For the first move of a chess game, you have 20 possible moves. After that, it only get's more complicated especially if you open up diagonals for the Queen and Bishops early. Remember, that you have evaluate your opponent's moves too. That's around to 100 total moves for a long game. So, 20+ to the power of 100 different positions to evaluate.
BTW, I'm also assuming that someone wins the game or a draw is reached byt agreement or stalemate instead of the 50 move rule. (After either player has made 50 moves, during which no pieces have been taken and no pawn has moved, the game is considered a draw. IIRC)
Memorywise the most efficent way of recording the position (that I can think of) of the pieces is recording the square it is on(1-64 for the bard squares from a1 to h8, with 0 being out of play), for each piece. thats 7 bits (practically a byte) for 32 pieces. So about 32 bytes.
So, in a very simplified way, we can say that the memory requirement for calculating a full game of chess is a lot more than (and we are rounding down a huge amount for simplicity sake here) 32 * N to the power of (about) 100 bytes, where N is the average number of available moves during a game. (Propably only 25-35 due to end games being simpler)
If you round that down further to a single integer followed by zeroes, the number would propably have around 140 zeroes in it. A gigabyte rounded in that fasion has only 9 zeroes. My hard drive's free space has only 11 zeroes in it.
No point in going on with that.
To be sure, chess AIs see a long way into the future. Freeware PC chess programs go 9 moves (for each player for a total of 18) deep in a few seconds, super computers built specifically for chess can do a lot better.
The advantage that humans have over computer (according to one of my computer science professors) is our ability to handle large chunks of related data (like a chess position to an experienced player) as easily as small ones. (like a number between 1 and 10) Also we can perform more complex operations on that data. For example, in chess, a human player has an easier time disregarding obviously bad moves than an AI, who has to study every move to a deep level before it can disregard it or it risks missing good sacrifices that can lead to a forced mate or forcibly recuperating material with interest.
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