Yes - badly worded.
How many different chess positions are there?
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Apr 7, 2022
0
#363
I meant my sentence. "What is empty, exactly?" might have been an improvement.
But at least I didn't say, 'What exactly is "empty"?'.
MARattigan wrote:
FREEtheBASE2018 wrote:
I was asking finally in the simplest case, how many dispositions are possible, and to make it even easier with the 32 pieces and allowing King side by side
It looks simple by the first thought, but even with that definition, it is already very complex
That would be B².K.Q.R.N.P
where
B=32.31 ways of placing the two light squared bishops (ditto dark squared)
K=60.59 ways of placing the kings after placing bishops
Q=58.57 ways of placing the queens after placing the previous
R=(56.55)/2 x (54.53)/2 ways of placing the rooks after placing the previous
N=(52.51)/2 x (50.49)/2 ways of placing the knights after placing the previous
P=⁴⁸C₈ x ⁴⁰C₈ ways of placing the pawns after placing the previous
i.e about 1.27 x 10³⁹
Am not finish with you, my keyboard doesn't react anymore, and I don't like to type on the visual keyboard.
#364
"What exactly is empty?"
++ The original post #1 above has just a title and no text.
"the latest estimate, accurate to 2 digits, is 4.8 x 10^44 legal chess positions"
++ Yes, that is correct. However, his 55958 samples found legal contain multiple underpromotions to pieces not previously captured and thus cannot occur in a reasonable game between humans or engines. Promotions to pieces not previously captured happen occasionally, usually to a 3rd or 4th queen. Underpromotions happen occasionally, usually to a knight. Multiple underpromotions to pieces not previously captured do not happen in reasonable games between humans or engines, say a game with > 50% accuracy.
So the positions are legal, but are no chess positions in the narrower sense.
There are even scientific papers on this in the field of psychology.
A group of chess masters and a control group of non-chessplayers were briefly shown positions on a chess board and were then asked to reproduce that position on a chess board.
When the positions came from real games between humans or engines, then the chess masters were much much better at this task than the control group.
When the positions were random, like the 55958 samples of Tromp, then the chess masters performed no better than the control group.
"Gourion's paper is entitled "An upper bound for the number of chess diagrams
without promotion" and that is what he gives."
++ A diagram is the same as a position, but with the side to move, en passant and castling rights not included. 'Without promotion' is short for without promotion to pieces not previously captured. That is more closely related to the number of chess positions as can occur in a reasonable game between humans or engines. This count excludes some sensible positions with 3 or 4 queens. This count includes many positions that cannot occur in a reasonable game between humans or engines, say a game with > 50% accuracy.
"Competition rules allow a draw claim under 9.2 when three positions that are considered the same (for the purposes of that rule and 9.6.1) have occurred on separate occasions."
++ Yes, indeed: From you beloved competition rules:
"9.2.1
The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not necessarily by a repetition of moves)"
When two positions are the same, then these are not different.
So they do not count as different positions.
The question of the original poster was: "How many different chess positions are there?"
Apr 8, 2022
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#366
tygxc wrote:
#364
"What exactly is empty?"
++ The original post #1 above has just a title and no text.
Text is difficult to decipher because it's almost the same as the background colour. Just select it to get readable text.
"the latest estimate, accurate to 2 digits, is 4.8 x 10^44 legal chess positions"
++ Yes, that is correct. However, his 55958 samples found legal contain multiple underpromotions to pieces not previously captured and thus cannot occur in a reasonable game between humans or engines. Promotions to pieces not previously captured happen occasionally, usually to a 3rd or 4th queen. Underpromotions happen occasionally, usually to a knight. Multiple underpromotions to pieces not previously captured do not happen in reasonable games between humans or engines, say a game with > 50% accuracy.
So the positions are legal, but are no chess positions in the narrower sense.
The "narrower sense" is obviously not what OP is asking about. Impossible to count without a reasonable definition of "reasonable".
