Tic Tac Toe

I suggest everyone to draw the question figure before reading the solution and mark the places from left to right as:

           1 2 3

           4 5 6

           7 8 9.

At 1 , 4 and 9 there is O and at 2, 7, and 8 there is X.

Alright , we do this question by finding out which person shall get the next chance by ruling out wrong possibilities.

Let us assume for once that X got the 6th chance . Let the X at 2 was his last chance. But we know he wants to win so instead of putting an X at 2 , he could have stopped O from completing the diagonal by putting a X at 5. So this possibility is ruled out.

Now let us assume that X put his last chance at 7. This might seem a legit move for stopping O to complete the left vertical. But instead of stopping O, he could have himself won by putting a X at 5 and completing the mid vertical , hence winning. So, this possibility is also ruled out.

The last possibility (putting an X at 8 ) is similar to our first assumption , he could have put a X at 5 and stopped O from completing the diagonal.

Thus ruling out all the possibilities of X getting the 6th chance , we conclude that only O could have got the 6th chance.Hence next turn is X's. He wins by putting a X at 5...

So much of typing!!! Tiring Solution :( !

This should be the intended solution:

Observe that both players can win by playing in the center position, and clearly since the players are playing for win, they won't leave such open line. The only possibility is that a person created two threats at once and his opponent blocks one threat, leaving the other, the one involving the center, open.

Suppose O created the double threat. Then he must have played on top-left, replied by X playing on bottom-left. But then in the position before that, X would have a threat on its own (vertical middle) which is not blocked by O, contradiction. Thus X created the double threat by playing on bottom-middle, and O blocked one line on bottom-right. Thus X will win by playing on the central square.

However , it has been proven that tic-tac-toe is a draw, so some player must have made a mistake. Indeed, whatever second move O played have let X to win by playing at bottom-middle, so O's second move (or perhaps first move) gave X the win, and thus this position shouldn't have occurred.

Here's the full analysis. Denote the spots with numbers, like in Pankaj's solution:

1 2 3
4 5 6
7 8 9

We also know X played 8 on his third move and O replied with 9.

If X played on 7, then whatever O's move was a mistake:

• 7 4? 5! 3 8 (double threat 2/9)
• 7 1? 9! 8 3 (double threat 5/6)

So X played on 2 as his first move. If O played on 4, it was a mistake: 2 4? 5! 8 3 (double threat 1/7). Thus O played on 1, and X played on 7. But then O should have played on 8 (blocking the double threat) or 5 (creating a threat on his own) instead of 4, so O's second move was also a mistake. (Note that 9 doesn't work: 2 1 7 9? 5! (double threat 3/8).)

if X was the last move then why wasn't X drawn on the middle square to stop O from completing its triplet........ there seems no conceivable motive behind this! on the other hand if O was the last move and if we omit the 9th square ( 3rd row 3rd column), it is apparent that X was sure to win! we call this situation a DOUBLE CROSS in common language. now there remain no grounds for doubt that O was the last move and hence X will be the next and also the victorious move!