@tygxc
In that paper I see the 64!/32! I expected.
The number of ways to have 32 squares empty - regardless of how many pieces captured or not and therefore missing from the other 32 squares in each case.
It also mentions the limits on bishops to 32 squares !
I forgot about that one. That's another big 'cutdown'.
But just a little attention to that 64!/32! is in order I'd say.
That's 64 factorial divided by 32 factorial ... with 32 factorial then cancelling out to produce (64 x 63 x 62 .... x 35 x 34 x 33) ... 32 numbers multiplied together.
That's a number greatly greatly in excess of 10 to the 32nd power - just by itself. 
Regarding the practicalities versus ridiculousnesses of determining a lowest possible upper bound for the number of possible positions ...
there is an issue of the amount of material on the board.
How many sub-situations are there?
I would say there are 31 such situations. At the most 'macro' level.
Its defined by a series of 31 terms - with a '1' at each end and a central term (15 pieces plus Kings on the board = 15 pieces taken also) ... the only one like that.
Would that term generate more positions than the final term? (which is all 32 pieces on board). I'm tempted to think the final term generates the most. But the final term has only one sub-sub-situation. Itself.
The expression looks something like this:
(1 + 10 (only 10 ways to have something with the 2 K's) + .... + M (maximum number of sub-sub situations - 15 extra pieces on but 15 off too - what would that number be?) + ... + 10 (one piece captured) +1 (no captures yet) )
This is all without even yet getting to positions generated ...
Relevance: Computer programmers would need to identify the sub-sub situations - and after that number is determined - then the computer would work on each - maybe in parallel - to determine how many positions on each.
Yes I know the paper has a breakdown ...
My point is that even the sub-sub situations within just the middle term of the 31 - could be a Daunting number ! And that's without getting to # of positions on each sub-sub. (could be a pun there)
Chess will never be solved, here's why
On the other 32 squares only 1 way to have none empty - and only one way to have 30 empty.
only 30 square ways to have one extra piece with the two Kings but with 10 possible pieces -so 300 ?
At the other end - 30 ways to have one of the 32 other squares empty but with any of ten possible pieces missing. So 300 again.
If I was doing this - I'd prefer to start with the material situations.
#229
There are other restrictions: no adjacent kings, no pair of same square bishops, side not having the move not in check, no illegal pawn structures...
Anyway, the paper has it right and provides an upper bound 3.8521*10^37.
#228
Yes 50 moves or 3 fold repetition has to be claimed. 75 moves or 5-fold repetition is automatic.
See article 9.
https://www.fide.com/FIDE/handbook/LawsOfChess.pdf
For the purpose of solving chess this can be ignored. When a position is a draw, then neither side has reason to object to the draw. When one side is winning, then the losing side has reason to claim and the winning side has reason to avoid 50 moves or 3-fold repetition.
#228
Yes 50 moves or 3 fold repetition has to be claimed. 75 moves or 5-fold repetition is automatic.
See article 9.
https://www.fide.com/FIDE/handbook/LawsOfChess.pdf
For the purpose of solving chess this can be ignored. When a position is a draw, then neither side has reason to object to the draw. When one side is winning, then the losing side has reason to claim and the winning side has reason to avoid 50 moves or 3-fold repetition.
that's very good to know. About the 75 and the 5-fold and so on ...
First I've heard of that !
according to the calculator 64!/32! is 4.8221992e+53
I imagine that could be typed as 4.8 x 10 to the 53rd power.
My keyboard refuses to offer its 'hat' symbol for exponents !
#224
The longest possible chess game with the 50 moves rule is 5898.5 moves.
However that involves sequences of 49 meaningless moves shuffling around before making an irreversible move: capture or pawn move.
The world record of the longest real chess game was Nikolić–Arsović, Belgrade 1989: 269 moves.
Which also involved a lot of meaningless moves shuffling around. (Nikolić missed several obvious conversions to a winning KRK endgame).
I could have sworn I already corrected you on this, but the 50 move rule doesn't limit the length of a game because a player must make a claim in order for the rule to be applied and there is no compulsion on either player to make the claim. It was in any case removed from FIDE's basic rules of chess in 2017.
At the same time a new 75 move rule was introduced into the competition rules which automatically terminates the game after 75 moves by each player, either in a win if the final move gives mate, otherwise a draw. This does limit the length of a game played under competition rules, but to 8848.5 moves, not 5898.5.
See https://www.wismuth.com/chess/longest-game.html .
A game played under basic rules is not limited in length. The 3-fold repetition rule was removed from the basic rules at the same time as the 50 move rule, but that also required a claim by one player to take effect and again there was no compulsion to do so.
The 3-fold repetition rule is retained in competition rules with the same constraints. A new automatic 5-fold repetition rule was introduced into the competition rules at the same time as the rules limiting game length were removed from the basic rules, but a game of 8848.5 is possible without that rule becoming applicable.
The longest possible forced mate under competition rules on the other hand is limited by the 50 move rule to at most 5898.5+50 moves. The actual length is likely to be much smaller.
#216
Again: see this scientific paper
https://arxiv.org/pdf/2112.09386.pdf
It gives the upper bound of 3.8521*10^37 legal positions.
