The Open File
by Life Master Mike Petersen (Zug)
Is Chess Infinite?
Chess games have to end. I've never heard of a chess game that ended because of any other reason than by checkmate, stalemate, draw, or resignation. Draws are the most important, here. They can happen when the position is repeated three times, there is a perpetual check, or by the 50-move rule.
So, that means that there is a finite number of moves in any chess game. The question is, how long is the longest possible chess game? The answer involves some math. It turns out that, after Black's 5948th move, White is able to claim a draw. Huh?
The calculation goes like this. (Pawn_Moves + Captures - Duplicates + Drawing_Interval_Grace_Period) * (Drawing_Interval), or 16*6 +30 -8 +1) * 50 = 5950. You can subtract two moves from this total because sequences of Captures/Pawn_Moves must have at least 4 alternations between the two players. Thus the number 5948.
Okay, now let's make some assumptions. As an extremely loose upper bound on the number of positions possible, allowing all illegal positions, and not differentiating between the various pieces, and noting that chessboards have 64 squares, both White and Black have 16 pieces, etc., we can now make a calculation. There are 64!/32! ways to place the pieces on the board. The (!) means "factorial", which is a shorthand mathematical way to indicate 64*63*62...3*2*1. For example, 5! = 5*4*3*2*1 = 120. So, 64!/32! = 4.8222 x 10^53. Read this as 4.8222 times 10 to the 53rd power. Keep in mind that this allows all positions, including illegal ones and positions which would clearly be impossible.
So, the absolute maximum number of games of chess = (4.8222 x 10^53) ^ 5948. Which equals...(are you ready?)
1.0516 x 10^270993
Uh, this is a very big number, folks. It is 10516 followed by 270989 zeros (I moved the decimal). Just how big is this number?
Well, there is a mathematical term for a pretty big number. It's called a googol, and it's equal to 10^100 (a one followed by 100 zeros). There is a bigger number. It's called a googolplex, and it is equal to a one followed by a googol of zeros.
Uh, okay, how big are these numbers? According to most astrophysicists, the total number of atoms in the observable Universe is somewhere between 10^72 and 10^82. While this is a mammoth number, it is far less than even a googol, never mind the mind-boggling googolplex.
Still having trouble wrapping your head around these numbers? Okay, consider this. A billion is a puny 10^9. If you counted to one billion at the rate of one number per second, how long do you think it would take you to count to a billion? The answer is 32 years. Starting to get the picture?
So, while the number of possible chess games is larger than a googol, which makes it far larger than the number of atoms calculated to be in the visible Universe, it is not larger than a googolplex.
So, chess is not infinite. But, it sure seems that way.
Click here for links to Mike's other work on Chess.com