Quantum Tic-Tac-Toe

A Mid-Game Puzzle

The following game was derived from some considerations of potential strategies for generating good openings. In this game, X has established a potent threat with his first two moves. O had to either directly block the threat, or arrange a counter-threat that would win earlier in time. She chose the latter strategem, and threatened to win in row 3. X had to block this, but his attempt to do so unwittingly provided O with another opportunity. Your challenge is to find O's opportunity and show how she can win.

Move   X   O 1   5 - 1   7 - 9 3   6 - 4   7< 9 5   8 - 5      7           9

O to move and win.

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Last month's puzzle gave you a game in progress, where X and O had filled the board with spooky marks, leaving three entanglements and a situation whose outcome is yet to be decided; your challenge was to answer five questions and thus evaluate X's position and find his best and worst scenarios.

Question 1 is actually easy to answer. There are three entanglements on the board, each entangling two moves. A two-move entanglement can be collapsed in any of three ways; thus X has 3 × 3 = 9 ways to cause an entanglement on move 7.

Question 2 is a bit tedious, but it can be answered by examining the 36 classical ensembles resulting from the 36 possible moves X has. It turns out that in 21 of those ensembles O has already won one classical game, in 14 of remaining ensembles O has two-in-a-row in one or more games, and only one ensemble has no possible win for O. That ensemble results from X marking squares 3 and 5 on move 7.

Question 3 asks whether X can keep O from winning without making a cyclic entanglement. On move 7, X has 27 possibilities that do not make an entanglement, but all of those leave O with a possible win, so X certainly cannot do this on move 7. X's next move will be move 9, and all his moves will certainly make a cycle. So the answer to this question appears to be, No.

Question 4 asks if X can force a win. The answer appears to be that X cannot. His best choice for move 7 (squares 3 and 5) leave him with two games in the ensemble with two marks in a row; O can easily block both of them by placing her next move in squares 1 and 2. X can still try for a win in row 2, but that can happen only if O makes the wrong choice in the final collapse.

Question 5 asks if X can give the game away. Given the number of possible wins O has in the various classical ensembles, it certainly appears that he can, if he is not careful.