"This time, I win," Inez announces.
You open your mouth to object but think better of it.
"Watch." She takes her pen back and adds an X in Square i:
"Now you have no choice but to block me by putting an O in Square a:"
… which makes me take Square c."
She leans back in her seat again, satisfied. "And I've won."
You frown down at the napkin. "No – I just put an O in Square f and I block you."
"And then I put an X in Square g and win."
Now you see it. "And if I block you at Square g, you just put an X in Square f –"
"– and win that way," Inez finishes for you. "In chess, they call this a 'fork' – where you are forced to block two attacks at the same time, which, of course, you can't. And I want to point out that you don't have to be an evil genius to set up this little trick. Once I placed my second X, you had to take Square a, which forced me to take Square c, which set up the fork. As soon as I placed that second X, there was no possible way for you to win."
Frowning, you snatch back the pen and pull a fresh napkin out of the holder. But a few minutes of trial and error shows you she's right.
"Tic-tac-toe is designed to make you feel like you have a choice, which makes it feel like a game," Inez says, "but you never really do, so it's not. It's the illusion of a game. Whoever goes first takes the center square and controls the board for the rest of the game."
You shake your head. "That can't be right. If it's impossible to win then no one would ever play it."
"Do you ever play tic-tac-toe anymore?"
"Point," you concede.
Inez pulls another napkin from the holder. "Let's go back to the very beginning:"
"Now, in both of the previous examples, the game wasn't decided until I placed that second X – in both cases, across and one column over from your O:
"And we know that the mirror image second moves – g for i and f for h – are also the same:
"… but that still leaves you with 10 alternatives:
"…which we know, thanks to mirror images, is really only 6 alternatives:
"So here's your next puzzle: of these 6 possible moves – a, d, h, k, l, and r – how many of them will ensure O the win?"
"Do we really have to do this?" you all but beg her.
Her lips purse into a wicked smile. "You don't have to keep playing the game if you don't want to."
And you have to admit that, for a few seconds, you consider just going back to the House and calling it a night. The only thing stopping you is that you'd have to live through this entire conversation a second time if you ever want to try again. "Fine," you sigh.
"Great! Now: we're going to assume perfect play here. I mean, O could win every single time if X makes it a point to try not to win:
"When I played this a kid, I lost only when I got sloppy, not noticing the other guy was about to get a tic-tac-toe. In perfect play, there's only one way to beat your opponent: you have to present her with two winning moves at once."
". . . and she can't block both," you finish for her. "A fork."
"Right. So in perfect play, any time you have a chance to build a fork you take it."
"Makes sense."
Inez shows you the napkin again:
"We saw that if X takes g or i on the first board, they're guaranteed a win. From that moment on, O can't do anything to prevent it." She looks up from the napkin and flashes you a mischievous smile. "In any of the other second moves available to X – a, d, h, k, l, or r – can O, not X, be guaranteed a win?"
You frown, studying the napkin. "This is going to take some trial and error."
"Well," Inez says, tapping the house key hanging around her neck, "time isn't exactly an issue, is it?"
"I guess not."
She waggles her fingers at you and then disappears. You take another panicked look around the room, but no one seems to have noticed. "Got lucky," you grumble. You turn your attention back to the napkin again.
You can determine the answer to Inez's question and find that number on the Solution Page. Or you can