There are even scientific papers on this in the field of psychology.
A group of chess masters and a control group of non-chessplayers were briefly shown positions on a chess board and were then asked to reproduce that position on a chess board.
When the positions came from real games between humans or engines, then the chess masters were much much better at this task than the control group.
When the positions were random, like the 55958 samples of Tromp, then the chess masters performed no better than the control group.
Not surprising. Not relevant.
"Gourion's paper is entitled "An upper bound for the number of chess diagrams
without promotion" and that is what he gives."
++ A diagram is the same as a position, but with the side to move, en passant and castling rights not included.
A diagram together with the side to move, en passant and castling rights are sufficient to determine possible forward play under FIDE basic rules and constitute a reasonable definition of "position" under basic rules.
They are not sufficient to determine forward play under FIDE competition rules. For competition rules a definition of "position" needs to include additional information as I detailed in this post.
To use the basic rules definition of position in a game played under competition rules would have many drawbacks.
* It would not be possible to say in general whether a position is won for one side, which side, or drawn.
* It would not be possible to apply the dead position rule
5.2.2
The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’. This immediately ends the game, provided that the move producing the position was in accordance with Article 3 and Articles 4.2 – 4.7.
because a position as defined would not be sufficient to determine whether either player could checkmate the opponent’s king with any series of legal moves.
* You couldn't talk about a chess engine's evaluation of a position, because the evaluation would vary depending on factors that are not a part of the position.
* You couldn't identify the nodes in your search space in this thread with positions, because there would be no way to determine from a position what nodes deeper in the tree it connects to.
No doubt there are others.
'Without promotion' is short for without promotion to pieces not previously captured. That is more closely related to the number of chess positions as can occur in a reasonable game between humans or engines. This count excludes some sensible positions with 3 or 4 queens. This count includes many positions that cannot occur in a reasonable game between humans or engines, say a game with > 50% accuracy.
As I already said - neither Gurion's figure nor your ideas about reasonable positions are relevant to OP's question. (And when challenged in this thread to show even a single legal position from Tromp's correct set that couldn't be reached with >50% accuracy you failed.)
"Competition rules allow a draw claim under 9.2 when three positions that are considered the same (for the purposes of that rule and 9.6.1) have occurred on separate occasions."
++ Yes, indeed: From you beloved competition rules:
"9.2.1
The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not necessarily by a repetition of moves)"
When two positions are the same, then these are not different.
So they do not count as different positions.
The question of the original poster was: "How many different chess positions are there?"
You could take the view that "position" is defined for the purposes of those rules by the attributes listed.
That's just a semantic quibble over the import of the phrase " considered the same". It's not important because it's obviously limited to just arts. 9.2 / 9.6.1.
#374
"Text is difficult to decipher because it's almost the same as the background colour. Just select it to get readable text." ++ I see, thank you.
"Impossible to count without a reasonable definition of "reasonable". "
++ I propose reasonable game = game with > 50% accuracy. Reasonable position = legal position that can result from the initial position by a game of > 50% accuracy. That is easy, objective, and verifyable.
"Not surprising. Not relevant." ++ Relevant: it provides another way of defining a sensible position: a position that a panel of grandmasters can reproduce better than a panel of laymen.
"It would not be possible to apply the dead position rule" ++ Of course it is possible to apply the dead position rule 5.2.2 of the basic rules also when competition rules are added.
"neither Gurion's figure nor your ideas about reasonable positions are relevant"
++ Gourion's number is relevant if we understand by chess position a position resulting from a game between human or engine opponents. See Laws of Chess:
1.1
The game of chess is played between two opponents who move their pieces on a square board called a ‘chessboard’.
"when challenged in this thread to show even a single legal position from Tromp's correct set that couldn't be reached with >50% accuracy you failed."
++ No, I did show one such position with its shortest proof game. No games of > 50% accuracy are possible.
