Quantum Tic-Tac-Toe

Another End Game

Consider the following game:

Move   X   O 1   4 - 1   1 - 2 3   3 - 5   5 - 7 5   8 - 9   9 - 6 7           9

X to move and win.

There have been no collapses, so X has 36 possible moves. There are three entanglements, so there are 27 realities in the classical ensemble. X and O each have won one game already within the classical ensemble, and O has five more games where she might win; in one of those, she has two different ways she might win, which is an ideal situation for a quantum move consisting of two spooky marks.

The challenges are:

• How many of X's moves result in a cyclic entanglement?
• Do any of X's moves, make it impossible for O to win, even if X tries to help O on his last move?
• Can X make it impossible for O to win (even with X's cooperation) without making a cyclic entanglement?
• Can X force a win?
• Can X give the game away?

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

I hope you had fun with last month's puzzle. The series of challenges proved somewhat difficult (the inventor of QT3 neglected to supply me with the solutions, so we all were in the same boat). But here is one set of solutions.

For part one, here is a game where O cannot get a mark in the center no matter where X places his last move:

Move   X   O 1   1 - 2   6 - 9 3   3 - 6   2 - 3 5   9 - 8   7 - 8 7   7 - 5   1 - 4 9

There being no spooky O in the center square, and O having no more moves, we will not see a real O in the center.

The game shown above also provides answers for the seven questions of Part 2:

• A move for X that leaves O with marks only in the side squares (2, 4, 6 and 8).
          5 - 7
• A move for X that leaves O with only one mark in a corner square.
          8 - 9
• A move for X that leaves O with exactly two marks in corner squares.
          3 - 6
• A move for X that leaves O with exactly three marks in corner squares.
          1 - 2
• A move for X that leaves O with the choice of four marks in the sides,
  or four marks in the corners.
          4 - 5   (There are two such moves; what is the other?)
• The only move for X that forces a cat's game.
          3 - 6
• A move for X that leaves him with a point and a half,
  regardless of how O collapses the game.
          4 - 7

Bonus 1: What minor change to the QT3 rules, would allow X to force O's marks to all be in the corners?
	On the last move, X may place both spooky marks in the same square. In the present rules, X can do this only if all squares but one have real marks in them. If this rule were changed, then X could place both marks of move 9 into square 4, and O's marks could only end up in the corners.
Bonus 2: What are the various probabilities of O having marks in corners?
	OK, did anybody else run the possible last moves, and collapses, and count all this out? Here are the results: In 2 games, O has three corners and a 50/50 chance at a fourth. In 1 game, O has three corners. In 2 games, O has two corners and a 50/50 chance at two more. In 4 games, O has two corners and a 50/50 chance at a third. In 1 game, O has two corners. In 2 games, O has one corner and a 50/50 chance at the other three. In 4 games, O has one corner and a 50/50 chance at two more. In 4 games, O has one corner and a 50/50 chance at a second. In 1 game, O has one corner. In 2 games, O has a 50/50 chance at four corners (or none). In 4 games, O has a 50/50 chance at three corners (or none). In 4 games, O has a 50/50 chance at two corners (or none). In 4 games, O has a 50/50 chance at one corner (or none). In 1 game, O has no corners.
Bonus 3: Did you find any other interesting combinations?
	If X moves in squares 2 and 3, then the outcome of the game is the same regardless of which way O collapses the entanglement.