Quantum Tic-Tac-Toe

A Multiple Part Puzzle

This month's challenge has several parts, of increasing difficulty. The solution to each part is used in the next part.

Part 1 is fairly easy: Devise a quantum game which gets to move 8 with no collapses. Move 9 will collapse everything, of course, but no matter where X places his last move, and no matter how O chooses to collapse the entanglement, O must not be able to get a real O in the center square.

Part 2 is harder: For the game you devised for part 1, find:

• A move for X that leaves O with marks only in the side squares (2, 4, 6 and 8).
• A move for X that leaves O with only one mark in a corner square.
• A move for X that leaves O with exactly two marks in corner squares.
• A move for X that leaves O with exactly three marks in corner squares.
• A move for X that leaves O with the choice of four marks in the sides,
  or four marks in the corners.
• The only move for X that forces a cat's game.
• A move for X that leaves him with a point and a half, regardless of how O collapses the game.

Bonus 1:
What minor change to the QT3 rules, would allow X to force O's marks to all be in the corners?

Bonus 2:
Assume that O would collapse randomly, if you played the same game over and over. You could then figure the probabilities for O having various numbers of marks in corners. What moves for X yield the various probabilities, from highest to lowest (that is, of the 36 possibilities for move 9, which one or ones give O the highest probability of marks in corners? Next highest? And so on down to no marks in corners at all.

Bonus 3:
X has 36 possible last moves, and O can collapse each of them in either of two ways. Are there any interesting combinations that we didn't identify above?

Be sure to include your name and email address! We want to give credit to the person that sends the best solution.

Last month's challenge, I'm sure, stretched your minds! You were to devise a single quantum game, that could end in both of the real games shown here:

Game 1		Game 2

Here is one solution.

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

This game ends with move 8, and produces one of the two real games shown above, depending on whether move 1 ends up being in square 1 or square 6.

"But wait a minute," I can hear you say. "There are eight moves. One of those games has eight moves, but the other only has five. Why?"

The answer to that question lies in the difference between perceiving the games as quantum games, or as real games. As quantum games that had collapsed to a classical state, there should have been eight moves in each game. But these are real games, and in one of them X won on move 5, and we don't keep playing after the game is over. So any moves after move 5, simply are not there in that game. We don't keep playing a quantum game after it is over, either; but in a quantum game the present has a certain "thickness", an extent into the past and future, and so a real move may exist that "happened after" the winning move, and a quantum move that happened before the winning move, may have no real existence at all.