On the number of chess positions

Wow!!

#102
"And how many sensible do you find in the larger sortedRnd1kResearch sample?"
There are 100 red FEN and 100 green FEN. I see several blatantly illegal, like adjacent kings, king in check from 2 sides of 1 diagonal...
Another simple illegality criterion: white doubled pawns <= 16 - number of black men
Which of those 200 should I look at?

Obviously, at the 200 positions in

$ grep -v Illegal sortedRnd1kResearch

5RQ1/q4P1N/P6K/1br1p1p1/n1P3r1/knb4N/2p1pp1R/5B2 w - - 0 1 No Checks wx 6 wp 3 wpr 0 wpx 5 maxuwp 12 minopp -9 bx 3 bp 5 bpr 0 bpx 3 maxubp 14 minopp -9
2Kn2rB/1ppp3q/1P5B/1pkP1Prb/1Rp5/1p3pQN/2p1PR2/2b5 w - - 0 1 No Checks wx 5 wp 4 wpr 1 wpx 3 maxuwp 5 minopp -1 bx 1 bp 8 bpr 0 bpx 0 maxubp 9 minopp -1

[197 lines skipped]

1k1K4/p4p1P/PP4rR/1NB1ppn1/2PpP2p/2PP2n1/3R2NP/Qb2b1qB b - e3 0 4 No Checks wx 0 wp 8 wpr 0 wpx 0 maxuwp 5 minopp 3 bx 3 bp 6 bpr 0 bpx 2 maxubp 3 minopp 3

When you arrive at a precise definition of "sensible" you should try to code it up in your favorite programming language so you can automatically filter for it.

#105
No 1 seems legal and sensible
No 200 is illegal

What about the other 198 lines in file sortedRnd1kResearch that don't say Illegal ?

If that was the last move in that game #1 then I do not see a sensible move before that. The rook on e1 must have been on the first rank and had the option to capture the black queen, but didn't. Do you allow for such obviously bad play and still call it sensible?

If you construct proof games which moves do you consider as not sensible? That is a very difficult criterion to define, I believe.

#108
You are right, this position is illegal

All the black men are still on the board so there was no way for the white pawns to reach their destinations b7 and g6.

It is hard to define what is sensible, but it is easier to determine what is not sensible.
9 excess promotions are not sensible
multiple underpromotions are not sensible
two dark square bishops at one side is not sensible
more than 3 doubled pawns is not sensible
a white king at the 8th rank with a board full of pieces and pawns is not sensible.

My approach is that the number of chess positions without any excess promotions, i.e. possible with one box of 32 chess men is a good value for the number of sensible positions.
I know some rare cases with 4 queens or with 3 knights are sensible, so I miss these.
On the other hand the number of positions without excess promotions still contains non sensible positions with 4 doubled pawns, two dark square bishops, a white king on the 8th rank on a board full of black pieces and pawns...
So the number of positions without excess promotions is an upper bound for the number of sensible positions.

That statement is almost meaningless unless you define what is sensible.

Will we ever see a precise definition of sensible from you?

Or would you agree that it's completely subjective and arbitrary,

and therefore trying to accurately quantify such a thing is of little value?

#110
Well here is another take and it can even be automatic.

What is legal?
A position is legal if and only if a proof game shows it to result from the initial position.

What is sensible?
A position is sensible if and only if engine analysis of its proof game shows > 50% accuracy.

The core of the problem is constructing a proof game.
That is why we often resort to negative criteria.

What is not legal?
The side NOT having the move is in check. (EDITED)
A king is in double check without a prior move to deliver that.
More captures required than ennemi men off the board.
Pawns cannot reach their positions.
Kings cannot reach their positions.
....

What is not sensible?
Multiple excess promotions.
Multiple underpromotions.
More than 1 bishop of the same color on one side.
More than 3 doubled pawns on one side.
A king in the rear with multiple ennemy men present.
...

50% is imho by far too constraint. Players with a chess.com rating of 1000 often are below an accuracy of 50% but they play "sensible" chess. And this value is also completely engine dependent. It can jump between 40 and 70 % easily as I have noticed frequently.

Also almost all legal positions can be reached in many different games. And you cannot check all games that lead to a given position. This approach won't work.

Engines are imperfect, and ergo, so are their accuracy scores.  So any leaps and conjecture made based on their analysis automatically disqualifies any proof claimed.

Here's what not sensible: claiming that there's some quantifiable notion of sensible position but refusing to pin it down, not doing anything serious with the data you asked me to provide, pondering even more ill-defined and even less-quantifiable notions of sensible...

PS: the side to move in check is perfectly legal:-(

#114
I did look at the sample of 30 you provided, though most are illegal. I am willing to look at your 200 sample, though most are illegal, I do it manually and I also have other things to do.

The problem is, that your number of chess positions contains most positions with 9 excess promotions, which are not sensible by any definition.

The number of chess positions without any excess promotions looks fine.
It excludes some sensible positions with 4 queens or 3 knights.
It includes some non sensible positions with weird pawn or king placements.

My question was: how does this number of positions without excess promotions break down into classes with 32 to 2 men?

Here is another proposition to quantifyably pin down the notion of sensible position:
A chess position is sensible if
There are no more than 3 queens at one side, and
there are no more than 2 rooks at one side, and
there are no more than 2 bishops on one side, not both on squares of the same color, and
there are no more than 3 knights on one side, and
there are no more than 3 doubled pawns on one side, and
the kings are no more than 2 ranks more advanced than their most advanced pawns.

If you find these thresholds arbitrary, then you can query a data base like ChessBase and quantitatively verify that less than 1% of real games fails to meet these criteria.

You seem to be the only one who has a problem with that. Many people are interested in accurate estimates of the number of legal positions. Which I managed to determine.

If you want to do the same for some arbitrary notion of sensible legal position, then go ahead. Just don't be surprised if few people will show interest...

#116
If everybody is interested in positions that look like this then I cannot help it
I find your posts #12 and #13 more valuable

You yourself have shown interest in this non-sensible position in

https://www.chess.com/forum/view/more-puzzles/shortest-proof-game-challenge?page=53#comment-63623519

and in dozens other non-sensible positions in that thread.

#118
Yes, but that is for fun. I do not need to know how many such positions there are. The number of positions that can occur in real games however is of genuine interest. That is why I consider your posts #12 and #13 as the most valuable in this thread.

No; there is no such number, as there is no way to define real games. The 4 positions you picked earlier [1] cannot occur in real games either.

The number of positions satisfying some arbitrary constraints such as in your #115 above is of no genuine interest.

[1] https://www.chess.com/forum/view/general/on-the-number-of-chess-positions?page=5#comment-64060779

I am totally on John's side. You are believing in "real" or "sensible" games. But this totally depends on your selection criteria. For instance the 50% accuracy which you proposed earlier or the maximum of 8 promotions, this is completely arbitrary and depending on these thresholds you would get completely different results. There is not 1 number of sensible positions, but there is ONE number of legal positions for which John has found remarkably small limits. And this is scientifically of high interest.

But yes it would be nice to have some numbers for how many positions can occur depending on some given criteria. That's why John invited you to take his positions and start working on them.

And even if this premise were used to produce some "proof", what would it be proof of?  Certainly not solving chess.