ZU-TTT

An Engineering Study of Scalability

"Sunday, 2024-05-19 - 00:34:52"

Tic-Tac-Toe

Traditional games
Most of us are familiar with the following games: Traditional (3x3) Tic-Tac-Toe Checkers Classic chess Traditional Bingo (game of chance) 3-Dimensional Bingo (game of skill) 4-Dimensional Bingo (advanced skill) 5-Dimensional Bingo (exceptional skill)
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.

Four-Dimensional Tic-Tac-Toe
Another game in the series - but with significant key differences.
Similarities	Differences
The game is still played sequentially. There are still a finite number of possible moves.	This game is played in four (4) dimensions. The game is much more difficult to visualize. There can now be more than two players. In fact, it is best played with at least four players. The number of possible moves increases significantly as you factor in the number of players.
Rules of the game
Two or more players (The more players, the more challenging the game.) Five-in-a-row wins the game.) Winning line can be in any direction - e.g., a straight line on a plane of any two of the four dimensions. (See examples below.)

Game Sheet (Print and play)

Mode Players

Beginner Two to three

Intermediate Four or five

Advanced Six to eight
Basic winning moves (Illustrated examples)
More winning moves (Illustrated examples)
Picking the winning moves.
Predictable next moves
Games of Greed vs. Defense
Winning vs. Not losing

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

General Exercises:	Dimensions
General Exercises:	2	3	4	5	6	7
How many possible first moves are there?	9	64	625	7,776
If there one player per dimension (the minimum recommended) how many possible first rounds are there?	72	249,984	151,127, 340,000
How many unique first moves are there?	3	3	6	6	10	10
How many unique first rounds are there?	12
What is the minimum number of rounds to win a game?	3
How many possible winning moves are there?	8	72
How many unique winning moves are possible?	3	3	4	4	5	6
What is the minimum number of moves (assuming the recommended minimum number of players) at which point it is impossible for anyone to win?
Do you prefer to play offense or defense?
Describe what would be your favorite strategy to win?