Indeed, he did effectively cut out adjacent kings only in the illegality tests.
Though I doubt if he could have generated
8726713169886222032347729969256422370854716254 positions.
In his earlier calculations of an upper bound adjacent kings were specifically excluded.
where did you find that?
If you take the 32 pieces and make all different positions, naturally the pawns doesn't count they are the 8 the same, same with the rook, night and, bishop, how many different positions on the board . (call it a picture) and take then 1 figure out one by one
Do you mean "all different diagrams" when you say, "all different positions"?
If so, does that mean you make no distinction between diagrams and positions. Would you for instance say that this
is the same position regardless of who has the move, and, if it's under competition rules, whether the ply count under the 50 move rule is 0 or 99?
yes, exactly no need to think about who moves, just all different PICTURES
If you want to know how many different diagrams can occur in legal positions, it would be very close to half the number of legal basic rules positions as estimated by Tromp. See this post.
That would be minute compared with the number of competition rules chess positions (but the diagrams are the same under basic rules or competition rules).
#341
"I doubt if he could have generated
8726713169886222032347729969256422370854716254 positions."
++ That is what his HASKELL program did. For each integer between 1 and 8726713169886222032347729969256422370854716254 he can give the position.
"In his earlier calculations of an upper bound adjacent kings were specifically excluded."
++ No, he generated positions including such with adjacent kings and then he sampled and then he eliminated illegal positions like such with adjacent kings. He started with smaller sample sizes and now he is at 1 million sample size.
The Gourion paper excludes adjacent kings from the start.
Indeed, he did effectively cut out adjacent kings only in the illegality tests.
Though I doubt if he could have generated
8726713169886222032347729969256422370854716254 positions.
In his earlier calculations of an upper bound adjacent kings were specifically excluded.
where did you find that?
If you want to know how many different diagrams can occur in legal positions, it would be very close to half the number of legal basic rules positions as estimated by Tromp. See this post.
That would be minute compared with the number of competition rules chess positions (but the diagrams are the same under basic rules or competition rules).
Is this the number of different legal positions with pawns promotion
Around 8x10^45 ???
#341
"I doubt if he could have generated
8726713169886222032347729969256422370854716254 positions."
++ That is what his HASKELL program did. For each integer between 1 and 8726713169886222032347729969256422370854716254 he can give the position.
"In his earlier calculations of an upper bound adjacent kings were specifically excluded."
++ No, he generated positions including such with adjacent kings and then he sampled and then he eliminated illegal positions like such with adjacent kings. He started with smaller sample sizes and now he is at 1 million sample size.
The Gourion paper excludes adjacent kings from the start.
Being able to calculate an upper bound and determine an indexing function doesn't appear to be the same as actually generating all the associated positions.
If you look at this link you will note that adjacent kings are specifically excluded. He judged it easier or more efficient to let the legality checks rule these out in the subsequent computation decribed here.
The Gurion paper has little relevance to the number of chess positions being concerned only with diagrams in which no excess promotions have occurred.
If you want to know how many different diagrams can occur in legal positions, it would be very close to half the number of legal basic rules positions as estimated by Tromp. See this post.
That would be minute compared with the number of competition rules chess positions (but the diagrams are the same under basic rules or competition rules).
Is this the number of different legal positions with pawns promotion
Around 8x10^45 ???
That would depend on which rules you are talking about. There are far more competition rules positions than basic rules positions if you assume that a position must determine possible forward play.
The number of diagrams under either set of rules will be around 2.4 x 10⁴⁴ (half the number of basic rules positions posted by @johntromp in the link I gave).
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
there are a lot of different chess position just to answer your question
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
I don't know. But coincidentally, I just watched a TV show and learned that a simple game of 'tic tac toe' can have something like 255168 of gameplay combinations. Seems like a lot.
I don't know. But coincidentally, I just watched a TV show and learned that a simple game of 'tic tac toe' can have 362,880 combinations. Seems like a lot.
Too many. That's 9! but a lot of possibilities will be terminated by wins before all squares are marked.
I guess that's what I get for basing my knowledge on an episode of America's Got Talent.
I edited my post to another number after searching around the interwebs. I'm seeing numbers from 19,683 to 362,880. Not sure.
#348
"Being able to calculate an upper bound and determine an indexing function doesn't appear to be the same as actually generating all the associated positions."
++ Of Course Tromp did not generate and store all 8726713169886222032347729969256422370854716254 positions, but for each integer between 1 and 8726713169886222032347729969256422370854716254 he can generate the corresponding position.
"If you look at this link you will note that adjacent kings are specifically excluded."
++ He allowed adjacent kings in his count of 8726713169886222032347729969256422370854716254. Then he sampled a million positions. Then he eliminated positions with adjacent kings, with both kings in check, with the side not having the move being in check... and then he determined legality of the remaining samples and thus he found 55958 samples legal, thus he arrived at his estimate of 10^44 legal positions.
The Gourion number 10^37 is more relevant, as nearly all of Tromp's legal positions contain multiple underpromotions to pieces not previously captured. The criteria of Gourion correspond better to positions in real chess games.
#348
"Being able to calculate an upper bound and determine an indexing function doesn't appear to be the same as actually generating all the associated positions."
++ Of Course Tromp did not generate and store all 8726713169886222032347729969256422370854716254 positions, but for each integer between 1 and 8726713169886222032347729969256422370854716254 he can generate the corresponding position.
As I said. You originally said he generated the positions.
"If you look at this link you will note that adjacent kings are specifically excluded."
++ He allowed adjacent kings in his count of 8726713169886222032347729969256422370854716254. ...
Indeed, but it's not the count in the link I gave.
The Gourion number 10^37 is more relevant, as nearly all of Tromp's legal positions contain multiple underpromotions to pieces not previously captured. The criteria of Gourion correspond better to positions in real chess games.
More relevant to what?
They're diagrams not positions and even if you take all of the positions to which the diagrams pertain, almost all of what is asked for in OP's question is missing (whether you're talking about basic rules positions or competition rules positions).
#360
The original post is empty, but the answer to the title question "How many different chess positions are there?" is: legal positions 10^44 per Tromp, positions that can occur in reasonable play with > 50% accuracy: less than 10^37 per Gourion.
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.
In 960Chess, the number refers to all different starting position someone told me?