Timeline for What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 2, 2013 at 5:51 | comment | added | The Masked Avenger | Of course, I need also to convince you that for any board without pawns, that there are many more with pawns. I suspect that can be done by showing that all but a small fraction of such boards are one or two legal moves away from a pawn capture or promotion, and that it can happen in more than one way. | |
| Aug 2, 2013 at 5:39 | comment | added | The Masked Avenger | What I have at present is pretty hairy and incomplete, but here is a sample. For any board B, look at B* which has all pieces removed except for white pawns. Computing the number of distinct B* is interesting with an upper bound of 64 choose 8 + similar terms = roughly 64 choose 8 or 4.4 x 10^9 (5.1 if you want less roughly). No pawn captures and no promotions means 6^8 positions; one promotion means 8x6^7. One pawn capture and no promotions is about 600x6^6; with promotion about three times as many. I am still enumerating feasible 8 pawn positions but have not reached 1% yet. | |
| Aug 2, 2013 at 5:28 | comment | added | Joel David Hamkins | I also expect such kind of reasoning to succeed (and I did vote up your answer), but can you carry through your analysis with a definite calculation? For example, there are many legal collection of pieces with no pawns, and so we would seem to need an analysis showing what fraction of the positions have sufficient pawns to carry through your idea. But as I've said, I think you will succeed---let's see some definite calculations! | |
| Aug 2, 2013 at 5:21 | comment | added | The Masked Avenger | You may find an additional factor of about 1/120 by analyzing pawns of one color. In addition to the fact that most of them inhabit just 3/4 of the board (at most one pawn can be promoted at any time), certain monotone arrangements are not allowed (e.g. a2,a3, and b2). I now think the actual ratio you seek is closer to 2^-24. I am finding a preliminary analysis by hand of feasible 8-pawn positions complex but well within reach of a computer analysis. | |
| Aug 2, 2013 at 5:11 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |