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There is a maxim that says there are more games of Chess than there are atoms in the observable universe. This may or may not be true... the mathematicians that came up with those numbers didn't really understand chess play.
If you assume "normal" play, then there are between 1 and about 3 possible, reasonable moved to be made. Yeah, you could move some crazy move, but that's going to end the game pretty fast... I'm not counting that game.
So the question becomes, what is the average number of "reasonable" moves available to a player during a game? If we assume a 50 move limit (as the original estimates made) and we don't bother thinking too much about transpositions (ending up in the same game positions through a different series of moves), then we might arrive at a number of possible game.
So what is your estimate of the number of reasonable moves on an average turn?
... by which I mean the number of reasonable games.
I absolutely agree that we couldn't currently store them. It's too big, but the number of "reasonable games" has to be much lower than the number non-players come up with. Reasonable games are games where both players are making sensible moves and many times those are forced.
I think you need to define "reasonable". When is it reasonable to sacrifice a Queen, for example, and when not. As with beauty, I suspect reasonable is in the eye of the beholder. We all agree it's a fantastically humongous number. However, I think with today's technology, most of these reasonable moves and games could be stored.
If the average is 2 (which is probably low), it only takes 10 moves (20 plies) to reach a million plausible unique games. At 20 moves, we're already at a trillion plausible unique games.
However, I think with today's technology, most of these reasonable moves and games could be stored.
The number of reasonable games, even by very conservative standards, is going to be very large, within an order of magnitude or so of the number of atoms on Earth. So you'll need a second planet handy if you plan on storing all these games somewhere.
Let's assume to start we have, on average, 2.5 reasonable moves in any given position. That's a ball park number I'm guessing from seeing situations in various games where there are upwards to 5 or 6 ideas to situations where there is literally 1 legal or "reasonable" move.
We know that on average there are about 40 moves in an average chess match.
Using these two numbers I could say something like 2.5^40 =
Now that's a big number. That's 8 quadrillion.
@neoliminal, you're off by a huge margin. There are about 40 moves in a typical game, but that's 40 moves for each side. Therefore, you need 2.5^80, not 2.5^40.
That's 68,422,776,578,360,208,541,197,733,559,078, or 68 nonillion.
Well that is a huge error on my part. Thanks for pointing it out. And that is a huge number of games!
6.8x10^31 vs 10^120...
Now we just need a "visual" representation of 6.8x10^31. Any thoughts?
The difference (10^120-2.5^80) is:
9.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999993157722342163979145880226644092206390233095986931075333217440020069379479072946281803524470888078212738037109375 × 10^119
Well, granted. I was referring to the cool mental picture that 10^120>10^80~atoms in the universe. 2.5^80 is smaller than 10^120 but it is still big, so!...do we know of anything that "is" 2.5^80 to help us visualize the number?
Nothing comes up from a casual google search. :-\
2.5^80 or the equivalent 6.8 x 10^31 is difficult to visualize. However, if you had a computer capable of processing one billion positions per second you would need approximately 160,000 times the age of the universe to process them all. This is why chess won't be mathematically solved in our lifetime though if AlphaZero or another engine reaches 3600 Elo many believe that would be a good approximation.
Another representation of this number would be the number of cubes of sand 1 mm on a side packed into a cube 25,000 miles (rounded) on each side.
Another representation is the number of molecules in 2 million liters of water, which would fill a cube 12.7 meters (41.6 feet) on a side.
Hope this helps.
Yes, that's quite nice. Thank you! I found this gem on Google with one of your examples:
"sending approximately 2 million liters of water per second over Niagara Falls"
My teacher told me one move per side. Does this help?