Draw | 34 | 68% | |

Win for White | 7 | 14% | |

Win for Black | 2 | 4% | |

No idea | 7 | 14% |

Win for White ?

OR

Its a zugzwang right from the begining and hence a win for Black .

Any possiblity of this in the near future i mean 60 - 70 years. so we live until then. !!!!!!!

At some point, chess might be proven to be a win or a draw, but you will never be able to build a chess AI that can do perfect play in all legal positions. "Mate in 292" from the starting position would be awesome, though =)

/* Steinar */

and secondly in this rapidly infact blitzly changing world every thing is possible who ever think just 100 years back what we have now.

may be some revolutionary new way to store data (e.g perhaps in the form of photons) make it possible to store such a huge number of moves.

`we don't know how big is the universe `

Yes, we know how big is the universe. Making those observation based calculations is called science. Although it is very sane to question science and not to make dogma of its statements, also is very ingenuous to do the contrary (disbelieve science just because its statements are beyond everyday perception)

>and even if they claimed some thing like that then i am sure thery are wrong.

Yes what those millions scientists know? You know better....

> The size of the Universe was basically arrived at by looking at how fast the Universe has been expanding since the Big Bang and integrating out the resulting metric.

Well i don't believe this big bang theory by scientists for a very simple reason which is

why these scientists have never tell us that from where that particular star or material have come which according to them have exploded in a big bang and universe is being created.

but according to known measurements and observations, there are some educated estimations.

For example, the estimation for the number of atoms in the known universe is interesting.

It is estimated at around 10^80. That's only an estimation, BUT,

There is an upper limit: 10^100.

More than that would contradict many past observations.

Now I don't really care if those estimations are correct or will they change in the future. So I'll just compare these numbers with figures known in Chess. Because the number of chess position is what we are interested in here.

This was actually discussed before in another thread on this forum, but I'll repeat the numbers here:

There are two different numbers: the game tree complexity of Chess, and the number of possible Chess positions.

The game tree complexity is much higher, since there can be many different paths in the game tree to the same position.

In his 1950 paper, Claude Shannon have reached a lower bound for the game tree complexity of Chess. 10^120.

This is a lower bound, so the number can be much higher (there have been other bounds discovered later but I'll keep it relatively short).

As to the number of different legal Chess positions:

This is certainly much lower than the estimated number of atoms, (un)fortunately.

Many estimations here too, but the upper bound is at around 10^52 different positions.

(Shannon reached an upper bound of 10^46, but without including many positions after certain captures and promotions).

That is still a huge number, and it will not be possible within the next few dozen years to have access to the full 32 piece endgame tablebases.

If you want a really entertaining tidbit not Chess related:

In the game of Go, the number of positions could only be bound by 10^171 (and the bound cannot be too far from the actual number).

I won't even mentioned the game tree complexity...

Anyone wants to construct a perfect database for THAT :)

Those are astronomical numbers anyway, so who cares how they compare to the number of Atoms/Molecules/Quarks/Gravitons/whatever in the observable universe.

>(disbelieve science just because its statements are beyond everyday perception)

+1

> (disbelieve science just because its statements are beyond everyday perception)

> +1

No just because they don't have the proof .

We believe earth is round because we have the proof.

>I have also read in some AI book that there are so many board positions in the chess that it exceedes the number of molecules in the universe >,but i don't believe that because we don't know how big is the universe then how could we make such a statement.

Yes we don't know exactly how big the universe is but we make estimations.

If you believe in human science you have to accept this estimations or provide better one's.

From:

http://en.wikipedia.org/wiki/Observable_universe

"The number of atoms in the observable universe is around 10^80"

Also in order to solve Chess, there is no need to store all possible positions in a hard disc drive or whatever storage system.

Connect 4 7x6 and Nim 50x50x50x50 for example, are solved without storing all moves somewhere. Only using some clever rules.

> Also in order to solve Chess, there is no need to store all possible positions in a hard disc drive or whatever storage system.