32 men 1.89 × 10^33
31 men 1.71 × 10^34
30 men 1.64 × 10^35
29 men 1.53 × 10^36
28 men 5.46 × 10^36
27 men 1.05 × 10^37
26 men 1.08 × 10^37
25 men 6.14 × 10^36
24 men 3.19 × 10^36
23 men 5.66 × 10^35
But are those limits on legal positions? Doesn't the author assume a maximum of 1 queen, two rooks etc. per player etc., ignoring the fact that these figures could increase by promotion?
I need to read the paper in more detail to decide that, but a collapse in the upper bound given by Tromp which I quoted earlier by a factor of around 300000000 seems most unlikely otherwise.
#237
The more recent paper gives an upper bound without excess promotions, i.e. max. 1 queen, 2 rooks, 2 knights, 2 bishops of opposite colors.
John Tromp counted all positions legal or not even with adjacent kings and then went on to sample positions to weed out 95% illegal ones and keeping 5% legal ones. However the vast majority of the randomly sampled positions by John Tromp are obviously without sense: 5 rooks, 3 queens, 4 same color bishops... That is why his number is too high.
The newer upper bound 3.8521*10^37 excludes some meaningful positions like with 4 queens, but still includes many many positions with meaningless pawn structures. That is why it is more accurate for the purpose of solving chess.
#237
The more recent paper gives an upper bound without excess promotions, i.e. max. 1 queen, 2 rooks, 2 knights, 2 bishops of opposite colors.
John Tromp counted all positions legal or not even with adjacent kings and then went on to sample positions to weed out 95% illegal ones and keeping 5% legal ones. However the vast majority of the randomly sampled positions by John Tromp are obviously without sense: 5 rooks, 3 queens, 4 same color bishops... That is why his number is too high.
The newer upper bound 3.8521*10^37 excludes some meaningful positions like with 4 queens, but still includes many many positions with meaningless pawn structures. That is why it is more accurate for the purpose of solving chess.
Exactly.
It's not an upper bound on legal positions.
To judge positions "obviously without sense" appears to be prejudjing the results of your proposed exercise.
#239
3.8521*10^37 is an upper bound on legal positions without excess promotions.
So it is a reasonable estimate for the number of legal and sensible positions.
Here is a random sample legal position as counted by Tromp. It is obvious that this never arises in a perfect game of chess.
Could you explain why that's obvious? I'm not very good at perfect chess.
#241
No human/engine players right in their minds would underpromote 5 white rooks, 3 black knights and a 2nd black dark square bishop.
All of Tromp's randomly sampled positions look like that.
At this point, I do not want chess to be solved. Or else it won't be fun anymore.
What I am more interested in is whether or how life will be solved or not.
>Life is struggle.
>I come to chess to relieve from it a little.
> see that chess has been solved.
> some stupid memorized all the solutions and has been paired with me.
> boring.
> I leave.
#241
No human/engine players right in their minds would underpromote 5 white rooks, 3 black knights and a 2nd black dark square bishop.
All of Tromp's randomly sampled positions look like that.
Final board layout in Crafty-Nakamura about a decade ago
As I said earlier, I'm not very good at perfect chess, but as far as I can judge all Nakamura's underpromotions were perfect moves.
#228
...
For the purpose of solving chess this can be ignored. When a position is a draw, then neither side has reason to object to the draw.
...
Is it not usual to ignore agreed draws in theoretical work?
#245
Nakamura was trolling and so are you.
#246
Most chess games are won by resignation, some by loss on time, very rarely by checkmate. When a player reaches the conclusion that a loss is inevitable it makes sense to resign.
Likewise most chess games are drawn by agreement, some by 3-fold repetition, some by no series of legal moves leading to checkmate, some by stalemate, very rarely by the 50-moves rule.
When both players reach the conclusion that a draw is inevitable it makes sense to agree on a draw.
#245
Nakamura was trolling and so are you.
But he won the game. Is Syzygy (playing perfectly) trolling here?
I suspect your accusation of trolling just masks an unwillingness to concede the point.
#246
Most chess games are won by resignation, some by loss on time, very rarely by checkmate. When a player reaches the conclusion that a loss is inevitable it makes sense to resign.
What on Earth has that got to do with #246 (or solving chess).
Likewise most chess games are drawn by agreement, some by 3-fold repetition, some by no series of legal moves leading to checkmate, very rarely by the 50-moves rule.
When both players reach the conclusion that a draw is inevitable it makes sense to agree on a draw.
But I repeat; is it not usual to ignore agreed draws in theoretical work?
#224
The longest possible chess game with the 50 moves rule is 5898.5 moves.
However that involves sequences of 49 meaningless moves shuffling around before making an irreversible move: capture or pawn move.
The world record of the longest real chess game was Nikolić–Arsović, Belgrade 1989: 269 moves.
Okay - good point - its a 49 multiplier rather than a 50.
But - does the draw have to be claimed?
The computers here make it automatic on 50?
I would think so. I can't remember. Maybe one has to hit the draw button?
But in an over the board tournament wouldn't you have to claim it?