a multiple of 10^32 moves in a game?
If the players could average a second to play each move then such a game would still take far more seconds than the total lifetimes of everybody who has ever lived on earth including those living now.
By the way Martin - if you happen to prefer proper superscript math text for copy/paste then its available through Chatgpt.
The so-called mathematica text. You have to push gpt to stop it stubbing its toe though. Copilot on the other hand insisted on stubbing its toe every time.
Anyway - with Chatgpt there's an alternative to the method of ^ symbol with no subscripts or superscripts. Is available.
I'd be curious to know how they address that in the schools.
In theory the long-used proper superscripting and subscripting should be made user-friendly available on all good keyboard displays without a lot of shift key and 'ASCII' buzztalk and buzz-presses or better yet minimal or no 'buzz-presses'.
Chess will never be solved, here's why
It would be longer than the age of the universe. But allowing any repetition in those "games" simply multiplies the length if every position can be repeated 4 times. I'm curious what the number is with no repetition at all, if both players actively tried to avoid it at any point during the game. Maybe more like 10^31?
Nobody knows the age of the universe.
Including because that age is probably infinite.
Those who want a finite age of the universe will make circular arguments for it and will want just One big bang in one location at one time.
But infinite big bangs in infinite space and time in the past and present and future take care of all the questions about entropy and what made it and what did it come from because it always was.
-----------------------------
There's no absolute 'something from nothing' no matter how badly people want that.
And they do want it. There's an enormous demand for it throughout history.
Its a commodity.
But website rules mean we're not allowed to discuss the nature of that demand here.
And I'm not surprised that EE would try to work 'age of the universe' into the discussion.
Just a tipoff to all chess.com members unfamiliar with EE - he has a tendency to delete all his posts every night - in various forums. Month in month out.
If all unfamiliar members are not tipped off - then they don't get it and he gets an advantage he should not get.
I was told when working on my math degree (having only in the last few months taken up chess) there exists a Shannon number that says the lower bound of games is 10^120. ...
You were told wrong (though that number is certainly a lower bound).
Shannon never said any such thing. See here (p.4) for what he actually said about the number 10¹²⁰.
The number of possible chess games depends on the version of chess.
Until 2017 the 75 move and 5-fold repetition rules were not in force in any version so there was nothing to limit the number of moves in a chess game (the 50 move and 3-fold repetition rules terminate a game only if they're claimed) so the number of possible chess games in versions prior to 2017 or FIDE basic rules chess since 2017 is infinite. It's easy to show that's ℵ₀ for finite games and ב₁ for games of length ω (maximum sensible length).
The number of possible chess games under current FIDE competition rules is claimed here to be between 10²⁹²⁴¹ and 10³⁴⁰⁸². (At first sight the upper bound appears to be surprisingly low to me, so it may be advisable to check the details of the working.)
Somebody unknown, subsequent to Shannon's paper, declared the number of chess games to be 10¹²⁰ and the false assertion has spread through the internet like dandelions on your back lawn. At least you gave the figure only as a lower bound.
Martin as always - very well informed.
Hi guys. Resume?? 😂
Nobody knows the age of the universe.
Including because that age is probably infinite.
Those who want a finite age of the universe will make circular arguments for it and will want just One big bang in one location at one time.
But infinite big bangs in infinite space and time in the past and present and future take care of all the questions about entropy and what made it and what did it come from because it always was.
-----------------------------
There's no absolute 'something from nothing' no matter how badly people want that.
And they do want it. There's an enormous demand for it throughout history.
Its a commodity.
But website rules mean we're not allowed to discuss the nature of that demand here.
And I'm not surprised that EE would try to work 'age of the universe' into the discussion.
Just a tipoff to all chess.com members unfamiliar with EE - he has a tendency to delete all his posts every night - in various forums. Month in month out.
If all unfamiliar members are not tipped off - then they don't get it and he gets an advantage he should not get.
Ya see MAR, this is the type of lunacy I was talking about that other time, this guy takes totally neutral and on-topic discussion comments and tries to twist them into some hidden agenda/conspiracy that I'm trying to get threads locked or derail the discussion. He does this in every thread I post in, like a stalker. And these universe comments are equally ridiculous (even though the topic was chess moves).
