Expected number of moves in Tic-tac-toe

Win_odd Dhamnekar

Junior Member
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Aug 14, 2018
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Hello,
Two robots play a game of tic-tac-toe. They choose their moves randomly. Let E denote the expected number of moves it would take for the robots to finish a game. Find 1000E rounded to the nearest integer.

Note: A move is defined as the action of one of the robots placing one X or one O. The game finishes after 9 moves no matter the result.

Solution:- I found it difficult to answer this question because it involves very complicated computations. If any member can answer this question in a step by step and easy to understand way, may reply with correct answer.
 
I believe the "complicated computations" are the point of the exercise.

1 move - 0%
2 moves - 0%
3 moves - 0%
4 moves - 0%
5 moves - >0%
6 moves - >0%
7 moves - >0%
8 moves - >0%
9 moves - 100%

There are only four tricky places to stop.

Move Number 1 can be in only three different KINDS of locations. Is that helpful? It's not like Chess where there are 400 possible different positions after two moves. TTT has only 72 after 2 moves - and these are substantially and practically reduced by the observation of symmetries on the box.

Move #2 (Number), depending on Move #1 (Text)
Corner + 5
Edge + 5
Center + 2
Thus, only 12 different conditions after two moves!

You don't have to follow this line. Let's see what line you do follow.
 
Hello,
I don't understand your answer.If you explain it a little bit more, it would be easy to understand to me as well as readers of free math help forum. I have computed total sample space of 9! moves.The game can only be finished in 5,6,7,8,9 moves. I have computed 34560 games to finish games in 5 moves, 31968 games to finish games in 6 moves, 95904 moves to finish games in7 moves. To finish game in 8 moves, I computed 25920 games for diagonal winning lines. But I am finding it little bit difficult to compute games for horizontal and vertical winning lines to finish game in 8 moves. Would you help me in this regard?
 
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