This is a game that deserves some close study, spending some time on,

and you will, if you study closely be able to beat everyone you play unless they have invested enough time

to learn what you have learned. It is simply the very same game, but we play it on

three boards instead.

The rules here are that a player on his move, his or her move, can put an X on any of the boards.

You can choose whichever board you like.

As players put X's on the boards eventually one of the boards will get three in a row.

And then that board - but only that board - is declared dead at that point.

You can't make anymore moves then, on that particular board, but you play on, on the remaining two.

Again, on one of the two remaining boards, once three in a row is made, that board dies,

it is no longer available for play. And you are left with whatever configuration of Xs may be on the third board,

and you complete that board.

Then whoever makes three in a row on the final board loses the game.

Okay, this game is difficult to master.

But it can be mastered.

Starting from this position

you do want to move first

now in practice

if you learn how to play this game

you can even let your opponent move first because you usually know enough you'll see a mistake they'll make and you'll be able to overcome that.

But if you're playing someone who happens to be really good then you do want to make sure that you move first.

Then there's another strange twist to this game that on three boards it actually doesn't even matter

what move you make. All the first moves are winning.

You can move where ever you want and it's still a winning position for you.

So you can pretend to be very magnanimous and say,

"Brady, absolutely, please just choose any cell. I will accept that as my move."

You don't say "winning move."

You say, "I will accept that as my move and then we can play the game.

So this is a nice trick you can play just when you're getting started.

Then in general you remember that on one board it was the first player who wanted to go

and on three boards it's also the first person. If you were to play this same game, I'm not recommending it, but if you were to play it on five you would also want to move first.

And on boards that if you have two, four, six boards you don't want to move first.

You want to move second.

Now, I have to be completely honest with you, in order to play well and to

be absolutely certain no one will ever beat you

you are going to have to know certain configurations of X patterns on the board and have them memorized.

And it's not just the pattern you need to memorize but you'll need to

memorize a particular

variable or symbol or fingerprint for each one.

Then what you need to do is multiply those symbols together which are just sort of like algebra things that you learn about in school

and then reach a certain safe configuration value.

That's basically all there is to it. The rest of it is just memorization.

This is small so I need to put on my glasses.

It is daunting but it's not quite so daunting as it looks.

This is a list of all the ways you can put up to nine Xs on a tic-tac-toe board.

It turns out there are exactly one hundred and two such ways.

No one turn off the video, okay?

There are one hundred and two of these but there are a lot of them that are very easy to remember.

For example if you look at the ones at the bottom here,

you'll see there are ones here that are all given the number 1 or the identity.

We're going to be multiplying these things together so

the ones that already have completed rows of Xs, they're already dead boards.

And so those are like the identity.

When we're multiplying things together, if a board is already dead it doesn't matter so it's just the number one.

You don't need to memorize dead boards because you look at them and it already has an X through it, you don't have to remember those.

So the only ones you need to focus on are the ones where there's still play available.

And for each of those it's going to be a symbol.

And the symbol is just going to be like a single algebraic term, like it might be an 'a', it might be a 'b squared'

it might be a 'bc', it might be a 'd squared'

or so on. And that's all you have to remember.

Plus a couple things having to do with reducing words and simplifying them.

Let me just show you some typical fingerprints.

Okay, one of them is say, just an X in the corner.

That happens to be the identity or the number one.

It's a fingerprint which does nothing.

It's the same as a dead board. If there's only one thing in the corner you can ignore it.

Another dead board is the one where the X is here. This is also a one.

All right. We might be systematic here so what is the last one, Brady? What would the last one be?

-An X in the middle.

It would be an X in the middle, exactly.

So if we have this one, that one is slightly different.

This I am going to call c squared.

There are certain positions that come up a lot more than others. When you look at the big table, you'll say,

"uh, I can't do this. This is too hard."

Well, no, it's not too hard. You just have to play more and just kind of get used to it.

Okay so let's take some typical ones that come up all the time.

One of them is this one.

This comes up a lot.