"That's just a semantic quibble over the import of the phrase " considered the same"."
++ It is not only the phrase "considered the same" in 9.2.2, but also the unambiguous phrase "the same position" in 9.2.1. and 9.6.1.
"It's not important because it's obviously limited to just arts. 9.2 / 9.6.1."
++ It is important as it is obviously not limited. It is the very definition of 'the same position' per Laws of Chess. You previously stated that there is no definition in the Laws of Chess, but there is one in 9.2.1 and 9.2.2. and 9.6.1.
xbrenbrenx wrote:
I only know missionary and the one with the cowboy 🤠
Know of them, you mean? 
Apr 8, 2022
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#369
xbrenbrenx wrote:
I only know missionary and the one with the cowboy 🤠
Watch the old movie, Quest for Fire. You'll learn some new ones.
Apr 8, 2022
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#371
tygxc wrote:
#374
"Text is difficult to decipher because it's almost the same as the background colour. Just select it to get readable text." ++ I see, thank you.
"Impossible to count without a reasonable definition of "reasonable". "
++ I propose reasonable game = game with > 50% accuracy. Reasonable position = legal position that can result from the initial position by a game of > 50% accuracy. That is easy, objective, and verifyable.
Not verifiable and definitely not what the OP is asking for.
"Not surprising. Not relevant." ++ Relevant: it provides another way of defining a sensible position: a position that a panel of grandmasters can reproduce better than a panel of laymen.
Again not what OP is asking for. Not relevant to this thread at any rate.
"It would not be possible to apply the dead position rule" ++ Of course it is possible to apply the dead position rule 5.2.2 of the basic rules also when competition rules are added.
Yes, of course it is possible. I said would not be possible to apply using your definition of "position".
When FIDE say, "a position has arisen ..." they mean simply a situation has occurred with all it's attributes. Under competition rules those attributes would include the ply count and the positions that have previously occurred; attributes that are necessary to determine if either side could mate, but which you exclude from your definition of "position".
"neither Gurion's figure nor your ideas about reasonable positions are relevant"
++ Gourion's number is relevant if we understand by chess position a position resulting from a game between human or engine opponents. See Laws of Chess:
1.1
The game of chess is played between two opponents who move their pieces on a square board called a ‘chessboard’.
OP is not asking how many diagrams have occurred in practical play. It's certainly irrelevant to the topic.
In any case many diagrams have occurred in practical play that are not in Gurions number.
When I used to play at primary school I was more interested in seeing how many queens I could get than how fast I could win. I think I got up to seven.
And this is a recorded game
All games referred to played in accordance with art. 1.1.
"when challenged in this thread to show even a single legal position from Tromp's correct set that couldn't be reached with >50% accuracy you failed."
++ No, I did show one such position with its shortest proof game. No games of > 50% accuracy are possible.
You reproduced one of Tromp's examples together with its proof game then failed to even show the game you reproduced didn't have accuracy > 50%, let alone that no such game was possible.
"That's just a semantic quibble over the import of the phrase " considered the same"."
++ It is not only the phrase "considered the same" in 9.2.2, but also the unambiguous phrase "the same position" in 9.2.1. and 9.6.1.
"It's not important because it's obviously limited to just arts. 9.2 / 9.6.1."
++ It is important as it is obviously not limited. It is the very definition of 'the same position' per Laws of Chess. You previously stated that there is no definition in the Laws of Chess, but there is one in 9.2.1 and 9.2.2. and 9.6.1.
The term "position" is used in the Basic Rules section of the FIDE laws which is self contained. Art. 9 is in the Competition rules section.
The only reason for including a description of when two positions are to be considered the same in art. 9.2.2 is that positions with the meaning of the term in the remainder of the document cannot recur. Déjà Vu is an illusion.
I would say FIDE's intention is that 9.2.2 applies only to the triple and quintuple repetition rules.
