On the number of chess positions

#82
A precise definition of "look like human play" is hard, though any chess player can decide at first glance if a position is sensible or not.

"No excess promotions" allows everybody to determine whether an arbitrary position satisfies it or not. It provides a practical upper limit: many positions without excess promotions do not look like human play and some positions with excess promotions did occur in human play.

My gripe with your count of possible and legal positions only is that the vast majority of these positions is not sensible, with no practical merit.

Looking at human play is not going to solve chess, and trying to force the solution to be limited by it is just bad science.

That's not specific to my notion of rankable positions.

That is pretty much inherent in any quantifiable notion of position.

This whole thread is about quantifiable results yet you keep complaining that it doesn't cater to your unquantifiable notion.

#85
As per your insightful post #13 most of your positions are with 9 excess promotions. Who cares how many such positions with about 9 excess promotions there are? My quantifyable question is: how many legal positions are there without any excess promotions for 32, 31... 2 men?

I think there are many many positions where no chess player can decide "at first glance" if a position is sensible or not. And what about chess positions where Magnus Carlson on purpose plays stupid moves to give his opponent a little chance? They do not look sensible at all.

While there are many positions on which all chess players will agree whether they are sensible or not, I imagine there is a set of a thousand or so chess positions such that no two chess players will agree on what subset is of them is sensible.

Which is to say that it is a highly subjective and somewhat vague notion.

#87
The vast vast majority of counted positions looks like this:

It is perfectly legal, but it will never ever happen in any real game of chess. Hence the objective and precise proposition to consider positions with no excess promotions.

What fraction of positions with no excess promotions do you expect to look like they could happen in a real game of chess? If it's tiny then what does your proposition achieve?

#90
I do not know what fraction of positions with no excess promotions would be real chess positions, but the fraction is surely more than the present fraction, so it at least comes closer to reality.

Then present number 10^44 is not a number of chess positions but rather a combinatorial number related to a chess board and 9 boxes of chess men.
The number of chess positions is the number of positions that can result from a game of chess.
Excess promotions happen in like 1 game out of 1000.
Underpromotions happen in like 1 game out of 1000.

Positions with multiple excess promotions and multiple underpromotions are no real chess positions. Nevertheless they make up the vast majority of the number 10^44.
Real chess positions occur in less than 1 per million.

At least that we agree on.

That is exactly what I estimated at 4.5*10^44

You're somehow unsatisfied that the vast majority of those look weird, and ask me to estimate the much less interesting number of positions with no promotions, which excludes tons of positions occurring in human game play, and which STILL has the vast majority looking weird. Just less weird.

Well, you're welcome to do the estimation yourself. I can even help you by preparing a random sample of 10000 positions with no promotions. You'll just have to determine how many of them are legal. Let me know if you want to proceed with that.

#92
Yes, I wellcome a random sample of FEN with no excess promotions, determine how many are legal and how many are sensible and maybe draft a new criterion for sensible.

See the noproms branch at

https://github.com/tromp/ChessPositionRanking/tree/noproms

In particular, I prepared samples of size 100, 1k, and 10k in sortedRnd{100,1k,10k}FENs

and labeled blatantly illegal ones in sortedRnd{100,1k,10k}Research

$ grep -c Illegal sortedRnd{100,1k,10k}Research
sortedRnd100Research:70
sortedRnd1kResearch:800
sortedRnd10kResearch:8056

Please tell us which of these 30 you "decide at first glance" to be sensible:

