There is an Ideal Opening Move in Chess

Introduction:I will build up my case through a series of theorems ending with a proof of the resolution. I will do my best to make it clear and easy to follow.Theorem 1: Chess is a finite game.Definition 1: Board StateA board state is an ordered 64-tuple set giving the occupier, if any, of every piece on the board. Unoccupied spaces are marked by blanks. It starts at A1, then goes to A2, etc... until it reaches H8. Some convention could be adopted for naming pieces (W for White, B for Black, R for Rook, B for Bishop, # for empty, etc.). Thus the starting position is {WR, WK, ...BR}. An integer can be assigned to each board state (1 for the starting position, 2 for some first move, 3 for some other first move, etc up to about 10^45 if we only worry about legal positions).Lemma 1: There are finitely many positions in chess.There are 32 pieces on a chess board (only 4 of which are unique) and 64 positions. There are only a limited number of ways to assign finite number of pieces to a finite number of spaces (though this number is large). A large number of these board states are not accessible from the starting position, are identical because they involve switching non-unique pieces, or involve less than 32 pieces (the minimum number of pieces on the board at the beginning of a turn is 3).Lemma 2: There are only finitely many possible turns in chess.A chess game ends if fifty moves have passed without a piece being captured. If a king is captured or only kings remain, the game ends. This means that no chess game can be longer than 1500 moves (the maximum time to exchange the other 30 pieces).Definition 2: Game StateA game state is an ordered pair giving the board state and turn number of a chess game. For example, {92385983475928740239756183610947261, 56} might be a game state. Not every game state is necessarily legal (example: {1, 2}). A game state is called an end state if it results in the game's ending (there is a checkmate or draw).Conclusion:The game states exhaust the possibilities for chess. A game with the game state given is fully described. The number of game states is finite (equal to the product of the number of positions and the number of turns). Therefore, the game of chess is finite.Theorem 2: If a position that ends the game is accessible from the position at the beginning of a player's turn, that player can guarantee that the game will end.Lemma 3: There is no chance in chess.There is chance in a game if a player does not know the outcome of a move before deciding to make it. When a player attacks in Risk, he or she does not know how many armies will remain at what territories afterward. In chess, a piece always moves to the desired position.Lemma 4: Chess is a game of perfect information.A game of perfect information is a game in which every player has access to all the information about the game. A game that hinges on imperfect information is Stratego. In Stratego, the value of each player's pieces is concealed from the other player. In chess, both players can see everything relevant to the game on the board before their eyes; there are no hidden variables.Lemma 5: Every position has finitely many positions that are accessible from it.This is entailed by Claim 1. No matter what moves were allowed, a move could only change the current game state to a new game state (and where the current turn number was n, the new turn number would have to be n + 1. Though time travel could make chess very interesting!).Conclusion:If a position is a game ending position, a player can determine that it is because there is perfect information. There is finite set of positions accessible from a given position in chess, so the player can always determine if one of them is a game ending position by analyzing each one. The player can also guarantee that a move will lead to the desired position because there is no chance. Final Proof: There is an ideal opening move in chess.Lemma 6: With each move, the number of accessible game states is reduced.This follows trivially from Theorem 1. Regardless of what moves are allowed, the turn number is increased by 1, and so the number of game states degrees by the cardinality of the set of board states. Because it is not legal to simply rearrange the pieces however one chooses each turn, the possibilities are in fact reduced far more rapidly.Lemma 7: An ending state is accessible from every game state that is not itself an ending state.This simply follows from the fact that the game does indeed end.Lemma 8: A player can only move into an end state that results in a tie or a win.This is a logical consequence of the rule that one may not put oneself in checkmate.Definition 3: Chess TreeFrom every game state A, there is a finite set of game states reachable from it (the legal max is 218 and the minimum is of course 0 for end states). Call the members of this set B1 ... BN. From each of B1...BN, there is a new set of game states accessible for that position. This process is recursive until the set of game states accessible from a final game state is empty. These can be arranged in a topological structure called a chess tree, which will be used to visualize the game. Such a tree would look like this, but it would be of much greater complexity: http://i.investopedia.com... Assumption: There is a path down the tree that will allow White to win or tie the game no matter what Black does.White can try tracing every path down the tree. If any path leads to a game state in which it is Black's turn and Black has an ideal move, that path is abandoned and a new path is attempted. If there is a path leading to a White tie or win regardless of what Black does, then every move in that tree, including the first move, is an ideal move for White (because, per the definition, it enables White to guarantee a victory). If such a tree exists, the resolution is true.Suppose that no such path exists for White. That means that if Black plays correctly, there is no way that White can win the game or intiate a tie. But the game must end, so that means that Black can guarantee a win or a tie from the starting position. Any correct move that Black makes from the starting position would be considered an ideal move.Conclusion:The result was first proved (more rigorously, but for different chess rules) by Ernest Zermelo in 1913. It should be stressed that no one knows what the ideal opening move in chess is or whether it belongs to White or Black (though it seems more plausible tha it is White's... if it is Black's then the starting positon is a Zugzwang), but we can be sure that such a move does indeed exist. If we are persistent and do not die in some terrible disaster, humans will eventually solve the game of chess just as we have solved the game of checkers. Any game that is basically finite can eventually be solved by looking at all the possibilities. This is obvious in the case of Tic-Tac-Toe, the solution to which can easily be visalized:http://www.csc.csudh.edu...Chess is a mathematically trivial game in the same way, it's just a great deal more computationally complex.