Nobody knows the age of the universe.
Including because that age is probably infinite.
Those who want a finite age of the universe will make circular arguments for it and will want just One big bang in one location at one time.
But infinite big bangs in infinite space and time in the past and present and future take care of all the questions about entropy and what made it and what did it come from because it always was.
-----------------------------
There's no absolute 'something from nothing' no matter how badly people want that.
And they do want it. There's an enormous demand for it throughout history.
Its a commodity.
But website rules mean we're not allowed to discuss the nature of that demand here.
And I'm not surprised that EE would try to work 'age of the universe' into the discussion.
Just a tipoff to all chess.com members unfamiliar with EE - he has a tendency to delete all his posts every night - in various forums. Month in month out.
If all unfamiliar members are not tipped off - then they don't get it and he gets an advantage he should not get.
Ya see MAR, this is the type of lunacy I was talking about that other time, this guy takes totally neutral and on-topic discussion comments and tries to twist them into some hidden agenda/conspiracy that I'm trying to get threads locked or derail the discussion. He does this in every thread I post in, like a stalker. And these universe comments are equally ridiculous (even though the topic was chess moves).
I would be reporting him if he harassed me that way. I'd like it if he stopped doing that too. It makes the discussion uncomfortable for everybody.
What is the number of unique 40 song playlists that can be drawn from a pool of 100 songs? Song order doesn't matter. Is there a formula to figure that out for any size playlist to be drawn from any number of songs?
That recent post by @shadowtanuki is not relevant to the forum subject.
But earlier EE complained about 'spam'.
Is there to be a double standard as to non-forum subjects?
Suggestion: EE and shadowtanuki should watch it about accusing people of 'spamming'.
"trying to get threads locked"
I never said that. That's slander.
But interesting that somebody else mentioned that?
---------------------------
And there's no reason why members should not be tipped off that EE is known to constantly delete his posts in various forums. Every night.
If he doesn't want that to get out - why doesn't he stop that practice?
Earlier he complained about 'spam' but just now started talking about 'age of the universe' - a controversial subject not relevant to the forum subject.
and just now from EE
"And these universe comments are equally ridiculous"
but he made one earlier.
He complains - about that which he does.
It would be longer than the age of the universe. But allowing any repetition in those "games" simply multiplies the length if every position can be repeated 4 times. I'm curious what the number is with no repetition at all, if both players actively tried to avoid it at any point during the game. Maybe more like 10^31?
before he can delete it. After him complaining about 'universe comments'.
-------------------------
And now - the Tromp number doesn't refer to repeated positions.
The games numbers and permutations do.
The Tromp number does not.
So there's no cutdown to 10^31.
Yes, an odd comment. Most people would not think the number of legal positions would magically shrink if people tried to avoid repeating them in a game.
Yes, an odd comment. Most people would not think the number of legal positions would magically shrink if people tried to avoid repeating them in a game.
It seems there is no true escaping from posters that arbitrarily eliminate a dozen (or two) orders of magnitude on a whim.
Ya see MAR, this is the type of lunacy I was talking about that other time, this guy takes totally neutral and on-topic discussion comments and tries to twist them into some hidden agenda/conspiracy that I'm trying to get threads locked or derail the discussion. He does this in every thread I post in, like a stalker. And these universe comments are equally ridiculous (even though the topic was chess moves).
I would be reporting him if he harassed me that way. I'd like it if he stopped doing that too. It makes the discussion uncomfortable for everybody.
I would agree that reporting someone if they continue to harass you once you have disengaged is warranted. But that requires actual, demonstrable and consistent disengagement.
I don't like it either, for the record. When I make pointed observations, they tend to be one and done.
For example, my other post...I could harp on the fact that Endgame and Tygxc share the issue of dismissing a dozen orders of magnitude without a thought, repeating this every single day for weeks on end even when everybody else has stopped talking about it. That is pointless, though.
Win, lose, or tie the battle and move on, without rancor.
Sorry, that was unintentional.
Ya see MAR, this is the type of lunacy I was talking about that other time, this guy takes totally neutral and on-topic discussion comments and tries to twist them into some hidden agenda/conspiracy that I'm trying to get threads locked or derail the discussion. He does this in every thread I post in, like a stalker. And these universe comments are equally ridiculous (even though the topic was chess moves).
