in here quite easily couldn't I)
True.
(So I'm gonna--I'm gonna come down to here...)
(...and then...)
(...fire my arrow here.)
Well that works to some extent
(Are you feeling good about your situation here?)
Yeah, I feel pretty good.
I'll go here...
So now the board is split into two games. There's one over here,
in which I'm way ahead cause I have a lot of space and
you have only one.
And there's another game over here. This divides the board into two pieces
(Brady: Is this common in, in, in games for games to get split into sub games like this?)
Certainly in the kind of combinatorial games that we study, yes.
Because that's a big part of how we manage to do an analysis.
Because in the mathematics we can do an analysis of this side of the board,
the result is not gonna be too complicated because this is fairly simple.
This over here is still fairly complicated. But we can do an analysis over here.
We do another analysis over here.
And then we have ways of adding these two analyses together to get an analysis of the whole game.
(Brady: Is there a name for this, this, this technique?)
Divide and conquer
(It feels like I'm I'm pretty dead over this side now right?)
Over here there's a lot more at stake.
Yeah, there's some squares back here you will probably get.
And there's some squares down here that more likely I will get.
There's some of the middle that we'll be fighting over.
So, okay presumably these two will be fighting for a while.
(Brady: It's my move at the minute though isn't it?)
Yes
(I'm gonna go here)
(and fire here.)
Okay, so I think I will just go here and shoot here.
So now I have one, two, three, four, five. You have one, two, three, four, five, six, seven.
Okay, unless I can move again which case I'm gonna come up here and argue with you.
But at the moment you have two moves more than I do here, probably,
roughly that.
And over here I have one, two, three, four, five, six.
Six or seven moves ahead of you.
(Brady: I'll go...here)
(and fire here.)
So, now, this is sort of interesting. If I come in after you, I,
you may cut me off.
So, I'm not sure what the best plan is here, I probably will do that anyway I'll come in and...
...yeah...I'm gonna be okay here too.
(Brady: The best I can do is - ) - You can, you can...
Well then I'll just kill you. And still get three points extra.
But if you don't I'll back up and get all these
(Yeah.)
(I'll go, I'll go here and fire...)
(here.)
So then I come here and shoot here.
(Okay, so I only have one possible move which is to go here and fire here.)
All right
And now I start using up my moves. So go here.
(And now I can't move -- I can't move that, I can't move that. And I can't move that.)
And I still have one, two, three, four, five, six, seven, eight, nine.
(Brady: Even if you had no more moves, you've won the game there)
That's right. That's right. But in some sense we say I've won by nine points.
There's only, there's only one game against a beginner I've ever lost out of somewhere like 40 now.
(Brady: You should have told me that before I would have, I would have tried harder if I knew the stakes were so high)
Jujo Jiang, who is a famous 9 dan go player, beat me the first Game of Amazons we played.
Although I challenged him to rematch and I won the second.
There are a number of levels at which, uh, it helps teach mathematics.
For very small children, the beginners, just observing that we have, we have addition here.
We have one two, three, four, five six seven moves for white here
and two moves here. So he wins by nine points.
Let's, let's, let's assume this end had some different way. Something like that, for instance.
Suppose, suppose the game it ended like this. See?
(Brady: Ooh, I would have preferred that.)
Well black has 1, 2, 3, 4, 5, 6, 7 moves,
white has 1, 2, 3, 4, 5, 6, 7 moves.
But white has another one so white is ahead by one point.
White is one move ahead. So he will win even if he has to go first.
Now if after this for instance--
it's now 7 vs. 7 and the mover will lose because this adds up to zero.
So in general we say a game is worth zero when the mover cannot win.
So a zero game is one which is a win for the second player.
In the original game, as prescribed by a Zamkauskas,
is on a 10x10 board and each side starts with 4 Amazons.
On an 8x8, we usually have 3 against 3,
and on a 6x6 it's usually two against two.
When, when we have an experienced player playing against the beginner, as I just did against Brady,
it'd be a more interesting game if I gave him three Amazon's against my two.
(Brady: So that's - sometimes people play that way, they handicap them?)
Sure, sure. That's one way to handicap.
The other is just: find a starting position in which one side's already more constrained than the other.
And so you can start from positions that are biased in one direction or the other.
Well, yeah, I think it's an interesting game. I, I think the opening's are -
I don't understand 'em. I don't think anybody does.
It's a game that is played. There have been international tournaments.
Back in the early days of Amazons they were played by fax and now I'm sure everything's computerised.
There are people do research in it, yes. There are,
Martin Muller at the University of Alberta has done a fair amount of work, as have some other people in,
in trying to evaluate sizes of various positions and
I think by now it's sort of known how to play on very small boards, maybe 4x4.
I don't think 6x6 is really understood yet.
The board has no direction. I mean all, all directions are equally good.
In that sense it's more like go than like chess or checkers where
there is a forward component built into the game. This game has no forward component.
Zamkauskas' starting position has the four blacks on one side and lined up against the four on the other.
Which I think is a little distracting for beginners because you - it sort of conveys this notion of
forward motion, coming at each other
but after a few moves each side gets around behind the other usually and it's, it's, it's up for grabs.
There is no - the board really has no direction.
[Preview] The region here is zero/zero because there's no Amazons,
but it turns out each of these positions
is worth a half a point for a black. Run a chain through here
and the only way you're gonna be able to stop it is to sacrifice a box.
Don't don't give away any sacrifice until you have to. Stop the side chains right away.