@4937
"rendering Tromp's estimate of 4.85304e+44 +- 3.9004e+42 as 10^44".
++ Each position with white to move has an up / down mirror position with black to move.
Each position with lost castling rights has a left / right mirror position.
That yields 1.21326e+44 +- 0.00975e+44, so 1e44 is close enough.
Likewise the Gourion estimate of 3 × 10^37 becomes 1.5 × 10^37
https://arxiv.org/abs/2112.09386
@4937
"rendering Tromp's estimate of 4.85304e+44 +- 3.9004e+42 as 10^44".
++ Each position with white to move has an up / down mirror position with black to move.
Each position with lost castling rights has a left / right mirror position.
That yields 1.21326e+44 +- 0.00975e+44, so 1e44 is close enough.
Likewise the Gourion estimate of 3 × 10^37 becomes 1.5 × 10^37
https://arxiv.org/abs/2112.09386
<<++ Each position with white to move has an up / down mirror position with black to move.>>
Wouldn't ++ "Each position with white to move has an identical position with black to move" be simpler and clearer? Does it lack any information?
By the way, the initial position hasn't! 
@4937
"rendering Tromp's estimate of 4.85304e+44 +- 3.9004e+42 as 10^44".
++ Each position with white to move has an up / down mirror position with black to move.
Each position with lost castling rights has a left / right mirror position.
That yields 1.21326e+44 +- 0.00975e+44, so 1e44 is close enough.
Likewise the Gourion estimate of 3 × 10^37 becomes 1.5 × 10^37
https://arxiv.org/abs/2112.09386
That ignores the fact that you're planning to use SF which has to keep track of triple repetitions. It can't do that if it take symmetries into account, because the equivalence classes under symmetries can occur three times without the game necessarily being drawn.
Gourion talks only about diagrams, not positions under either basic or competition rules.
It can't use basic rules in the first place.
If you check the positions after move 34 in the mainline and variation here you will find the FENs are identical so the basic rules positions are the same. The mainline is a draw at that point while the variation is a mate in 17. Unfortunately it appears that your workhorse is not up to mating with a king and pawn against a lone king.
By the way that's another of your 1 in 10^zillion cases where SF's top four moves lose half a point.
White to move
Black to move
@4942
"That ignores the fact that you're planning to use SF which has to keep track of triple repetitions."
++ No, I do not use Stockfish to track triple repetitions, I just compare FENs.
"equivalence classes under symmetries can occur three times"
++ No they do not.
The up/down and left/right mirror images do not occur and certainly not in the same game.
"Gourion talks only about diagrams, not positions."
++ Tromp calculates diagrams and multiplies by 2 to get positions.
That is why the Tromp count is a factor 4 too high for this purpose
and the Gourion count is only a factor 2 too high for this purpose.
@4942
"That ignores the fact that you're planning to use SF which has to keep track of triple repetitions."
++ No I do not use Stockfish to track triple repetitions, I just compare FENs.
You can compare FENs 10^9 times a second and restart SF with a different move without breaking step. Impressive!
But it sort of ignores the fact that the number of FENs is not divided by 4 either.
"equivalence classes under symmetries can occur three times"
++ No they do not.
The up/down and left/right mirror images do not occur and certainly not in the same game.
That seems to account for all cases. Interesting! Youve just proved that no positions can ever occur in a chess game.
"Gourion talks only about diagrams, not positions."
++ Tromp calculates diagrams and multiplies by 2 to get positions.
That is why the Tromp count is a factor 4 too high for this purpose
and the Gourion count is only a factor 2 too high for this purpose.
You don't say what purpose, but if it's anything to do with playing chess you need to take into account more than just diagrams. Otherwise fights would break out when both players are trying to move at the same time.
It'll be solved because AI will calculate everything.
@4945
"You can compare FENs 10^9 times a second"
++ No, not at all. FEN is only compared every 17 s. A comparison is a simple operation.
"That seems to account for all cases."
++ No, the mirror positions cannot transpose into each other.
"You don't say what purpose" ++ For the purpose as a starting point to estimate the number of relevant positions in weakly solving chess.
Throughout this thread, some have mistakenly considered extremely large finite numbers to be the equivalent of infinity....."for all practical purposes". This is a mistake. No matter how large a finite number is, it is not the equivalent of infinity....or even close.... period. And any logic or conclusions that follow that assumption are false.
You don't get any large finite numbers, they're all completely miniscule compared with practically all the rest. Come to that you don't get any large infinite numbers either.
Well, some infinite numbers (let's presume that means "cardinals" in this context) are large compared with a lot of other infinite numbers. So they can be considered large in the same way as a googol is often viewed as a large integer.
A major example would be a (strongly or weakly) inaccessible cardinal). Any inaccessible cardinal is bigger than any cardinal whose existence is provable in ZFC!
ZFC itself does not imply the existence of an inaccessible cardinal. The existence of one implies the consistency of ZFC, which itself implies that it cannot be proven that the existence of a large cardinal is consistent with ZFC (most mathematicians believe it is)).
That the linked article includes the phrase " Therefore, inaccessible cardinals are a type of large cardinal" and there is a page on said large cardinals shows mathematicians are not averse to the idea of large infinite numbers. Worse, it turns out that there is a whole litany of distinct ways in which cardinals can be large!
@4949
10^17 is a huge number, but it is finite. Chess is finite.
@4945
"You can compare FENs 10^9 times a second"
++ No, not at all. FEN is only compared every 17 s. A comparison is a simple operation.
By which time SF will have used positions rather than equivalence classes of positions 17x10^9 times on you figures.
