∫i First moves
By Tom Brown
The perfect game in Chess would be that of the best first move answered by the best move, replied again by best and then the best once more. The age old question is who wins? Is it white, or black, or is it a stalemate, a draw?
These essays are rather technical but in not too much detail. One should be able to understand the ideas and follow the reasoning.
The stage is a chessboard and there are 32 chess men that is 8 white pieces and 8 black, 8 white pawns and 8 black pawns. The game is played on a checkered square board 8 x 8 squares of black and white. There are rules for allowable moves (restrictions) for the pieces and pawns. White has the first move and then they take turns.
A "feasable position" is any imaginable arrangement of all or some men and empty squares, irrespective of any further meaning. By a "possible position" we mean a position that can be reached in a real game, in other words played according to the rules. Therefore each possible position is also a feasable one, but by far most of the feasable postions are not possible. Thus the collection (set) of all possible positions is strictly contained in the collection (is a subset of) of the feasable ones.
Counting the feasable positions is an exercise in combinatorics although elaborate, in principle a method in itself could be quite straightforward. According to my reasoning you see this number is enormous but if you had unlimited time you can actually numerically calculate it as an exact number and write it all out. It is a fixed finite number. The tools involve things like combinations and permutations and in fact only methods of very elementary discrete mathematics are needed.
Obviously all this has very little to do with actually playing chess.
For white's first move there are 20 different available, replies for each move for black is another 20 each. As a giant tree's branches spread we will choose twenty candidates as an estimate for each subsequent move.
The starting position gives 20 and for black another 20 each, that is 20 x 20 = 400. After black's second move you're already on 20 x 20 x 400 = 400 x 400 =160 000 possible positions. The number increases exponentially.
To get an idea after only four moves by black our estimate will be 400 x 400 x 400 x 400 = 6 400 000 000. The possible positions quickly become probably far more than the number of electrons in the universe and it will take you many many hundreds of orders of the estimated age of the unverse to write it out. The numbers quoted are by careful guess work.
I can't really think how one should go about calculating a "perfect game" there must be far more distinct games than all the atoms in the known universe, probably more even than a Googleplex! All confused and totally haywire and for a human mind or computer quite completely perfectly not possible. These are numbers you can only dream of and cannot comprehend in any way and so, in essence are practically meaningless.
How many possible positions are there?
In principle then you would be able to write a calculation of feasable positions out but in practice you cannot numerically evaluate it. So by brute force and some very crude estimates we know the number of possible positions exists and is finite. In fact we have a fixed upper bound as the total of feasable positions and theoretically we know what this is. Since every possible position is also feasable we do know the number of possible positions is finite and in fact will be orders less than the feasable number and by total overkill.
We are testing the limits of infinity. The keys to the Universe.