As I already said, it can't be used as a definition of "position" in the Basic Rules section, because that is not the way the document is structured. But neither can it apply to the remainder of the Competition Rules section. Art. 6.9 is in that section and includes the sentence
However, the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves.
With your definition of "position" under competition rules a position could not be such that etc., because your definition of "position" doesn't contain enough information to decide whether or not the opponent can checkmate. With FIDE's usual meaning of "situation occurring", there is no problem.
#379
"not what the OP is asking for"
#1
"So that means the max possible number of positions would be 7^64 (number of squares) which is somewhere around 1.2 e 54, or 2.4 e54 if you count whose move it is."
++ There are better estimates: 10^44 legal positions (Tromp), less than 10^37 sensible positions (Gourion).
"Assuming there is a perfect single move for every position, how many would have to be known to create a theoritical perfect chess computer/player?"
++ That would be the number of legal, sensible, reachable, and relevant positions estimated around 10^17.
I didn't read all what has been written already, I just wanted to add one maybe interesting fact : The only kind of pieces who reduce drastically the number of legal positions possibilities are the pawns.
When a pawns move there is no way to get that position back in the play again as he can't go back, which isn't the case with all other pieces. Meaning each time a pawn moves implicates a huge number of reduced possible positions left
#381
Yes, that is correct: each pawn move and each capture reduce the number of reachable positions.
Apr 10, 2022
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#375
tygxc wrote:
#379
"not what the OP is asking for"
#1
"So that means the max possible number of positions would be 7^64 (number of squares) which is somewhere around 1.2 e 54, or 2.4 e54 if you count whose move it is."
++ There are better estimates: 10^44 legal positions (Tromp), less than 10^37 sensible positions (Gourion).
"Assuming there is a perfect single move for every position, how many would have to be known to create a theoritical perfect chess computer/player?"
++ That would be the number of legal, sensible, reachable, and relevant positions estimated around 10^17.
Tromp's estimate of the number of legal positions, which assumes basic rules (no limititation on the number of moves nor the number of times a position may be repeated) is 4.8 x 10^44 (see this post)
That is what OP is asking for. He probably didn't take into account that the number of legal positions under competition rules (with 50/75 move and triple/quintuple repetition rules) is vastly greater, but he'd probably also be interested in that figure.
OP's hypothetical assumption that there is a a perfect single move for every position is obviously false if you read it as "a single perfect move" and the answer would be indeterminate.
For example the Syzygy tablebase is a theoretically perfect player in any position with up to 7 men on the board so long as the position doesn't include castling rights.
Both of these variants (and many others) are perfect play from the position shown according to Syzygy:
White to play, ply count=0
White to play, ply count=0
The positions you say are relevant are relevant only to yourself, not to the topic.
#383
"OP's hypothetical assumption that there is a a perfect single move for every position is obviously false" ++ Yes, that is right: in most positions there are several perfect moves.
"That is what OP is asking for." ++ No: he asked:
"how many would have to be known to create a theoritical perfect chess computer/player?"
Certainly not all 10^44 legal positions per Tromp and not even all 10^37 positions without promotions to pieces not previously captured per Gourion would have to be known to create a theoretical perfect chess computer / player. The answer to that question is the number of legal, sensible, reachable, and relevant positions, which I estimate at 10^27.
Apr 10, 2022
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#377
tygxc wrote:
#383
"OP's hypothetical assumption that there is a a perfect single move for every position is obviously false" ++ Yes, that is right: in most positions there are several perfect moves.
"That is what OP is asking for." ++ No: he asked:
"how many would have to be known to create a theoritical perfect chess computer/player?"
Certainly not all 10^44 legal positions per Tromp and not even all 10^37 positions without promotions to pieces not previously captured per Gourion would have to be known to create a theoretical perfect chess computer / player. The answer to that question is the number of legal, sensible, reachable, and relevant positions, which I estimate at 10^27.
