Explaining Tic-Tac-Toe using Entropy
- The state space of a game is 9!
- there are 9 moves per game
- on average there are 5 degrees of freedom for any given move
- on average there are 4 previous actions before the 'current turn'
Defining the Board
We can define a board as membership in any of 7 sets:
1 set == The empty Set 3 sets for row membership (A, B, C) 3 sets for column membership (D, E, F)
Defining the gamepiece
A gamepiece is composed of 2 elements: Location and State
Location is defined as the gamepiece being a member of 1 row and 1 column set or the empty set.
State is represented as a membership in a set describing all places containing an 'X' or an 'O'.
Defining Game Branching
Gameplay can be modeled using a branching tree notation.
We can model past moves as:
3^4 # entropy of move history
We can model the X/O alternating game 'choice' (when playing as a single player game):
By combining all these factors we have shown that the defined set memberships must be correct.
This can be directly applied to Type theory as a means of demonstrating that a given type model correctly describes some target problem set.
9! = 3^4 * 2^5 * 2 * 2 * 5 * 7