Quantum Tic-Tac-Toe

Interference

The challenge this month has two parts. First, devise two partly-completed games that are fully collapsed; that is, all the marks on the board so far, have become real. The games should be similar but not identical, and the best response for the next move in each game should be similar but not the same. The best responses should be something where, in both games, one spooky mark is in row a and the other is in row b, or one is in column a and the other in column b. If that move were collapsed randomly, there would be a 50/50 chance of the mark being in row/column a or row/column b.

Second, make a superposition of those two games. If a move is in the same square in both games, there is a real mark in that square in the superposed game. If a move is in different squares in the two games, then that move is uncollapsed (spooky marks in each of the two squares) in the superposed game. In the move corresponding to the best response in the original two games, the chance of landing in a should be 100% and in b, 0% (or vice versa).

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 was an end-game puzzle: X to win on move 9. Here is the game up to move 8, and a solution:

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

X has 9 moves that will positively lose, 18 moves that can win the game, 6 moves that result in a cat's game, and 3 that will win regardless of what O does. One of X's guaranteed wins is to mark squares 4 and 7. The other two I'll leave for you to find.