Tic tac toe
Miriam emerged from her bath not at all ready for bed but looking for someone to play with her. I agreed she could stay up and that we could play together while Robby was getting ready for bed. The game was my choice. My objective was to induce Miriam’s copying my successful gambits and her re-applying them against me (cf. vignette 9).
Miriam began with her currently favorite opening to produce this game, recorded in the following dialog (her moves are letters, mine are numbers):
1. B | 3 | C ------------- | 1 | ------------- | 2 | A
Miriam Me first, please. Bob O. K. You first. Miriam Will you go in front of me? Bob What do you mean? Miriam Like here, if I go here [at opposite diagonal]. Bob Well, let’s try it and see. . . . Suppose I go over there? [at opposite diagonal] Miriam No. Don’t. Bob Suppose I go there? Miriam O. K. Bob That’ll be number 1. . . . Now I’ll put 2 right there. Miriam [placing her third X] Two ways to win! Bob Um. Do you have any ways to lose? Miriam Yeah [in a small voice] Bob You’re going to lose. Miriam I’ll put — Bob TIC TAC TOE. Miriam [complaint — wah wah wah!] You stupid. Bob I’m not stupid. Miriam Yes you are. Bob No. I’m pretty good at tic-tac-toe. How did I beat you? Miriam You went to, to, to [noises match her gestures to the places I moved].
Miriam’s description of my winning play was not illuminating to me. I hoped replaying game one in reversed roles would help decenter her focus. In game two Miriam refuses to replay game one, preferring to block my third corner move (contrast games two and one). Her putative blocking attempt fails because of the symmetry of the gambit. Game three replays game one with the original roles maintained. When I call attention to the place of forced moves in my play, Miriam follows that lead in modifying her failing three corner strategy.
2. 2 | | B 3. B | C | 3 --------------- --------------- C | A | 4 | 2 | E --------------- --------------- 3 | 4 | 1 D | 1 | A
Bob You watch. I’ll play the same game you played. I’ll put my 1 there. Where are you going to put your piece? [center square X move] Oh. You don’t want to play my game, huh. How ’bout I put my 2 up here? [Miriam then puts 2nd X in opposite corner] Are you watching now? what have I got? Miriam Two ways to win. Bob How did I do that to you? Miriam You went to, to, so you can have a way to win. Bob Could you do that to me? Miriam Yeah. Bob Let’s try it again. Miriam Me this time first. Bob You want to go first? Miriam Are you going to go in front of me? Bob I don’t think I’ll let you beat me. . . . You’re afraid I’ll go over in this diagonal corner here? Right there? Well, I won’t do that. I’ll go some other place. But remember: in this game  I did not go in the diagonal corner and still had you, didn’t I? Yeah. I’ll go right here. Miriam Oh. You’re trying to play your dirty trick. Bob I don’t play and dirty tricks. I play good tricks. . . . Now. You have one way to win there. I am forced to move here. Miriam [tooting noises — continuing intermittently] Bob Do I have one way to win? Yes. You are forced to move down there. You have one way to win there. I am forced to move there. Miriam X. Bob So that’s a tie.
Game four proceeds as my attempt to show Miriam what is expected of a player whose initial plan is frustrated, i. e. one should not gripe nor negotiate turn takings at victory but should adopt the best expedient one can.
4. A | 4 | C -------------- D | 3 | B ------------- 2 | B | 1
Bob Let’s play game #4. I’ll go first now. I’m going to go right there. Are you going to go across from me? Are you going to block my move? Go ahead. Can you block me so I don’t do that? Oh phooey. Now I’ve got to figure out some other way, because I know I can’t use that good trick that you know, so I have to figure out some other trick. I will go here . Now I have one way to win. Miriam [blocking row] None way. Bob O. K. You blocked me. Ha. I will go here . Now I have one way to win. . . . Hum. Right here, I see you have a way to win. I will go there .
In game five, I attempt to exhibit the purposes behind each of my moves, specifically showing that I think of her responses to my moves as well as my own objectives. Instead of attempting to negotiate a victory, I assume she will move to block my plan and adopt a different gambit on that basis.
