Well, the game is Tic-Tac-Toe. Some people know it as Noughts and Crosses. Nine
cells, and we have two players, X and O. Let's say X goes first. A good move is
typically to take the center here, like that, and then O might go over here,
for example. X might follow up here, threatening to get three in a row, and
then O will typically play here. Then I've got some choices here as X. I might
try to open a new threat by moving over here, and now O is compelled to play
over here. I'm X—maybe I make one final threat here. O moves here,
threatening to win, but then I just block it over here. Now this game ends in a
draw almost every time amongst people who even have the slightest understanding
of how to play it. I mean, you might be able to beat a small child, but anyone
who has any competence can draw a game of Tic-Tac-Toe. In other words, manage
not to lose it, and this I view as a flaw of the game, and that's what we're
here to talk about. How can we improve Tic-Tac-Toe?
I think the natural human tendency is to say, "well, the game is too simple. We
need to make it more complicated somehow," so people have tried playing on a
4x4 board, for example, and if you google "3-D Tic-Tac-Toe" you'll probably
find a few things there, but that's not the point of view we're going to take
here. We're going to look at the exact problem with this game, Brady, which in
my opinion is that we can have draws, right? This is the thing that we don't
like—draws. So we propose to fix that problem forever by having both players play X.
So let's just look at this game, because it kinda sounds kinda stupid, right?
We're going to play Tic-Tac-Toe where both of us play X, and whoever makes
three in a row wins. Let's say that I take the middle cell here. Well now you
have to respond, and you immediately have a big problem because wherever you
respond—let's say here on the side or in the corner—I'm immediately going
to win. What we've just shown is that the game is totally useless, right,
because the first person is always going to take the center, right? I mean,
there are other possible moves, right, but I mean you can win by just simply taking the center.
Let's say you did make a different move, OK, so let's at least look at that.
Let's say that I moved on the side here. That would be stupid because you know
this wins. Alright, so now, the other person doesn't wanna make two in a row,
but you've sort of occupied one column and one row here, so I don't wanna move
in either of those, so I'm gonna move over here, for example. Now it's the
person who moved first's turn, and they still have the intersection of the
third row and the first column to play in—here—but now I have an X in every
row and an X in every column, and that's going to compel whoever plays next to
make two in a row and then the game's going to end. Player one wins again.
We've analysed it completely and we haven't exactly found an interesting game
here, so this is not yet what we want to have.
What we wanna do is create an incentive for people to not make three in a row,
and so we're going to play what's called the Misère, or losing form, of the
game. So here we're going to say that instead of the person who makes three in
a row the winner, we're going to make that person the loser. This game ends
when someone makes three in a row, and that person loses. It's to string it out
as long as possible and force your opponent to be the first one to make three
in a row. So let's say, for example, that someone takes a move in the corner.
Moving in the corner is not the move you wanna make in this Misère game. So why
is that? OK, so let's say I move in the corner. What is a good response to
this? Well, what you want to do is move in the opposite corner. It's not clear
that this is a good move, but let me explain why it is. By taking the opposite
point here, you have essentialy made the middle square something that neither
of us want to play in, right, because you immediately will lose now if you move
in the middle square. So now it's back to the first player. He can't play in
the middle, so let's say he plays here. Ah, but now the second player can now
again play exactly opposite on the other side of the center, so now they go
there. Now the first player has another move—doesn't wanna move in the
center. Now let's say he certainly doesn't wanna move here because that loses,
so perhaps he plays here, but then again, on the opposite side of the board,
there's an empty square. So the opponent will play here and now the first
player has to lose—every move makes three in a row. The take-away from this
is that the first player must not choose a cell that is on the boundary of the
board, because then the first player will be vulnerable to exactly this
mirroring strategy, and ultimately will have to be the first one to complete
three in a row. —What happens if the first player plays in the middle? —Yes,
right, that's the only other possibility, so should the first player move to
the middle, and does the first player win? So let's look at that one.
OK, so here's a first player who moves in the middle. Now I claim this is a
good move and this wins—that I can force my opponent to make the three in a
row. So let's take any typical move by my opponent. Let's say my opponent moves
here. OK, so this is my opponent's move, and the secret sauce here is that you
move a knight's move away from where your opponent just moved. —A knight, as in
a chess... —A chess knight. So in other words, two down and one over, or one up
and two over, sort of an L-shape away. So here my opponent has just moved here
so I have two choices. This would be two down and one over this way, or I also
have two down and one over this way, so I could play in either one of these.
Let's say I play here. I claim that is a good move. Now my opponent has to make
some move, and doesn't want to make three in a row so they certainly won't play
here. What if they play here? OK, so let's say they play here, then I claim,
again, a good response is a knight's move away, and now here... I've already
played here. I can't do this one, but I can do this one. So I go here, and now
I have this interesting Y-shape or Z-shape here, plus a little blob in the
opposite corner, and now this—if you stare at it, you can see—wherever you
play, you're gonna make three in a row and lose.
If you're playing X-only Tic-Tac-Toe where the person who makes three in a row
loses, you definitely want to move in the center, and then after that it's very
simple—you just copy your opponent's move, moving a knight's move away. Done.
You are world champion on one board.
Here's the game I really wanna 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...
— Numberphile wouldn't be possible without support from the Mathematical
Sciences Research Institute—the MSRI. That's the building right there. If you'd
like to find out more about them, I'll put some links under the video, but for
now, let's just admire the view they have over the San Francisco bay. It's
incredible. The Golden Gate Bridge, Alcatraz, San Francisco city... amazing.