Vn85.1 9/6/77

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?
Miriam

Yeah.
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?
Miriam

Yep.
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?
Miriam

Unh-uh.
Bob

No? Do I have a forced move?
Miriam

Yeah. . . . Actually, I have [beat you]. You have a forced move.
Bob

Then what?
Miriam

I’ll move there [the alternate adjacent corner] and get two ways to win.
Bob

So you’ve beat me already.
Miriam

I know.
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?
Miriam

[smiles] Yeah.

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
[gleeful laughter].
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.
Miriam

Uh-huh.
Bob

Did you know that? Were you just trying to trick me?
Miriam

Unh-uh.
Bob

You probably didn’t know it really.
Miriam

Right.
Bob

Do you know it now?
Miriam

[smiles]
Bob

Yeah.

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.
Miriam

Right.
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?
Miriam

Unh-uh.
Bob

Will you beat me if I go here? [corner]
Miriam

Yeah.
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.
Miriam

Arrgh.
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.
Miriam

How?
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]
Bob

Oh brother.

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.

Relevance
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.

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