To be precise he asked, " Assuming there is a perfect single move for every position, how many would have to be known to create a theoritical perfect chess computer/player?". Given that the antecedent is false any number would be an answer.
Assuming the antecedent were omitted then he'd be asking for a strong solution. The number of positions would be as noted by Tromp under basic rules but vastly higher under competition rules (though nowhere near as high as the upper bound of 4.8 x 10^44 x 5^(4.8 x 10^44) I gave here which could easily be vastly reduced.)
Your figure of 10^27 is the result of very flawed analysis. (See this thread.)
#385
"Assuming the antecedent were omitted then he'd be asking for a strong solution."
++ No, he would be asking about a weak solution.
"The number of positions would be as noted by Tromp"
++ No, none of the 55958 sampled positions Tromp found legal would have to be known by the theoretically perfect chess computer/player as these would never occur in any game he plays and hence only a tiny fraction of the 4.79 * 10^44 legal positions.
If the theoretically perfect chess computer/player defends 1 e4 e5 and 1 d4 d5 as black and opens 1 e4 as white, then he does not need to know any positions resulting from the Dutch Defence, the King's Indian Defence etc.
The theoretically perfect player would never underpromote to multiple pieces not previously captured and when his opponent tried to do so, that opponent would be checkmated before he could do so multiple times.
"vastly higher under competition rules"
++ No, your observations about competition rules are not relevant at all as previously explained with help from the Laws of Chess, which define when positions are the same and when they are different. The original poster asked about different positions. Even aside from that, the perfect chess computer/player would find means to avoid the draw in a won position and would find other ways to draw in a drawn position. That is what happens in ICCF WC drawn games, 99% of which are perfect play.
Apr 10, 2022
0
#379
Well that's all b*llocks, but I know from experience there's no way to stop you from continuing to spout it, so I'll just leave you to do so.
For those wondering about the amount of possible moves, this video sums it up :
https://www.youtube.com/watch?v=Km024eldY1A
But it is not about the number of positions. The number of positions is 10^44 according to John Tromp and Peter Österlund. But it is the number of total positions, if we want the number of "perfect positions", I think we can't really answer, because if there was 1 perfect move, there would be only 1 game for each opening during the computer championships :
https://www.chess.com/computer-chess-championship
Can we really talk about the "perfect move" when games only differ in our personal preferences (or how the computer is set up) ?
#360
The original post is empty,
No idea what you mean by that. I've been back a couple of pages and my links seem to be working.
What exactly is empty?
but the answer to the title question "How many different chess positions are there?" is: legal positions 10^44 per Tromp,
Tromp says in his post here (a few pages back):
the latest estimate, accurate to 2 digits, is 4.8 x 10^44 legal chess positions
That is the most accurate estimate of basic rules chess positions available to date.
positions that can occur in reasonable play with > 50% accuracy: less than 10^37 per Gourion.
Gourion's paper is entitled "An upper bound for the number of chess diagrams
without promotion" and that is what he gives. Nothing very strongly related the number of legal chess positions under either basic or competition rules (and in any case his best estimate of legal chess diagrams without promotions is close to 3 × 10³⁷).
It's obvious from the vast discrepency between the two figures, that he's not talking about the number of legal positions.
The comments about reasonable play and accuracy are entirely of your own manufacture. When invited to prove these conjectures in this thread you produced nothing substantive in support.
There are no more positions under competition rules than under basic rules. Competition rules allow to claim a draw when the same position occurs three times, hence those 3 positions are the same and are not different.
Competition rules allow a draw claim under 9.2 when three positions that are considered the same (for the purposes of that rule and 9.6.1) have occurred on separate occasions.
Any concept of "position" that leaves the possible continuations undefined is of limited use for the purpose of analysing chess.
Because more attributes of a position are necessary to determine possible forward play under competition rules than are necessary under basic rules it follows that for any useful definition of "position" there are more positions under competition rules than there are under basic rules.
(See the example and comments in this post.)