When we started playing tic-tac-toe, I asked Miriam how many different ways can you start when you move first. She claimed 9 ways, one for each block in the frame. I pushed the point further by inquiring whether these three frames were really different or the same:
X | | | | | | | | | | | | | | | | X X | | 1 2 3
She judged the first two to be the same and the third different from them. My intention in today’s play was to work through the range of all game Miriam could see as different responses to the corner opening. We pursued this by my letting her move first in every game with the specific objective of finding those responses which would not lead to my immediate defeat.
Game 1: Miriam moves first (letters)
A | D | C | 2 | 3 2 | | B
|Bob||If I go here [the middle of an outside row, not adjacent to A],|
can you beat me?
|Bob||There, in that side place? Or if I go in the corner?|
You don’t want me to go in the corner? [opposite diagonal to A]
|Miriam||I want you to go some other place.|
|Bob||How about if I go here. Can you beat me? [the adjacent corner|
where move 1 is made]
|Miriam||No. Don’t go there. . . . O. K. You can.|
|Bob||How about if I go over here, in this other corner? [the alternate adjacent corner]|
|Miriam||It doesn’t matter [the moves are equivalent].|
|Bob||Oh. If I go there, the moves are the same?|
|Bob||I’ll go in one of these corners here that are the same. . . .|
You think you can beat me?
|Miriam||I don’t know [moves B].|
|Bob||You think you beat me already?|
|Bob||No? Do I have a forced move?|
|Miriam||Yeah. . . . Actually, I have [beat you]. You have a forced move.|
|Miriam||I’ll move there [the alternate adjacent corner] and get two ways to win.|
|Bob||So you’ve beat me already.|
|Bob||Actually, so long as I made that move there (1), you beat me already.|
And you told me you didn’t want me to move there. . . . Did you know you could beat me
when i moved there? . . . You did? Did you trick me?
Game 2: Miriam moves first (letters)
A | D | C 2 | 3 | B | | 1
I recapitulate the last game, identify both adjacent corners as responses with which I can get beaten, and recall Miriam’s assertion she can beat me anytime. I respond with a non-adjacent, middle row move.
|Bob||That means I should either move in this far corner [opposite to the opening]|
or in the middle, or here or here [in the two adjacent, middle of row moves]. Let’s suppose
I move here [opposite corner]. Will you beat me?
|Miriam||I don’t know.|
|Bob||I’ll try it [moves 1].|
|Miriam||[laughs] I’ll put my B here!|
|Bob||Oh. Oh-oh. Do you have me beat already?|
|Miriam||Yep. See. I go there [alternate adjacent corner] and I’ve got two ways to win|
|Bob||So, as soon as I put my 1 in there, you knew you could beat me,|
because you didn’t have a forced move.
|Bob||Did you know that? Were you just trying to trick me?|
|Bob||You probably didn’t know it really.|
|Bob||Do you know it now?|
Game 3: Miriam moves first (letters)
A | C | 3 4 | 1 | E D | 2 | B
This game begins with the moves Miriam originally sought for the execution of her ‘dirty trick.’
|Bob||If I go here [center], can you beat me?|
|Miriam||I think so.|
|Bob||I’ll put my 1 right in the middle. How are you going to beat me now?|
|Miriam||[moves B] Whichever side you go [she gestures toward the corners],|
I’ll go on the other side [the alternate corner] and get two ways to win.
|Bob||Ah ha. That’s a good strategy.|
|Bob||But it assumes I make a move in that corner or the opposite corner.|
|Miriam||I. . .I know what you’re going to do.|
|Bob||What am I going to do?|
|Miriam||You’ll go here [bottom row, middle].|
|Bob||[pointing to the others in turn] Or here or here or here. Does it matter|
which of these four I go in?
|Bob||Will you beat me if I go here? [corner]|
|Bob||I don’t like to lose all the time. I’ll go here [moves 2].|
Game 4: Miriam moves first (letters)
A | 2 | B 1 | C | D | | B1-> 3
Beginning this game, I review the moves I made and where I’ve been defeated. I cite the adjacent middles of rows as the only locations I haven’t attempted and select them as the next trial.
|Miriam||I’m gonna beat you I think [moves B1 ].|
|Bob||Why do you think you’re going to beat me?|
|Miriam||‘Cause. . . . Oh no, I can’t if you go there [in the center].|
|Bob||The move you made is not a winning move. I have a forced move in the center.|
|Miriam||I’ll go here [adjacent corner move].|
|Bob||Then I’ll win because you would miss your forced move there.|
|Bob||If you want to take that B out, cross it out and try some other move; maybe you should.|
|Miriam||Where else? . . . Here? [move B in adjacent corner] Is that O. K.?|
|Bob||Let’s see. The problem with the other corner [now crossed out]: if I went in the center|
you have a forced move in the side. . . but now I must move here [move 2] and you have me beat.
|Bob||Where are your chances to win?|
|Miriam||Here [from A through the center] and here [from B through the center].|
|Bob||If you move where they cross you get two ways to win.|
|Miriam||[laughs, moves C]|
At the end of this game, I summarize: “If you start off with a corner opening, you can beat the other guy no matter where he goes — almost — unless he plays in the middle and side as I did in game 3.” Miriam ran off to announce her victories to Gretchen.
These data show Miriam and me working through all the responses (except one: see vignette 71, games 3, 4, and 5) to a corner opening. They provide a good sense of the range of Miriam’s strategic thinking.