> Connect 4 7x6 and Nim 50x50x50x50 for example, are solved without storing all moves somewhere. Only using some clever rules

Yes and by the way my favourite hobby is to sit infront of a chessboard for 2 -3 hours with a computer as a helper and tries to grab the sequence and secret in the chess not only by mind power but also by spirtuality !!.

**I have also read in some AI book that there are so many board positions in the chess that it exceedes the number of molecules in the universe ,but i don't believe that because we don't know how big is the universe then how could we make such a statement.**

and secondly in this rapidly infact blitzly changing world every thing is possible who ever think just 100 years back what we have now.

may be some revolutionary new way to store data (e.g perhaps in the form of photons) make it possible to store such a huge number of moves.

and secondly in this rapidly infact blitzly changing world every thing is possible who ever think just 100 years back what we have now.

may be some revolutionary new way to store data (e.g perhaps in the form of photons) make it possible to store such a huge number of moves.

Well, then let me put it this way: Certainly you know the legend of the chess board and the wheat corns?

Well, if we take the wheat production of the whole Earth in 2005, we would need the complete one-year-harvest of roughly about 2500 planets like Earth to fulfill the claim.

So it's really extremely unlikely that there will ever be something like a 32-men-tablebase. I guess in 2050 we haven't completed the 8-men bases...

No, that's definitely not true--just take 64 permute 32 and you get 4.822 x 10^53. Now, if you have a quantum computer with some sort of quantum hard drive that is able to store positions based on different up/down arrangements of electrons, this might be somewhat feasible. At first you might think, "okay, we need the number that I put above times 64 electrons times the mass of an electron--that massive of a quantum hard disk, which is 2.81 x 10^25 kilograms, or 4.7 times as massive as Earth. However, keep in mind that you can do all kinds of clever things with entangled electrons, and one could probably use various up/down combinations to make gigantic short-cuts. With 128 electrons, for example, you would be able to store far, far more than just two positions--how many more, I don't know, but I'm sure it's at least in the thousands, if you're sufficiently clever. It gets even better as you increase the number--256 electrons wouldn't double, or even quadruple the number of positions you'd be able to store. My guess is that it would at least square it, and possibly much more (it might even "factorial" it).

I would guess that we're about a century away from having a quantum "hard disk" that is capable of storing all of the positions for a 32-man tablebase. Next comes the problem of generating these positions in the first place. That would be a job for quantum computers, which are quite naturally suited to such tasks. I would guess that we will have a quantum computer that is able to generate a 32-man tablebase before we have a quantum hard disk that is capable of storing the data--but in any case, this would seem to imply that chess should be solved in a hundred years. Assuming that I'm not woefully wrong, of course...

> No, that's definitely not true--just take 64 permute 32 and you get 4.822 x 10^53

For correction the possible board position are approx 10^120

The actual number is lower than this: divide out 8! for identical pawns, another 8! for the other side's identical pawns, and do the same in dividing out 2! 6 times for each side's identical rooks, knights, and bishops. The answer then is 4.6347 * 10^42 as the number of placing the 32 chess pieces on 64 squares. This doesn't even take into account that many of these positions are either illegal or impossible to get to from the initial board setup. The best strategy for figuring out the fraction of illegal/impossible positions would be to do a simulation of randomly placing the 32 pieces on the board and see what percentage are illegal after doing this, say, 1000 times.

You might be concerned that I haven't calculated the number of ways of putting 31 pieces on the board, or 30, 29, etc. You'll find upon doing a similar calculation to above that these are smaller enough such that adding them up will show that my answer is correct to within one or two orders of magnitude. Anyway, we were talking about a 32-man tablebase, which will be the largest one by far.