$ grep -v Illegal sortedRnd100Research | sed 's/^I.*//'
n2N4/1P1pQrB1/2b2rP1/bpp3pp/Kp3p2/PN1PP3/2k3pn/3RR2q w - - 0 1
3q1nR1/p2b3P/1p2PB1r/1r1kp1bB/1P2p1n1/p3P1P1/2ppK1P1/QN6 w - - 0 1
5R2/pK1p4/1pNbp1bn/PpB3np/3QP1NP/B4qpP/r2P1pk1/3r3R w - - 3 6
b2R4/P1pRp1Kn/k3r1p1/2p1Pr1p/4p3/N1PqpnQ1/1P1NPPB1/2b5 w - - 1 3
n1n3qk/p3p3/P2QPpp1/1B2p2p/P3PK2/N3PpRB/Rr2rp1P/4b3 w - - 0 5
3q4/2nn1rpp/2p2bpP/rPP2p1N/2P2P1p/pk1bRK1p/4B1P1/5QRN w - - 3 4
n2BQ3/p2B1b2/PPpNp2p/1r1RPK2/r5p1/qRN2p1b/1P2PkP1/6n1 w - - 0 1
br1B4/2pQ2r1/2pPP1R1/B2pP1pp/2p5/p3NP2/1k1qN1PP/R2Knn2 w - - 1 2
1K5N/p1p1PP2/1qP3rp/1Bb2RPP/bp1p2P1/2P2R1p/1k1p3r/4nQN1 w - - 0 1
K7/Q1p1k1p1/p1P3np/N1P2r2/1B1Pb1bP/1pRpp1PP/1N1P1B2/q4n2 w - - 0 6
1r3nn1/1pp1p1P1/pPbpN2p/4B3/P2P1P2/pkP1RB1P/2p5/2Qqb1NK w - - 3 5
3Rb1nq/1prB1R2/2P1pK1k/n4Ppp/P6N/pP1PPN1P/2p3Pp/5r2 w - - 3 4
8/p1NPr3/2Kp2b1/1PpP1Pb1/1n2p1Rp/PQ1R1Pqp/p2nPr1P/3N1k2 w - - 0 1
4b3/2P1pnRn/P1KpPNP1/1p1Pp3/R3rP2/1r1PQ3/B1b5/1N1B1k1q w - - 0 1
R5Q1/pPP1RP2/N1PB1P2/3kP2n/K4pNp/4pr2/qp1rnP1B/4b2b w - - 0 1
4Bk1n/3r1p1p/1K2qR2/8/NP1ppP2/ppb1PNbQ/PPpP1P1R/2rnB3 w - - 3 5
2br4/1p2R3/1p3p1K/3P1P1n/P2pPNrP/p2Pk1Pp/1QqbN1nP/B4R2 w - - 0 1
B1b3n1/4P3/R5Pb/2Pp1pq1/2nrpkNp/p2Pp3/PNQ1PPP1/1r2K1B1 w - - 0 4
k1N1RK2/P1ppp3/B1bpp3/1r2P1pP/1q1Rp2P/4pn2/2P1rQ2/1B2n1N1 w - g6 1 2
1B4nb/2p2r2/1P1Q4/B1pP1Pp1/ppPNP3/6r1/NPp2RPk/1qK1bR2 w - g6 1 3
2Q4N/R4B2/1K1B1bpp/P2P1p1k/PNp2p1n/1qrP1Pn1/rpR5/3b4 b - - 0 1
2N5/pb5p/P1pp3P/p2P1Rqn/KB1kB1Q1/bN1n1p2/2Pr2P1/r3R3 b - - 0 1
4K3/1r4p1/BP6/P1B1kpnQ/2bRP3/1p1Pp1Pq/Npr2Pp1/1n1b3N b - - 0 1
K4nN1/p1pp4/bb2pPp1/rNB1P1PR/4np2/p5PP/PR4Qk/2Br4 b - - 1 3
2N1n1Bq/pp1N4/P1Rn3b/3Ppb2/rP1k1p2/1pB3rp/1KQP1pPP/2R5 b - - 2 3
4N2k/QP2qp2/1nPPp3/1PRprPp1/4Pb2/p3PR2/rpN1B1b1/1n1K4 b - - 0 2
1k1r4/3prNP1/RbqP1pN1/Q1P5/1p1P1n2/1BP1R2b/pKpP1P1n/4B3 b - - 0 1
1N1bn2B/b1pBNnpK/1kQ3P1/2R5/P2pppP1/3RPP2/p1pPrP2/1r6 b - - 3 4
8/rPN3pn/bp1b1PPp/nN4PP/r3PPp1/PB2Q1q1/KR1R3B/2k5 b - - 0 1
1NB3b1/4pk1P/2pp1pr1/P1q1RKQ1/1pP2pp1/1B2P1p1/R4NP1/n5nr b - c3 0 3

