I took your suggestion and looked it up. Without being able to say definitively, it would appear as if this is a problem that has not yet been solved. Oddly enough, between several of the websites that I came across, the number of estimated possible positions varied by over 80 orders of magnitude!!! The highest that I saw was 10^120 (10 to the 120th power) but I believe that does not take into account rules of the game and some of the other constraints that we've already addressed in this thread. Most other estimates were in the 10^43-10^49 range, which is still a very wide range of possible positions.
Any else have better luck pinning down an exact number?
Just to give some kind of idea on the size of number that is being talked about...the number of atoms in the universe is thought to be around 10^80 (correct me if this is wrong)
More on that here: http://pages.prodigy.net/jhonig/bignum/qauniver.html
Just to give some kind of idea on the size of number that is being talked about...the number of atoms in the universe is thought to be around 10^80 (correct me if this is wrong)
This is often quoted when people are talking about large numbers. And I don't know why.
Perhaps it is because peple think that this is the largest number that really "exists" in our universe or something like that. Or perhaps people want to show with this fact that it makes no sense talking about so high numbers.
But in fact the opposite is right: It just shows that there is nothing special about a=#{x|x atom in our universe}, because there are even higher numbers that describe facts we may understand (bad example, because too smal: the number of different chess positions...)
Originally posted by thire This is often quoted when people are talking about large numbers. And I don't know why.
Perhaps it is because peple think that this is the largest number that really "exists" in our universe or something like that. Or perhaps people want to show with this fact that it makes no sense talking about so high numbers.
But in fact the opposite is right: It just ...[text shortened]... ts we may understand (bad example, because too smal: the number of different chess positions...)
At the same time, I think it's reckoned by psychologists that we only have an instinctive 'feel' for the numbers 1 to about 5. If you want really big numbers, try the following:
Let F1,F2,F3... be functions with the following properties:
F1(x) = x + 1
Fn(x) = F(n-1)^x(x), where '^x' means 'compose the function with itself x times'
If you see nothing distressing about these functions, how about this one:
G(x) = Fx(x).
Point taken. Psychologically, I would say that the largest number a person has a real "feel" for is the largest number such that when they look at a collection of that many small enough objects, they can tell immediately how many there are. So probably about 20 is typical. People don't generally bear in mind that counting POSSIBILITIES almost always leads to numbers astronomically larger than numbers obtained by counting things (even "astronomically larger" is the wrong phrase there, based as it is on counting distance, or stars, or something...). Also, it is always very irksome the way people interchange "very large" and "infinite". To most people, "infinity" begins at about 3000 (about the number of stars readily visible on a clear night) 😉.
Originally posted by royalchicken So probably about 20 is typical. Originally posted by Acolyte At the same time, I think it's reckoned by psychologists that we only have an instinctive 'feel' for the numbers 1 to about 5. I think Acolyte is right: when your pack of cigarettes is at about halft empty - you don't count the cigaretts to know how many are left? People use their fingers when counting, ... I guess five is the upper bound...
Originally posted by Acolyte ...
Now work out G(4) and G(5) 🙄 The smiliy there is very well chosen!
I would agree. In a similar vein, I am also of the belief that mathematical statements etc. are discovered, not invented. However, I still maintain that larger numbers are indelibly etched in our psyches (then again, maybe I should just admit to myself that nobody else involuntarily and instinctively finds prime factorizations of licence plates on cars 😉).
I think Acolyte is right: when your pack of cigarettes is at about halft empty - you don't count the cigaretts to know how many are left? People use their fingers when counting, ... I guess five is the upper bound...
Sorry, I don't smoke 😉. I think this is all a matter of how used to dicking around with numbers one is. The star bit was a joke. Specifically, I was making fun of certain poets.
There are positions which are not breaking any rules but they are not possible (example: startposition, but the white queen on the black's place.)
so the problem is much much more comlicated if we want to figure out the nuber os different positions which may be result (one time) of a chess game.
Originally posted by thire There are positions which are not breaking any rules but they are not possible (example: startposition, but the white queen on the black's place.)
so the problem is much much more comlicated if we want to figure out the nuber os different positions which may be result (one time) of a chess game.
Yes, that was the idea. I was hoping to see if we could figure out the total number of legal positions that could result from legitimate play in a chess game. And you're right, it's absurdly complicated to try and figure it out.
Originally posted by royalchicken Sorry, I don't smoke 😉. I think this is all a matter of how used to dicking around with numbers one is. The star bit was a joke. Specifically, I was making fun of certain poets.
I was referring to what limited 'hard-wired' number sense we have, eg there is evidence to show that most 6-month-old infants believe that 1+1 does not equal 3 (although the proof that they're right is apparently far from simple 🙄)
Originally posted by Acolyte I was referring to what limited 'hard-wired' number sense we have, eg there is evidence to show that most 6-month-old infants believe that 1+1 does not equal 3 (although the proof that they're right is apparently far from simple 🙄)
Possibly we should listen to what thire said Goedel said. Because the 'hard-wired number sense' of which you speak is certainly very real, but I think it is very fundamental in that it manifests itslef through actual counting of physical objects, etc. otherwise, it would be meaningless to 6-month-olds, because the number sense of which I speak is the abstract, less fundamental type, which I think is rooted in some kind of communication, and, as such, is not possessed by infants. The difference between the two is that anyone can mentally tell you that 4927/379 = 13 (the time varies from person to person, but anyone with an abstract notion of numbers can do it). However, no one that I've heard of can determine, at sight, how many objects there are in a pile beyond maybe 20 or 30. So we need to go to slightly bigger numbers to differentiate between the two manifestations of number sense. which is probably why we wondered about the number of ways of playing a game of chess anyway.
Number of positions
24 Jan '03 03:10
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