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  • 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

  • likedraws. 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

  • respondlet'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 inherebut 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 movedoesn'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 loseevery 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 winsthat 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 thisif you stare at it, you can seewherever 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

  • simpleyou 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 willif you study closelybe

  • able to beat everyone you play...

  • Numberphile wouldn't be possible without support from the Mathematical

  • Sciences Research Institutethe 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.

Well, the game is Tic-Tac-Toe. Some people know it as Noughts and Crosses. Nine

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嘀嗒嘀嗒(僅限Xs) - 數字愛好者 (Tic-Tac-Toe (with Xs only) - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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