> No, there are 32 pieces, and 64 squares on which to place these pieces. Thus, to calculate the number of ways to place 32 different pieces on 64 squares, you do "64 permute 32", which is 64!/32! = (64*63*62*61*...*4*3*2*1)/(32*31*30*...*4*3*2*1) = 64*63*62*...*35*34*33*32 = 482219923991114978843459072919892677776312893440000000 ~= 4.822199 * 10^53. But even this number is too high.

here is a link for possible chess moves.

http://computer.howstuffworks.com/chess1.htm

**number of ways of putting 32 chess pieces on a board with 64 squares**, which is a calculation done above that is quite straightforward and should be easy to follow. You'll also note that it matches the answer arrived at by mathematicians a long time ago (thus the motivation for doing a more precise calculation, as I knew that my 10^53 answer was far too high).

The only possibly legitimate concern you should have is that I have not added in the number of positions with 31 pieces, 30, 29, etc. Okay, that's easy enough to do: take the answer that I gave you, i.e. the 4.6347266... x 10^42, which is EXACTLY the number of ways to put 32 chess pieces on a chess board (counting illegal positions), then divide by 32, write that number down, then divide that answer by 31, write that number down, then divide that answer by 30, etc. all the day down to dividing by 3. Add all of those numbers together, and you will arrive at an UPPER limit for the number of ways of arranging between 2 and 32 pieces on a chess board. I have already done the math for you: the answer is 4.784395... x 10^42, which is only negligibly higher than my original answer. Wow, that was even closer than I thought! So now you have a precise UPPER limit for the number of ways of putting between 2 and 32 pieces on a chess board. The actual number is smaller since most of those positions are illegal.

> I checked out their report, and it's wrong--they make some big assumptions in getting to their answer

But i have a Book infront of me named "Artificial intelligence (structures and strategies for complex problem solving) " by george f luger and william a stubblefield and they also have the same number 10^120

another link

http://www.geocities.com/explorer127pl/szachy.html

**positions**is a very simple matter of combinatorics--why don't you try going through that argument? I know that I'm not the first person to produce this argument, as it's a well-known result in mathematics, though I did produce is "from scratch" as "an exercise for the reader". I don't see why you're having trouble understanding it--is there are particular step that I didn't explain well enough? If so, say something, and I'll be sure to fill in the details.

> I can assure you that 10^120 absolutely does NOT refer to the number of possible chess positions. It's possible that it refers to the number of possible chess GAMES, in which case that number is fairly reasonable. The number of chess

positionsis a very simple matter of combinatorics--why don't you try going through that argument? I know that I'm not the first person to produce this argument, as it's a well-known result in mathematics, though I did produce is "from scratch" as "an exercise for the reader". I don't see why you're having trouble understanding it--is there are particular step that I didn't explain well enough? If so, say something, and I'll be sure to fill in the details

Well i have acknowledged your explanation almost 76 minutes before your response.

I don't think you can reasonably expect me to always have unlimited leisure time to spend in front of the computer trying to respond to all points in a particular chat room.

> I checked out their report, and it's wrong--they make some big assumptions in getting to their answer, assumptions such as all positions being non-identical, assumptions that there will always be such and such number of moves in a position, etc.--and that really doesn't get at the

number of ways of putting 32 chess pieces on a board with 64 squares

Well at last i find the one which is according to your theory and now i admit that u are correct about the chess position

here is the link by the way

http://www.chessbox.de/Compu/schachzahl1b_e.html

another one

http://www.harvardmagazine.com/on-line/1102196.html

There have been many attempts, advances, and some regressions (and many articles which were later shown to be wrong)

with this issue over the years.

First of all, the calculation given a few posts ago is a bit wrong. The mistake has been done before but was acknowledged by the article writer.

So A little order into the chaos: 10^42 does not include positions occuring after promotions. It also does not include positions after some capturing moves (a little trickier to understand what captures than to understand the promotion problem).

The number of positions missing is a lot higher than the number of illegal positions that calculation contains.

So, a few better estimates have been made, but they are pretty much all between 10^42 to 10^52 (with 10^46 as a lot more probable that 10^42).