And this happens to be b squared.

Okay, that's a good one.

One of them is when you have two Xs on the corners, opposite corners like this, then 'a'.

That comes up a lot.

The one that has ones like this, and that one also happens to be an 'a'.

Okay so let's say you go to the trouble of learning all these fingerprints, how is that going to help you?

Well, let's see. Let's start a hypothetical game. Let's say that I've made this first move X.

I look at this position and I like it.

Why do I like it? Because

this happens to be a one.

This one happens to be a 'c'

and this one happens to be a 'c'.

and when I multiply these two together

in my head I get c squared.

And c squared happens to be one of the magical

polynomials, or just monomials that I want to leave in every position to make sure I win.

c squared. So we have to remember that I like c squared.

So what are the other ones? There's c squared,

there's also the little letter a

there's b squared, and there's b times c.

And that's it. Those are the four you need to remember.

Those four ones, if you make a move so that the fingerprints multiply to one of those possibilities

then there's no way for your opponent to regain control; you will still win the game.

So at each stage you are always trying to find a move

that equals an a, a 'bc', a 'b squared', or a 'c squared'.

Well, let's say that our opponent decides, they are free to of course to move on any of these free boards,

but let's say that my opponent decides to move on the same one here and moves here.

Okay, what's a good reply here?

Well, there's actually quite a nice reply.

Remember c squared was itself a good move

and if I kill this board by moving here then it's completely out of play and I'm only going to be left with these two.

So that's going to be a c squared too. Boom!

I do this. I'm still under control and I've killed one board also so we're getting closer to winning also.

When you're playing this game you tend to look first at the options that involve killing boards

Right, it's better. Think of it like an samurai

it's better to kill your opponent sooner rather than later because if you let them linger

they may rise up and all of a sudden surprise you.

I always look first, is there a good move that involves killing something that's already on the board.

So now let's say my opponent plays to shall we say let's play here.

Okay, so that one, if you remember, that's a one now so I cross out the c and I make this a one.

Now we multiply these two together. It's no longer c squared, it is now multiplying to c.

-That's bad.

That is bad. That is not one of the -

I mean it's good for me because I need

if it were a c squared that would be bad because I'd have no winning move, right. Remember that's what I'm trying to do to someone else.

But if it's a c, I know that there is a winning move, I just need to figure out what is the winning move.

Another one that I've happened to memorize and it's a very simple one is just that

if you have two like this, one in the centre and one on the edge connected

it can be on the diagonal or oriented however you like

that's always a 'b'.

K, so this is a 'b'.

That's a 'b'.

And of course all the reflections and rotations and so on, these are all 'b's.

All right, so that's what I'm going to do here I'm going to change this

away from a one. It's no longer a one. It's now a 'b'.

And now it's b times c and that's one of the magic ones.

So now I'm no longer at b, I'm at bc and I'm still happy. I'm still winning.

If I left my opponent facing a losing position at bc, now let's say, "what do they do?"

Well one thing they could do is, you know, I killed a board, maybe they're dumb enough to, you know, kill a board too. So let's say they kill this one.

Boom!

Okay, so now this is now, what? It's the identity, it's dead.

So it's just the number one. Right, remember from algebra, multiply by one has no effect. So all the dead boards, they're always one.

Now you've got a c. Well we've already talked about this, haven't we? What is the winning move when we have one board?

Bryan, what is the winning move? The winning move is not to move on the edge, remember? It's to move in the middle. So we do this.

Okay, so now we still have to win this.

But we know how to do that too.

We just do the knight's move strategy.

Okay, so my opponent

my opponent might move here

I move a knight's move away here

now my opponent maybe they move here

I move a knight's move away there

and now my opponent

is dead

and I have won, demonstrating

my mastery

that I hope all the numberphile video people will also become my rivals although I don't look forward to it, of tic-tac-toe on three boards in misère play. So good luck to everybody.

This video, and all Numberphile videos you watch are supported by the Mathematical Sciences Research Institute here in Berkeley, California

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