ZU-TTT

An Engineering Study of Scalability

"Sunday, 2012-05-20 - 13:48:18"

Tic-Tac-Toe

Traditional games
Most of us are familiar with the following games: Traditional (3x3) Tic-Tac-Toe Checkers Classic chess
Similarities	Differences
All three games share the following in common: Each is played with two players.^* Each game is played sequentially, with the two players alternating moves. Each has a finite number of possible moves. Each is played in two dimensions. ^*Yes, one of the players might be a computer.	The number of possible board positions increases rapidly from the traditional two-dimensional (3x3) game of Tic-Tac-Toe ... to Checkers (half of 8x8) ... to traditional chess (8x8). Indeed, the greater the number of possible positions, the more complex the game, the more challenging it is, and the more difficult to win (or at least to "outplay" your opponent.) One of the most frequent pieces of advice when playing chess is, "Look at the whole board. Consider the big picture."

Taking things "up a notch" -- the scalability of the game

Increasing the "dimensions" increases the level of abstraction -- and the level of complexity._<Note 1>

Tic Tac Toe Dimensions cells Winning move^<Note 2> Players

Standard game 2-dimensions 3 X 3 3 in-a-row 2

Improved game 3-dimensions 4 X 4 X 4
^<Note 3> 4 in-a-row
^<Note 3> 2 or more
^<Note 1>

Even better game^<Note 4> 4-dimensions 5 X 5 X 5 X 5
^<Note 5> 5 in-a-row
^<Note 5> 2 or more
^<Note 1>

And better yet 5-dimensions 6 X 6 X 6 X 6 X 6
^<Note 6> 6 in-a-row
^<Note 7> 2 or more
^<Note 1>

Keep going 6-dimensions 7 X 7 X 7 X 7 X 7 X 7
^<Note 8> 7 in-a-row
^<Note 8> 2 or more
^<Note 1>

... and going 7-dimensions 8 X 8 X 8 X 8 X 8 X 8 X 8
^<Note 8> 8 in-a-row
^<Note 8> 2 or more
^<Note 1>

... ... ... ... ...

Doctorate thesis n-dimensions (n+1)^{n^th} (n+1) in-a-row 2 or more
^<Note 1>

Notes:

^<1> Increasing the number of players also increases the complexity. While the absolute minimum is always two players, a more pleasurable minimum would be one player per dimension, e.g., at least four players for the four-dimensional version.

^<2> The number to win in true Tic-Tac-Toe is always one more than the number of dimensions of the game space, e.g., 3-in-a- row for the original 2-dimensional game.)

^<3> Not to be confused with a physical cube implementation of the game, using an "abbreviated" and "simplified" 3x3x3 cell structure.

^<4> It can even be used to implement an exhanced version of bingo, requiring more skill than luck.

^<5> Not to be confused with an "abbreviated" and "simplified" version. The full game structure is illustrated here.

^<6> The value of sufficient "visual bandwidth" becomes increasingly apparent if one tries to implement this on a limited display media (e.g., computer screen)

^<7> Even the use of computers becomes a significant challenge. There are 6x6x6x6x6 = 6⁵ = 7,776 possibilities for the first move alone. If there are 5 players (the minimum recommended) there are 7776 x 7775 x 7774 x 7773 x 7772 = 28,393,742,898,980,409,600 possibilities for just the first round.

^<8> See the Scalability Matrix for Tic-Tac-Toe.

Go to Four-Dimensional Tic-Tac-Toe exercise.