5. A | C | 3 -------------- 4 | 1 | -------------- D | 2 | B
Miriam Me first. Will you block me? Bob Maybe. But even if I don’t block you, it still seems I do pretty good, don’t I? Miriam Yep. Bob Did I block you here? [in gane 1] No. But I beat you. . . . If I go here [at perpendicular diagonal corner on 2nd move] you can block me and get two ways to win. Right? Miriam Right. Bob If I go here and you block me, do you get two ways to win? No, you can’t. I am going to go here [move 2] and I have one way to win. You made a forced move [C]. You have one way to win, so I am forced to move  [Miriam blocks 3 – 1] and you have one way to win again. So I have another forced move and it’s a draw.
In games six and seven, after defeating Miriam, I again attempt getting her to re-apply an opponent’s successful strategy against him. (My opening in game six, Miriam’s in game seven; dialog describes game 7):
6. | A | 2 -------------- | 1 | B -------------- C | 1 | 3
7. 3 | A | C -------------- B | | 4 -------------- 2 | | D
Bob You move first. Let’s see if you can beat me the same way I just beat you. O. K. You’re starting with an X. I’m going to go right where you went. Let’s see if you can beat me just the same way I beat you. Miriam Wish. Bob Is that the same way? Miriam Did you go here? Bob Yes. O. K. So you’re going in the corner now. Now this  is a forced move, because you have one way to win, so I have to go here. Miriam Two ways to win. Bob Yes, you do. And you went over here . So I will too, and you beat me. . . . O. K. Miriam [cheering herself] Yaaaa. I won for the first time. Hooray.
The interest in game eight is that it shows Miriam more intent on blocking the opponent’s next move than winning directly. Her failing to notice a winning move leads into my codifying the order in which she should apply her decision principles.
8 B | 1 | -------------- C | A | D2 -------------- 3 | D1 | 2
Bob Would you like to first again, Miriam? Miriam O. K. Yeah. Bob That one’s yours. Let’s see if you can beat me a different way. I will go there again. But see if you can beat me some different way. Oh. O. K. I have a forced move. I have to go here . Miriam [gets two ways to win] Bob I have a forced move here. So I must go here . Miriam [starts to block 2 – 3 row] Bob No, no. Miriam I blocked you. Bob But look. Is it better to block me or better to win? Miriam Win, win. Bob But one of the things you have to figure, Miriam, every time, you have to ask yourself: does the other guy have a way to win? Can I beat him, first? ‘Cause if you can beat him, first, you don’t have to stop him from winning, ’cause you won already. Miriam Right. Bob So let’s see. The number 1 thing you look for [writing list], you say: can I win? Miriam Can we stop for a while? Bob Yeah. The second thing is: forced moves. And the third thing is what? Two ways to win! O. K.? Miriam O. K. What’s the seventeenth thing? Bob No, they’re the only three things you have to look for, Miriam. . . . Can you tell me what the three things are you look for? Miriam Yeah Bob The first thing is what? Miriam Can I win. Bob What’s the second? Miriam Forced moves. Bob And the third? Miriam Two ways to win. Bob Which one do you look for first? Miriam Can I win. Bob Second? Miriam Forced moves. Bob Third? Miriam Two ways to win. Bob You got it. That’s all there is to tic-tac-toe. If you always use those three rules, in that order, you’re going to be a winner. O. K.? Miriam Yeah. Bob Or else maybe you’ll come to a draw. I think you’d better wash your face and go to bed. Miriam Good night. Bob Good night, sweety.
I expect tic-tac-toe to serve Miriam as a simple model of a bi-polar activity, i. e. one wherein at each step of your activity you must attend to your previous actions and a response to that action. (By a model, I mean a framework in terms of which one may conceive of other activities, such as putting questions to nature.) The features of tic-tac-toe which I see as useful are: its interactivity; the opening gambit may be yours or your antagonist’s; there are a set of good tricks one can learn; there are pitfalls to avoid; when one does not see a sequence of forced moves to game end, there is an ordered set of heuristics to follow.
If Miriam can reflect on her own procedures in playing tic-tac-toe and uses tic-tac-toe as a model for exploring phenomena, reflexive abstraction will be a natural consequence .