A Mid-Game Puzzle
|
Last month, you were given a game in progress
where O has an opportunity to win, if she can find it.
To the right is the quantum game and its classical ensemble,
made from the move list given last month.
Brute-force examination of the possibilities would be daunting;
there are seven open squares, so O has 21 choices for move 6.
Fortunately, she can use the ensemble to guide her choice.
X has open 2-rows in four of the six games in the ensemble,
while O has open 2-rows in only two.
It seems reasonable to eliminate as many of X's 2-rows as possible.
O can eliminate four games where X has a 2-row, by placing
move 6 in squares 5 and 8. X has no real choice about the resulting
collapse, because one choice is an immediate win for O.
|
|
| |
|
The result looks good for O: two games left in the ensemble,
O has open 2-rows in both games,
X has nothing in either game.
However, X can prevent O from winning in several ways.
For instance, X can play move 7 in squares 3 and 6;
O cannot move into 3 and 6 herself, because X
will win. She wants to choose the next collapse, so her only
choice is to move in squares 3 and 2.
X's last move can be in squares 4 and 6, or 3 and 6;
either way, it's a cat's game;
X doesn't win, but neither does O.
|
|
| |
|
O's strongest choice for move 6 does not look like a winning move,
at first glance. However, if she places move 6 in squares 5 and 6,
look at the classical ensemble:
there are five games, and in all five
O has an open 2-row, while X has a 2-row in only one game.
Moreover, if O makes the correct choice at move 8,
X can no longer win.
On move 7, X will either make a cycle, which will leave real marks
in all squares but 2 and 3, or he will not make a cycle, and all squares
but 7 and 9 are still available.
If X makes a cycle, O must move in squares 2 and 3,
and so must X, but O chooses the collapse and wins.
If X does not make a cycle, O's correct move is in squares 3 and 8;
this may or may not make a cycle, depending on where X had moved.
If X gets to collapse a cycle after move 8, he does not have
a choice that gives him a win.
If there is no cycle to collapse after move 8, there will be one
after move 9 – but O gets to choose the collapse and she wins.
|
|
|
|