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Vn005.01

Tic Tac Toe

5/13/77

Years ago I bought a tic-tac-toe game for playing with Robby. The board is 12 x 12; the X’s and O’s are large yellow pieces. At that time I taught Robby a single strategy for playing the game: look for two ways to win. Robby quickly became quite good at the game. Miriam, at age four, learned that one took turns and that winning was getting your three pieces in a row. She did not take the strategy instruction so readily. The children played with the game with different competences: Robby winning with his strategy; Miriam winning frequently enough when he made errors to be satisfied. As Robby came to make fewer errors, Miriam played less, the pieces falling to other uses: the dog chewed one of the X’s, the play group used an O as a hockey puck.

Tic-tac-toe came forward again as a game at a recent visit to the Children’s Museum. Miriam lost to the computer when it, moving first, chose the center square. When she moved first in the center square she never did better than a draw. Robby beat the computer with a first move corner choice (let Robby’s sequence be 1, 2, 3, 4, and the computer’s A, B, C, D):

 1  |     |  3
--------------
 |  A  |
--------------
 B  |     |  2

By move 3, Robby had forked the computer, had “two ways to win.” Since neither Miriam nor his friend John had done better than a draw, he gladly showed them the gambit on his next turn.

Tonight, after dinner, Miriam asked me to play tic-tac-toe with her. She was quite familiar with the terminology of two ways to win and implicit victory whenever she achieved a forking pattern. She understood and accepted the terminology of a forced move.

Whenever Miriam had the first move and chose the center square, and I chose a non-corner, she consistently won. This was true regardless of orientation of the board. Let these two games represent the rest as well (Miriam’s moves are the digits, mine the letters):

    |     |  2       2 |  A  |
 --------------    --------------
    |  1  |            |  1  |
 --------------    --------------
  B |  A  |  3       3 |     | B

In both, her “two ways to win” victories were not noted as being ‘tricks’ of any sort.

I gradually altered my responses to her first move center choice until I chose a corner square in responsse to every center square first move. Making no mistakes, we come inevitably to a draw, thus:

 2 |  C  |  5        3 |  B  |  5
 --------------     --------------
 D |  1  |  4        4 |  1  |  D
 --------------     --------------
 A |  3  |  B        A |  2  |  C 

Miriam began then a new gambit, the first move corner choice. When I responded with a center square choice (as did the computer at the Children’s Muuseum), Miriam had her two ways to win by the third move. Miriam described this new gambit as her ‘dirty trick.’ I was quick to tell her I thought it was not a ‘dirty trick’ but a ‘good trick,’ and we both later referred to it that way.

After being beaten several times by her new gambit, I blocked its effectiveness by refusing to make the center square response to her opening move. When I responded thus

  1 |      |
 -------------- 
    |      |
 -------------- 
    |      |  A   

Miriam complained vociferously. It was clear she did not know what to do. She eventually proceeded by placing her piece anywhere. In another variation, she was quite surprised at my victory:

  1 |  B  |  3
 -------------- 
    |  C  |
 -------------- 
    |  A  |  2

She was so intent on her ‘good trick’ she failed to see the simple victory she might have achieved on her third move. This was the only ‘mistake’ I recall Miriam making.

I conclude from these observations that Miriam’s strategies are very specific in nature. When consolidated, they may be orientation insensitive, but not even orientation insensitivity is immediate. Further, Miriam made her ‘mistake’ because she had not yet integrated her ‘good trick’ with the primary rule of tic-tac-toe: make all forced moves. One might better conceive of her “two ways to win” less as a strategy than as a more complex and various, more immediate objective, more immediate than the victory criterion of three pieces in a row.

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