Vn10901

Vn109.1 Tic Tac Toe 10/4/77

These 5 games are revealing of Miriam’s knowledge and ignorance both. Game 2 reveals more of my failings than I am happy to admit, but its contrast with game 3 permits a central revelation of her thinking about tic-tac-toe. These two together show by how much good fortune (when it occurs) is preferable to good planning. Throughout this session I prompted Miriam to think out loud and make predictions, hoping that she would thereby illuminate her representation of the game. The consequence is evidence how well articulated is her knowledge of what she does in specific cases.

Game 1: Miriam moves first (letters)

         A | 3 | C    
           | 2 | D    
         1 |   | B 
Bob

[after Miriam’s opening] I’m going to ask you some questions. Will you answer them?
Miriam

Yeah.
Bob

[placing 1] Can you beat me?
Miriam

Think so.
Bob

Go ahead.
Miriam

[moves B]
Bob

Do you have me beat already?
Miriam

Almost.
Bob

Can you show me how?
Miriam

If I put one here [at ‘C’], I’ll get two ways to win. . . 3 ways to win. One [B – C], two [A – C], three [A – B].
Bob

Can I go anywhere to stop you from getting those?
Miriam

I don’t know.
Bob

Suppose I go up here [at’C’]; could you still beat me?
Miriam

Yes.
Bob

How?
Miriam

[places her index finger on 2]
Bob

Ah, yes. The way things are [gesturing from A to B], do I have a forced move? . . . So I have to go here [moves 2], and you still get two ways to win.
Miriam

[moves C] C. Go!
Bob

Go, huh? Hum. All right [moving 3], all right.
Miriam

[moving D] D! [pointing to C] You know why I went there?
Bob

No.
Miriam

If I went here [pointing to 3], you would put yours down there [pointing to C].
Bob

That’s right. I guess you had a forced move too.
Miriam

Yeah [agreeing that such was her reason]. Yay! I win!

Game 2: Bob moves first (numbers)

After Miriam’s center response, I realized I was myself so unfamiliar with games of this opening I didn’t have any specific plan to follow. I was confused and not wanting to keep Miriam waiting, moved aimlessly at 2. The game thus becomes pointless but does exhibit Miriam’s defensive play without confusion by any aggressive objective (hers or mine).

         c  | 2 | b    
         1  | a | 5    
         3  | d | 4  

Game 3: Miriam moves first in the center

         3 | D1 | 2    
         1 | A  |
         B | D2 | C  
Bob

If I move here (1), can you beat me?
Miriam

It will be sort of like the same game.
Bob

Same game as what?
Miriam

The last game. Go!
Bob

You think it will?
Miriam

Yeah.
Bob

How?
Miriam

[moves B — after hesitating and moving her hand between corners B and 2; laughs]
Bob

Let’s see. I have a forced move now [moves 2]. How do you figure out where to go next?
Miriam

I just pick a space [moves C].
Bob

Why is that a good space?
Miriam

I don’t know.
Bob

You have no idea?
Miriam

I just pick a space.
Bob

Why don’t you move here [pointing to side opposite 1]. I think that would be a good place.
Miriam

Nahh. I want to move there [pointing to C].
Bob

Is there any reason?
Miriam

No.
Bob

You just don’t want to tell me. Here. . . . I’ll stop you [places 3] along your way to win there.
Miriam

[quickly moves D1 between 2 and 3]
Bob

Did you block me?
Miriam

Yeah.
Bob

‘Cause you thought I had a way to win?
Miriam

Yeah [it’s obvious] 3 and 2.
Bob

That’s right. I had a way to win. Do you think it’s better to block somebody who’s got a way to win or do you think it’s better to win yourself?
Miriam

Win.
Bob

Do you think you have a way to win?
Miriam

No.
Bob

May I call your attention, Miriam, to a way to win you could have had? [points to D2]
Miriam

[moves D1 to D2]
Bob

That’s why I asked so many questions. I wanted to know if you knew you had two ways to win.
Miriam

No, I didn’t. . . . Tic-tac-toe, three in a row.

Miriam became angry when I argued her victory ‘didn’t count’ since I had to show it to her.

Game 4: Bob moves first

This game shows Miriam’s confusion of move 2 in a game of form VII-B (the only safe response to a corner opening) with move 1 of game form IV (cf. Learning: Tic-tac-toe ). This is an explicit example of configuration dominating to the exclusion of serial information.

         1 |    |     
         b | a2 |     
         2 | a1 |   
Bob

Can you go any place at all so I won’t beat you? If I move in the corner [moves 1].
Miriam

One place.
Bob

Is there a safe place? Where is it?
Miriam

[moves a1]
Bob

You believe that’s a safe place?
Miriam

Yeah.
Bob

Well. . . shall I prove you wrong?
Miriam

Yeah.
Bob

[moves 2] What now?
Miriam

[making forced move b] Hold it. I want to have him. [cheats: she moves a1 to a2]
Bob

That’s not fair. You moved here [removes a2 to a1].
Miriam

No [replaces a2 in the center].
Bob

Let’s back off, then, if you don’t want to play that game.

Game 5: Bob moves first (restarting game 4 with his opening marker at 1 and Miriam’s at a1)
RESET the figure in a sensible fashion

         1 |   | b3          1 |   | b3   
           | a1|               | 3 |  
         b2| b1| 2           a2|   | 2 
Bob

Let’s say you moved there [a1] right off. If I move here [2] what do you do? Can you move any place?
Miriam

[removes a1 to b1]
Bob

Miriam! That’s just not fair.
Miriam

[reluctantly replaces a1]
Bob

Now, where can you move?
Miriam

I know [moves b1].
Bob

[pointing to b2] Why didn’t you move there?
Miriam

Good idea! [moves b1 to b2]
Bob

O. K.? You want to do that?
Miriam

What? [moves b2 back to b1]
Bob

Go ahead and move here. I’ll show you what I’ll do.
Miriam

[moves b1 to b2] Win?
Bob

Take a look at my chances to win.
Miriam

None.
Bob

Do my chances to win come together?
Miriam

Unh-uh.
Bob

[gesturing to the fourth corner] No?
Miriam

[grabbing Bob’s hand] No! [moves b2 to b3]
Bob

You think that’s a good defense?
Miriam

[laughing, points to the empty space b2] Here.
Bob

Yes, they do.
Miriam

[moves a1 to a2 as in the second frame] No. I didn’t want you to go there.

When I move 3 in the space just vacated, Miriam sulks and we give up tic-tac-toe for another game.

Relevance
Miriam exhibits her extensive and flexible command of games of the form of game 1. Her comment, after the opening two moves of game 3, that it will probably be the same as game 2 renders explicit the absence from her thinking of the concept of move order variations as significant in tic-tac-toe. I consider it staggering that anyone could play so well as Miriam does and yet not have a well formulated idea of opening advantage. Game 3 also appears to show a game whose play has (may be interpreted as having) degenerated to a serial procedure with loss of an original, configuration-oriented forking objective. Game 4 shows strong confusion between the 2 move of game VII-B and 1 of game IV. These games permitted no show of table-turning because Bob never clearly won any games.

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