"That seems to account for all cases."
++ No, the mirror positions cannot transpose into each other.
Well not by themselves. Somebody has to move the pieces.
"You don't say what purpose" ++ For the purpose as a strating point to estimate the number of relevant positions in weakly solving chess.
In which case the Tromp figures are not too high for basic rules chess by a factor of 4 if you're planning to use SF they're correct, The Gurion count is not fit for purpose - your purpose that is, not Gurion's.
Tromp's figure is humongously too low if you plan to use SF for solving competition rules chess.
Let's imagine what a solution of chess requires. First it requires a big tablebase (in all seriousness, one that is too big to be infeasible. Let's pretend not and say we have a puny 12-piece tablebase. This requires about 10^24 bytes. That's about 10000 times bigger than the total storage capacity of all computers on the planet. Ridiculous, but technologically plausible. If you have a few trillion dollars)
Next you need two strategies. The first forces a drawn position in the tablebase with white, the second forces a drawn position in the tablebase with black.
The problem is that it seem rather unlikely you could avoid needing to rely on drawn positions not in the tablebase, if the opponent fails to co-operate in exchanging pieces (eg he blocks the position and shuffles pieces).
But can you somehow get round the need to use a large state space by considering that only opposing moves that produce new positions matter. If the opponent against a drawing strategy moves to a position previously reached he has achieved nothing.
So you don't need the full state for the positions, rather you check whether new positions are already in your strategy.
Thus I think @tygxc is correct to believe that FEN states are adequate for a solution of chess, even if he is woefully wrong on the computing power needed to solve this rigourously, as achieved for checkers.
Any flaws in that reasoning?
@4951
"By which time SF will have used positions rather than equivalence classes"
++ Yes, the 3-fold repetition rule is about repetition of a position, i.e. the same FEN without move#.
"Somebody has to move the pieces."
++ You cannot transpose the 2 mirror images @4953 into each other by moving the pieces.
Only 1 of the 4 mirror images can be relevant.
"In which case the Tromp figures are not too high" ++ Tromp gives the number of legal positions, but in weakly solving chess only 1 of the 4 mirror images can occur.
"The Gurion count is not fit for purpose" ++ The Gourion number is a better estimate, as the Tromp positions cannot result from optimal play. Look at the 3 random samples.
https://github.com/tromp/ChessPositionRanking
They all have underpromotions to rook or bishop on both sides. The only reason to underpromote to rook or bishop instead of queen is to avoid stalemate i.e. avoid a draw. It makes no sense for both sides to avoid a draw. So at least one side made a mistake by underpromoting instead of queening. So these positions cannot result from optimal play.
"solving competition rules chess"
++ The plan is to solve chess with the 3-fold repetition rule (or even simplify to 2-fold) and without the 50-moves rule, which plays no role.
The solution for chess with the 50-moves rule is the same as without.
@4952
"First it requires a big tablebase"
++ No, that is not true. There are much more chess positions around 26 men than around 12 men. A 7-men endgame table base is enough.
In TCEC the engines frequently hit their 7-men endgame table base as early as move 10.
"Next you need two strategies."
++ No, you only need 1 strategy for black to achieve a draw against all white opposition.
"The first forces a drawn position in the tablebase with white" ++ No, unneeded.
"the second forces a drawn position in the tablebase with black"
++ Can also be a prior 3-fold repetition, e.g. perpetual check, or a known draw.
"if the opponent fails to co-operate in exchanging pieces"
++ There is always compelling reason to move pawns or exchange pieces.
In ICCF WC Finals the 50-moves rule is never invoked.
Such games are > 99% sure to be perfect games with optimal play and no errors.
Intuitively solving chess with 2-fold repetition draw is equivalent to solving it with 3-fold repetition.
The solution for chess with the 50-move rule is not the same as a solution of chess without this rule. A strategy for the former could happily reach a tablebase position where the opponent couldn't mate quickly enough, so a 50-move draw was inevitable. This strategy would fail without the 50-move rule. This is trivial.
Fortunately the 50 move rule is not difficult to deal with and FEN does so.
...
++ You cannot transpose the 2 mirror images @4953 into each other by moving the pieces. Only 1 of the 4 mirror images can be relevant.
...
You're speaking for yourself. I find it quite easy.
@4955
"solving chess with 2-fold repetition draw is equivalent to solving it with 3-fold repetition."
++ Yes, that is correct.
"The solution for chess with the 50-move rule is not the same as a solution of chess without"
++ Yes they are not necessarily the same.
However, if chess is solved to be a draw without the 50-moves rule, then that same solution also applies with the 50-moves rule. The reverse is not necessarily true.
"A strategy for the former could happily reach a tablebase position where the opponent couldn't mate quickly enough, so a 50-move draw was inevitable."
++ It is not necessary. In ICCF they can claim 7-men endgame table base wins that exceed 50 moves without capture or pawn move.
Such endgame table base win claims do not occur.
Endgame table base draw claims are common in 10% of games.
"This strategy would fail without the 50-move rule."
++ Yes, but the strategy without the 50-move rule applies also with the 50-moves rule.
@4956
"I find it quite easy."
++ I am speaking of middlegame positions of around 26 men, where most positions are.
See table 3: https://arxiv.org/pdf/2112.09386.pdf
For positions without pawns there is even 8-fold symmetry:
up / down, left / right, and mirror images along a diagonal.
Positions with 7 men or less are not relevant: those are looked up in the endgame table base.
Oh it is wrong, sorry. Missed a4. I don't like chess.com puzzles. They won't allow you to think for yourself so I treat them as trivial. I've done two and got them both right, though.