So, it's a HIGH number... whatever it is exactly

I don't want to be an ass, but I studied physics a long time ago. I don't understand why you are suddenly talking about the big bang when the question is about the size of the universe in contradistinction to its mass. No one really has a clue about what preceded the big bang. The most reasonable thing we can think of is that the big bang resulted from a black hole. As to the laws of physics, we do not have a set of coherent laws which explain the physical behaviour of the universe, it's just what we know right now. There are so many things we do not know, even at the level of simple electrons (heavy, non-elusive particles) - we all use electronic devices which are based on the applications of insights from quantum mechanics but this does not mean we understand the behaviour of electrons, they are doing things no one understands, engineers just work around it.

I don't know what you mean when you say "they are doing things no one understands"--actually, that's one thing that we understand very well! That's quantum electrodynamics, one of the first "practically exact" quantum field theories. We can figure out what the electrons are doing to as many significant digits as our measuring equipment can measure (something like 14 significant digits in many cases). You might be about to mention Feynman's quote, "nobody understands quantum mechanics"--what he means is that we have all of these mathematical laws resulting from quantum mechanics that make these astoundingly correct predictions, and we don't completely understand

**why**. Perhaps what you mean is that nobody understands the behavior of large groups of electrons, i.e. in complex atoms.

You are correct in that we don't have a complete set of coherent laws for the Universe--but we have a large enough set of laws that are good enough to explain almost anything; there are many areas where the bridges between different sets of laws are in open debate, i.e. quantum gravity, but that doesn't mean that times didn't exist when quantum gravity was relevant (and, for that matter, quantum gravity is obviously still relevant not only near the central regions of black holes, but even outside of those, as their event horizons give rise to Hawking radiation). That's a bold statement that "the most reasonable thing we can think of is that the big bang resulted from a black hole"--I think you'll find that most cosmologists would not say that, though it's certainly a possibility. The point there is that while a black hole collapsing might be sufficient, there is no reason to suspect that it's necessary, as there are plenty of other plausible pre-big bang scenarios (though that's certainly not to say "known", as of the whole host of plausible scenarios, we have no idea which one gave birth to

**our**Universe).

Back to chess, chess is most likely a draw with best play. If it was otherwise, I think the perfect game would have been discovered by now. Every novelty that has ever been put forth till now has been neutralized ... so that would lead me to guess that chess is a draw.

Hey, don't ignore me yet! I'm saying it again: The universe is infinite.

Now, the matter on the universe is certainly finite, and then people will call that universe, and study its length and then reach its size, and calculate for how many years has it been there, etc.

However, I'm also including the whole empty "space" that surrounds our "universe", to where our "universe" is expanding, a part of it.

Imagine a balloon that can get infinitely bigger with air, and that is the only thing that exists, if we're inside of it, and we have a word that includes "everything" (the universe), then getting the size of the balloon isn't going to tell us the size of the empty space around it.

If the universe is the balloon then what's the outside of it? If I can tele-port to "outside" the known universe, then I'm not on the universe anymore? (Of course I'm still on the universe! That's why I'm counting the empty space around it).

The multiverse theory says that maybe there are other universes out there, but their light hasn't reached us yet...

"The multiverse theory says that maybe there are other universes out there, but their light hasn't reached us yet..."

The multiverse ideas say something quite different unless some popular author has sufficiently destroyed the true idea of which physicists speak so as to make it unrecognizable. When one speaks of the multiverse, one

