Name: Secrets of 3-Board Tic-Tac-Toe - Numberphile
Uploaded: 2020-03-30T04:25:19.000Z
Duration: 10 min 54 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

To-infinitive

Here's the game that we really want to talk about.

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

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

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,

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

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

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.

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.

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 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?

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

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

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

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.

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 b squared, and there's b times c.

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

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Secrets of 3-Board Tic-Tac-Toe - Numberphile

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