I would be reporting him if he harassed me that way. I'd like it if he stopped doing that too. It makes the discussion uncomfortable for everybody.
I would agree that reporting someone if they continue to harass you once you have disengaged is warranted. But that requires actual, demonstrable and consistent disengagement.
I don't like it either, for the record. When I make pointed observations, they tend to be one and done.
For example, my other post...I could harp on the fact that Endgame and Tygxc share the issue of dismissing a dozen orders of magnitude without a thought, and I could repeat this every single day for weeks on end. But that is pointless.
Except that tygxc constantly repeated it.
And so was constantly talked back to about it.
When EE deletes posts people keep asking
'what are you referring to'?
At at least one point a moderator told me there was a problem because EE's constant deleting of his posts left at least one person with more than five consecutive posts. No sign that EE is 'disengaging' on that.
-------------------------------------
This forum has no opening poster because that poster cancelled his account after making this forum.
That means there's no such poster here to block anybody spamming.
Is relevant.
Worth mentioning from time to time.
So that people know.
----------------------------------
chess 'will never be solved' but every chess game could be regarded as an attempt to do exactly that.
Except that tygxc constantly repeated it.
And so was constantly talked back to about it.
When EE deletes posts people keep asking
'what are you referring to'?
At at least one point a moderator told me there was a problem because EE's constant deleting of his posts left at least one person with more than five consecutive posts.
-------------------------------------
This forum has no opening poster because that poster cancelled his account after making this forum.
That means there's no such poster here to block anybody spamming.
Is relevant.
Worth mentioning from time to time.
So that people know.
From time to time is good, yes.
If Tygxc ever does come back for another round of the same old reduction of 10^44 down to 10^17, then he will end up facing the same refutations all over again, whether by new posters or old. No point in worrying about it until then.
Regarding the reduction of that 5 x 10⁴⁴ number that's something they're trying to do by tablebasing. Starting from just two kings on the board and then adding.
But even just 8 pieces on the board is daunting.
They've been trying to solve all positions with just eight men for years.
And now it occurs to me that to get information on such tablebasing maybe Copilot or Chatgpt would be useful.
Tried it just now. Chatgpt was not user-friendly on it.
But after being asked about the tablebase projects Copilot was very obliging.
Although I had to prompt it to get it to talk about the Syzygy project.
For a while Copilot did okay.
Stated it took the project about five months to solve 4x 10^14 legal positions to solve all chess positions with seven pieces on board.
Which worked out to an average of about 32 million positions per second.
--------------------------------------------
Now the tablebase projects have not 'solved' for 8 pieces on board.
But they do have an estimated upper bound for the total number of legal positions with 8 pieces.
About 2.79 trillion positions.
So I then asked Copilot to compute the time needed to solve all of those at the already-computed rate of 32 million positions per second.
It came up with 24 hours.
After already admitting that the project has been working on 8 pieces for three years.
-------------------------------------
I pointed out the contradiction but as is often the case with AI it had no good explanation.
But now I see one of its flaws.
Its report on the number of 8 piece positions is less than the number for 7.
Which couldn't be.
----------------------
I confronted it and got this:
"Yeah, I get what you’re saying. The number of total positions indeed increases with more pieces. For seven pieces, the number should be 423 trillion, but when you add another piece to make it eight, it shouldn't drop to 2.79 trillion, as I mentioned.
Let me correct that: The estimated number of legal positions for 8 pieces is much higher, reaching into the quadrillions. The initial value I gave you was incorrect."