**usually**speaks of this idea resulting in the many-worlds interpretation of quantum mechanics, in which one says that at each instant, our Universe "takes the path" in reality of one of nearly infinite possible Universes. However, this has relatively little meaning in terms of any sort of topology, though it's possible that this interpretation of quantum mechanics might have slightly different consequences in some string theories than the "canonical" Copenhagen interpretation--I would need to check that out. You will also occasionally hear use of multiverse in referring to the group of Universes that came about as a result of the process that gave birth to our own Universe in the group of inflation theories known as "eternal" inflation, in which you have some perturbation in this "empty 'space'" of which you speak (this stuff is extremely unstable, I should mention), causing a Big Bang, which also causes more perturbation, causing another Big Bang, which causes more...etc. until you have a whole host of different Universes in this space (are you worried about collisions between Universes and thus an end? Physicists in the 1980's were, but they eventually determined that the dimensionality of this "outside space" was so complex that such collisions would probably never happen; string theorists have revived the idea of these sorts of collisions as being the thing that caused what we think of as the "true" Big Bang that gave birth to our present Universe.). In terms of predictive capability, eternal inflation theories make specific predictions about the power spectrum of the cosmic microwave background, and if my understanding is correct, they are the inflation theories that are currently closest in their predictions.

Even if this "extremely unstable empty 'space'" cannot be considered as part of our universe, do you agree that it may be infinite? (That the empty 'space' can be infinite?).

**VERY**infinite, so to speak--in fact, more infinite than one could possibly imagine, as the dimensionality of such a space is quite unknown (and definitely greater than three) :-). However, it does not have any resemblance to the sort of space in which our superclusters, galaxies, stars, planets, people, and groundhogs inhabit.

Then if one Big Bang happened where we are now, and the 'space' is very infinite, I bet the same requirements to have a Big Bang happened somewhere else as well (In this 'space'), and I don't see why wouldn't it happen several times at several places.

So, I stay on my idea that there are plenty of universes around, even if they are separated by an infinite distance and will never reach each other.

The point is, that once we find a way to surpass the upper limit of the speed at which information can travel (As easy as stopping the universe's expansion, I guess), we can communicate with the other universes, and share the 32 men chess tablebases, since it doesn't matter how big they are, we'd get enough universes to store them :)

The number of 32 pieces positions is clearly smaller than the number of 31 pieces positions .

with 32 pieces you know that there are no captures so in every file there are 2 pawns and the number of options for pawns is only 15^8

You also know that there are no promoted pawns and you have not these restiction with 31 pieces.

number of legal positions with 32 pieces is smaller than 15^8*48*47*46*45*(44*43/2)*(42*41/2)*(40*39/2)*(38*37/2)*(36*35/2)*(34*33/2)

Uri

I do not divide by 32 if I remove one piece.

It does not work like that.

Even if I remove only one pawn I have more positions with 15 pawns then with 16 pawns even if I assume that the pawn is captured not by a pawn but by a piece.

There are 15^8 possible pawn structures with 16 pawns(15 possibilities for every file).

If I remove one white pawn I need to choose a file of the pawn that I remove(8 possibilities) and later I need to choose square of the opponent pawn(6 possibilities)

In the rest of the 7 files I have 15 possibilities.

It means that I have 48*15^7 possible pawn structures even without including the case that there are 2 pawns on the same file that can happen in case of pawn capture pawn.

If I have only 15 pawns on the board

I also have more 49 squares to put the other pieces so I have more possibilities to put the other pieces relative to the case of 16 pawns.

Uri

So a corollary of your idea would be that the largest number of possible positions would be with one piece or zero pieces. Of course, this is rubbish--there are 64 possible positions with one piece, per piece. The error in your idea is that you're not taking into account the fact that 64 permute 32 is much larger than 64 permute 31 (that's why I can take my answer and divide by 32 to get an approximate result). And notice that I'm saying 64

**permute**32, not 64

**combine**32--this takes into account the fact that the chess pieces are distinguishable. However, for each indistinguishable chess piece, we must divide by the factorial of the number of such identical chess pieces. Thus, when we remove a piece, we do the same thing as with 64 permute 32 and then divide by factorials of the number of all indistinguishable chess pieces, except this time with 64 permute 31. Thus, the very

**WORST**thing we can do is remove a pawn because now instead of dividing by 8! = 40320, we're dividing by 7! = 5040; if we remove some other piece, we simply go from dividing by 2! = 2 or dividing by 1! = 1, which doesn't change things much.

