The game of tic-tac-toe

[Kent]

Today was terribe, because i had to attend a 8 hour traffic school presentation. The presentation was boring, but i had try to made the most of it by play tic-tac- toe by myself. At some point, i realize that the game of tic- tac - toe could be represented using a Possibility tree. There should be 2^9 ways to play the game. Can any provide a link to such possibility tree? or perhaps somthing similar? Are there ways to optimize your chance of winning?

[Aki]

Shouldn't there be 9! = 362,880 ways to play it?

[Kent]

Originally Posted by Aki

Shouldn't there be 9! = 362,880 ways to play it?

1) Perhaps there are 9! ways to play the game between two player.

2) There number of arrengment of O and X in the diagram is 2^9

3) The number of possible win of the game perhaps is C( 9, 3) =9!/(3!6!) <---i think this is wrong, but can anyone find the solution ?


3 much not be it. If one label the space of a tic tac toe game by a, b, c, d, e, f, g,h,i

To win a game, you need need 3 letters chosen from the set of {a, b, c, d, e, f, g,h,i}

ex: abc, bcd, ahi, cef ...etc


The thing is that not all 3 letter combinations are possible. the letter on the rim have( not corners) have 2 ways. The corners have 3 ways. the letter at the center have 4 ways. If they are all disjoint, then perhaps i could add those suckers together, but since those letter combine to form a win then it is really not that helpful.

[Bo]

1) Perhaps there are 9! ways to play the game between two player.

2) There number of arrengment of O and X in the diagram is 2^9

the problem is more complex then you both propose here
there are 9! ways if each player in turn puts down a piece, and the game wouldn't stop if a player has three in a row (and the complete board isn't filled yet). on the other hand there are 2^9 ways to fill a complete board with X's and O's, but -apart from the point above; such a situation doesn't always occur when you actually play-, in this way the number of X's and O's doesn't have to be equal.

Bo

[pgrmdave]

but you don't take into account that there are only three different places you can put the origional piece. A corner, a middle of a side, and the center - the board is rotatable.

[pgrmdave]

x|_|_
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is the same as

_|_|_
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_|_|x

[Thomas]

Originally Posted by pgrmdave

x|_|_
_|_|_
_|_|_

is the same as

_|_|_
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_|_|x

This is a very interesting point, Considering this, the 3 spaces on top cancel out the bottom 3, one side cancels on side, and the middle space counts as on possibility, which leaves five spaces, so, my logic is:
the number of way to play is 5x4x3x2x1 = 120

[Kent]

delete

[Kent]

Originally Posted by pgrmdave

x|_|_
_|_|_
_|_|_

is the same as

_|_|_
_|_|_
_|_|x

I don t know what you are trying to say. Perhaps you can eloberate a bit?

Let us suppost such label:

[a][b][c]
[d][e][f]
[g][h][i]

abc, ghi constitude a win .

adg, cfi constitide a win

etc.....

I think there are 8 ways to win a game abc , def, ghi, adg, beh, cfi, aei, cei

What do you propose?

[pgrmdave]

I propose that abc and ghi are the same thing, in effect. that a=c=i=g and b=f=h=d The orientation of the board is unimportant.