Tablebases solves aren't exactly the same in that respect as not every single literal combination of that number of pieces or less has to be tested. For example if it was determined that 3 rooks can always beat 3 knights, then it is automatically known that 4+ rooks could beat the same number of knights. I once had a burning curiosity about this with queens vs knights, and I literally had stockfish play itself (not as accurate as a tablebase) in all 90 possible combinations of queens vs knights. I recorded the results in a table I may have posted in one of my earlier threads last year. At a certain point an increasing number of knights draws a fixed number of queens, and then after that, eventually turn into a win against the queens. Once say, 6 or 7 knights beat 2 queens, any higher number is also automatically a win. Once, say, 3 queens beat 4-5 knights, no higher number of queens even needs to be tested. And I found the drawing threshold is usually a fine line, meaning at a certain point adding one more pieces changes it from a loss to a draw, and adding 1 more after that bumps it up to a win. Very rarely are 2 different combinations consecutive draws. Then I did a little basic math and found that a ratio of 2.5 to 2.6 knights, is needed to hold a draw against x queens. Now obviously 15+ piece tablebases are impractical. But this logic and research method can be reduced down to smaller piece counts. For example a rook and knight usually beat 2 knights. Therefore adding any other pieces to the rook side doesn't need to be developed into a tablebase (basic engine analysis can take into account immediate forks/cramped positions..etc). That's what I was talking about with exchange-down conversion tablebases, knowing what piece combinations will almost always cancel out down to an already known tablebase/win draw. And such transitions/conversions can be handled by a conventional chess program like stockfish, and then the tablebases can take it from there after that point.
I should add that I went back to Copilot and made it correct itself.
And it corrected to 4.84 quadrillion positions for 8 pieces.
Then I made it divide in the 32 million positions per second speed.
And it came up with 4.8 years to solve for 8 pieces.
Which is plausible since the project has been on it for 3 years already.
--------------------------
But one should not assume this means that 8 pieces will be solved in 1.8 years.
As more pieces are added - complexity increases.
So that 32 million positions solved per second might decrease substantially with time.
It could be looked into as to how the speed has varied so far as each piece is added.
But whatever the pattern - that doesn't mean it will or won't extrapolate.
I was told when working on my math degree (having only in the last few months taken up chess) there exists a Shannon number that says the lower bound of games is 10^120. ...
You were told wrong (though that number is certainly a lower bound).
Shannon never said any such thing. See here (p.4) for what he actually said about the number 10¹²⁰.
The number of possible chess games depends on the version of chess.
Until 2017 the 75 move and 5-fold repetition rules were not in force in any version so there was nothing to limit the number of moves in a chess game (the 50 move and 3-fold repetition rules terminate a game only if they're claimed) so the number of possible chess games in versions prior to 2017 or FIDE basic rules chess since 2017 is infinite. It's easy to show that's ℵ₀ for finite games and ב₁ for games of length ω (maximum sensible length).
The number of possible chess games under current FIDE competition rules is claimed here to be between 10²⁹²⁴¹ and 10³⁴⁰⁸². (At first sight the upper bound appears to be surprisingly low to me, so it may be advisable to check the details of the working.)
Somebody unknown, subsequent to Shannon's paper, declared the number of chess games to be 10¹²⁰ and the false assertion has spread through the internet like dandelions on your back lawn. At least you gave the figure only as a lower bound.
I wasn't even including any repetition at all in my thinking about it. Not even 2 fold. Not even triangulation pseudo-repeating positions. I was careful to not include any repetitions in that example game lol. The real interesting thing is removing the 50 move rule from consideration! With it the longest possible game is set at something like 5300 or 5500 moves due to a limited number of pawns moves and captures. The number of possible games is still beyond astronomically high but without the 50 move rule, the more interesting question becomes how long a chess game can continue before repetition inevitably occurs due to the finite number of squares and pieces. Is that 10^50 or something on its own, the longest length of a game without the 50 move rule?
As mentioned neither the 50 move rule nor 3-fold repetition rule limit the length of games under FIDE rules because the players are not forced to claim them. The length of games under FIDE competition rules has been limited since 2017 by the 75 move and 5-fold repetition rules which apply without any claim by the players.
The figure of 10^50 is far too high for a game played under FIDE basic rules with the addition of a mandatory 5-fold repetition rule. Tromp gives an upper bound on the number of legal basic rules positions (which correspond with equivalence classes of positions considered the same under FIDE art 9.2.3) of n=8726713169886222032347729969256422370854716254. Then a game of length >= 4n+1 would necessarily contain a 5-fold repetition. That gives 3.5x10^32 as an upper bound on the longest possible game under such rules.
(I don't have any good idea of what the actual maximal length would be, but the maximum for FIDE competition rule is 8848.5, which would also be an upper bound for the game specified.)