#94
At first glance:

legal: 1 3 4 7 8 22 23
sensible: 1 7 8 22

So, taking a page from your own book and extrapolating from something completely unwarranted ...4/30 = 13.3% of positions are sensible.  That moves the order of magnitude by...

<1

(not 6 as you posited with your 1-in-a-million claim)

Congratulations.  But you're still 25 orders of magnitude away from backing up your claim that chess will be solved by traversing only 10^18 positions out of the full 10^44.  It's been pretty clear since the start that you don't really grasp exponents, and don't seem to understand how much larger 10^44 is than 10^18.  You keep trying to shave 20+ orders of magnitude away by positing various position culling criteria...but you haven't figured out that you would have to eliminate 999,999,999,999,999,999,999,999 out of every 1,000,000,000,000,000,000,000,000 positions just to reach your 10^18 number.

That's a tad bit more than "sensible".  If you said that only one single solitary position in all of mankind's chess playing history to this point was "sensible", that would actually still be far less outrageous than what you are claiming now.

#96
No nonsense please:
19201527561695835455154058755564594798074*4/30 = 2.6e+39
No hijacking please: this thread is about the number of positions. Solving chess is another thread.

So these are what you consider sensible:

1

7

8

22

Can you imagine these occurring in human play?

#98
At first glance yes. I first determined which ones with reasons to be illegal, presuming the rest legal without a proof game. I then eliminated those with reasons to be not sensible, presuming the remainder sensible without closer inspection.
1. Last move was a6xBb5+
7. Last move was e6+
8. Last move was Qd2+
22. Last move was Nb3+
Closer inspection might give reasons to declare some of these non sensible as well, but these are at least much more sensible than the previous positions with multiple excess promotions and underpromotions.

Now we're in agreement, this is what I've been saying to you since you showed up in all the threads.  Now all you have to do is comply .  The reason I post in this thread is because I don't like the way you extrapolate and misrepresent Mr. Tromp's fine work.  I notice you are careful not to do that here, but you do it everywhere else on the forums.  But as you will notice, I don't comment here that often.

What well-defined criterion would set those 4 positions apart from the others?

And how many sensible do you find in the larger sortedRnd1kResearch sample?

#101
"What well-defined criterion would set those 4 positions apart from the others?"
To brand a position without excess promotions as not sensible requires multiple criteria.

A 1st well defined criterion is: no 2 bishops of the same color on one side.

A 2nd well defined criterion could be to limit the number of doubled pawns per side, counting tripled pawns as 2 doubled pawns, quadrupled pawns as 3 doubled pawns, quintupled pawns as 4 doubled pawns, sextupled pawns as 5 doubled pawns. A legal position with a sextupled pawn exists, but is not sensible. I propose a maximum of 3 doubled pawns per side.

A 3rd not yet so well defined criterion would relate the positions of the kings to the positions of the pawns. A legal position with e.g. 6 white pawns on the 2nd rank and a black king on the 1st rank is not sensible. This requires more thought. Maybe define 3 zones:
1: below the white pawns,
2: between the white pawns and the black pawns,
3: above the black pawns.
The white king starts in zone 1 and the black king starts in zone 3. In the course of a game as pawns advance and men get traded, kings enter zone 2. In the further cause of a game and with more pawn advances and men traded a white king occasionally gets to zone 3 and a black king occasionally gets to zone 1. So maybe: X white pawns and a black king in zone 2 or Y white pawns and a black king in zone 1.

"And how many sensible do you find in the larger sortedRnd1kResearch sample?"
I have not yet looked at the larger sample.