Probability in chess • page 3/3 • General Chess Discussion • lichess.org

Hmm...ok how many possible ELIGIBLE POSITIONS are there which could be reached in a chess game

"John Tromp estimated the number of legal chess positions with a 95% confidence level at 4.48 * 10^44 +- 0.37 * 10^44, based on an efficiently computable bijection between integers and chess positions."

So about 10^44.

https://en.wikipedia.org/wiki/Shannon_number#Accurate_Estimates

@Aj123673 said in #18: > Hmm...ok how many possible ELIGIBLE POSITIONS are there which could be reached in a chess game "John Tromp estimated the number of legal chess positions with a 95% confidence level at 4.48 * 10^44 +- 0.37 * 10^44, based on an efficiently computable bijection between integers and chess positions." So about 10^44. https://en.wikipedia.org/wiki/Shannon_number#Accurate_Estimates

dboing

#22

hmmm. i should read this thread... read the op. It seems to be put in some finite set combinatorics universe. I would suggest that once one is not afraid of the word probability any more, that we might include plunging into bigger spaces and asking infinitesimal questions too (and back up to finite at some point, but the passage into bigger might be helpful). Think markov decision processes perhaps, for those who like alpha-zero and leela implementations, that would be the container of their mathematics. Probability does not mean clueless. it can also be made very compact support. like a dirac distribution.....

now saving a link to this thread on my puter for reading backlog in some folder about underlying notions in chess theory.
Good title and reasonable op restriction to start with (let the others fly off the handle, or just me).

hmmm. i should read this thread... read the op. It seems to be put in some finite set combinatorics universe. I would suggest that once one is not afraid of the word probability any more, that we might include plunging into bigger spaces and asking infinitesimal questions too (and back up to finite at some point, but the passage into bigger might be helpful). Think markov decision processes perhaps, for those who like alpha-zero and leela implementations, that would be the container of their mathematics. Probability does not mean clueless. it can also be made very compact support. like a dirac distribution..... now saving a link to this thread on my puter for reading backlog in some folder about underlying notions in chess theory. Good title and reasonable op restriction to start with (let the others fly off the handle, or just me).

edot12345

#23

so the longest possible chess game can last 5,870 moves, I won't even try guessing how many possiblities there are

dboing

edited

#24

now how many patterns of "similar" positions could be estimated from such a big set of different positions? Not asking how many can a human learn, although that is a valid question, and has been estimated based on linguistic experiments (if i am not mistaken), but maybe chunks, patterns, and groups of similar but different positions are not the same kind of position subsets.

on the subject of game depth. there have been computer experiments involving notions of branching degrees, that show that one needs the 50 or 70 moves rule to get finite (but still very big game depths). without that such an N move rule, it could be infinite depth (even if postions are finite, but huge).

The N fold rule not being about path repetition only, one can take as long as needed to repeat a position.
Perhaps best play might argue against previous points. and i may have misunderstood past forum thread posts pointing to such computer experiments. Anyone could make confirm or infirm, or fix the last 2 paragraphs or mine?

now how many patterns of "similar" positions could be estimated from such a big set of different positions? Not asking how many can a human learn, although that is a valid question, and has been estimated based on linguistic experiments (if i am not mistaken), but maybe chunks, patterns, and groups of similar but different positions are not the same kind of position subsets. on the subject of game depth. there have been computer experiments involving notions of branching degrees, that show that one needs the 50 or 70 moves rule to get finite (but still very big game depths). without that such an N move rule, it could be infinite depth (even if postions are finite, but huge). The N fold rule not being about path repetition only, one can take as long as needed to repeat a position. Perhaps best play might argue against previous points. and i may have misunderstood past forum thread posts pointing to such computer experiments. Anyone could make confirm or infirm, or fix the last 2 paragraphs or mine?

dboing

#25

I think that I was meaning infinite=order of magnitude of number of legal positions, for my N move rule versus N fold rule for termination of games as bounding argument.

not same order of magnitude, the N move rule induced bound happens way lower than that approx . 10^40 related finiteness argument for paths containing N fold repeats.

I just wanted to fix my understanding of what the computer experiment may have been about in focusing on 50 move rule.

I think that I was meaning infinite=order of magnitude of number of legal positions, for my N move rule versus N fold rule for termination of games as bounding argument. not same order of magnitude, the N move rule induced bound happens way lower than that approx . 10^40 related finiteness argument for paths containing N fold repeats. I just wanted to fix my understanding of what the computer experiment may have been about in focusing on 50 move rule.