Let's count from the very beginning now: first, how to arrange 32 chess pieces. We start off with 64 permute 32 = 4.822199 x 10^53. Now we must divide by 8! twice since the pawns for each side are indistinguishable, and this now gives 2.966225 x 10^44. Now we must divide by 2! six times, since the rooks, knights, and bishops for each side are indistinguishable (to take into account that bishops must be on opposite color squares, we would actually divide again since each can only be on one of 32 possible squares, but I won't do that here). Doing this yields 4.6347 x 10^44 as the number of ways of putting 32 chess pieces on 64 squares.

Now for the number of ways of putting 31 chess pieces on 64 squares. We start off with 64 permute 31 = 64!/(64 - 31)! = 64!/33! = 1.46 x 10^52. You'll agree that the very worst thing we can do is remove a pawn--this will give by far the largest number of positions. Now here is the crucial point--by doing it this way with the permutations, I'm already taking into account all of the possible ways of removing a pawn--it's just one of the theorems in combinatorics (with the proof left as an exercise for the reader :-) --actually, the proof is relatively straightforward). Thus, we take my previous answer and divide first by 8! (since one side still has all eight pawns) and now by 7!, and then again six times by 2!, yielding 1.12357 x 10^44 as the number of ways of putting 31 chess pieces on 64 squares. Note that it doesn't matter which color is the pawn--that's already taken into account by the fact that I already double-counted by saying "32 chess pieces" at the beginning instead of "16 chess pieces of each color". You might come back and say, "well we need to add in what happens if we remove a knight, a bishop, or a rook, etc.". For each of those, the answer is smaller by AT LEAST a factor of four, and with removing the queen, the answer is smaller by a factor of 8. Thus, adding these in gives an answer that is still smaller than what we had with 32 pieces. In addition, I'm quite certain that we're at least double-counting in here somewhere--at least, that's what my combinatoric gut tells me.

Either way, I'm sure you'll agree that the number of possible chess positions is 10^(40+n), where n < 6, and thus certainly not the ridiculous number of 10^120 that was being quoted earlier (though that might be the number of possible branches in a search tree), which was the whole point of this.

When all the pieces are different then it is correct that more pieces give more possibilities but when part of the pieces are identical it is not always correct.

Imagine the case of 64 queens(you have only one possibility to put them when you have

more possibility to put 32 queens on the board).

2)I obviously did not consider illegal positions of 32 pieces when white has pawns at a2 and a3

in the calculation in the previous post when your calculation count them as position.

A file has only 15 possibilities for legal pawn structure in that file assuming no pawns were captured:

namely:

1)white a2 black a3

2)white a2 black a4

3)white a2 black a5

4)white a2 black a6

5)white a2 black a7

6)white a3 black a4

7)white a3 black a5

8)white a3 black a6

9)white a3 black a7

10)white a4 black a5

11)white a4 black a6

12)white a4 black a7

13)white a5 black a6

14)white a5 black a7

15)white a6 black a7

Same is for every file and it means 15^8 legal pawn structures.

With 15 pawns you have more possible pawn structures even if you only remove a pawn because the file that you remove a pawn from it has 6 possibilities and you have 8 possibility to choose a file to remove a pawn from it.

Uri

**legal**positions with 31 pieces than with 32 pieces. However, it would still seem more likely that you'd have the largest number of legal positions with 32 pieces than with any other number of pieces, whether greater or less--think Pascal's triangle. But if we're talking about

**legal**pawn structures, I'm guessing that not only is my 4 x 10^42 number a large overestimate, but it's a VERY large overestimate.

The number of legal positions with 32 pieces is far smaller than 10^40 but the number of legal positions with less pieces when pieces may promote to every piece(and pawn capture pawn can allow 3 promotions) may be higher than 10^40(I do not know if it is higher than 10^40)

Uri

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