[QUOTE=Brian-E;258865]For example with the rook, is the configuration below legal?[/QUOTE]

I see that this was answered as a perfectly legal position with black to move, i.e. that white has just moved bxa8(R)+. It would also have been legal for the pawn to have simply moved without capturing, i.e. from a7 to a8(R)+.

But I thought I would point out one more thing: The problem of mirroring. Mirroring can only be done side to side where a pawn is on the board or it has just moved in the case of promotion like this. So this would be a legal position if it was mirrored to the g & h files but not legal if it was mirrored to the 1, 2, and 3 ranks.

It's quite an interesting issue...the problem of mirroring to shorten the problem creates additional difficulties.

[QUOTE=gd_barnes;259031]
One question : When determining "different" chess positions, is it considered a different position if it is white to move than if it is black to move even if all of the pieces are in the same place? I would think so because it makes a big difference in whether it is a legal position. If it is white to move, than white can legally be in check -or- even in checkmate. Obviously it would be illegal if it was black to move in the same position.

Gary[/QUOTE]
Gary:
Mr P-1 defines, precisely, a configuration (we would call it colloquially a position) as the arrangement of chess pieces on the board, and a legal position as one that *could* have been reached by two players according to the commonly understood rules of chess. This is what I am working with when I count positions. (note: many of those positions require cooperation to reach; this is not illegal in chess, even if it won't win any games).

He also defines a "position" as the configuration plus all required information to determine all possible outcomes of the game. That includes who is about to move, which rooks in their starting corners might castle, and any pawns at risk of being taken "en passant".

The document I have been uploading *WILL* solve the exact number of legal configurations for 3 pieces on the board, and I will make an effort to check it, but getting to 4 is much more problematic, as there are already a plethora of special cases, much less dealing with all 32 pieces on the board!

[QUOTE=Christenson;259038]Gary:
Mr P-1 defines, precisely, a configuration (we would call it colloquially a position) as the arrangement of chess pieces on the board, and a legal position as one that *could* have been reached by two players according to the commonly understood rules of chess. This is what I am working with when I count positions. (note: many of those positions require cooperation to reach; this is not illegal in chess, even if it won't win any games).

He also defines a "position" as the configuration plus all required information to determine all possible outcomes of the game. That includes who is about to move, which rooks in their starting corners might castle, and any pawns at risk of being taken "en passant".

The document I have been uploading *WILL* solve the exact number of legal configurations for 3 pieces on the board, and I will make an effort to check it, but getting to 4 is much more problematic, as there are already a plethora of special cases, much less dealing with all 32 pieces on the board![/QUOTE]

I wonder if a binary vector would help. 1 bit for different things.

[QUOTE=science_man_88;259042]I wonder if a binary vector would help. 1 bit for different things.[/QUOTE]

SM88, it's not hard to REPRESENT a chess configuration using bits...all those pictures you guys have been presenting actually waste quite a few bits showing us exactly what we are talking about, giving the big picture, as it were....the problem is the sheer complexity of keeping track of and counting all the different special cases.
For an exercise, count the number of illegal 2-King and one bishop positions. The document I've been working on gives an idea of the flavor. Or give a list of all the possible sets of pieces on the board with 2 kings and 2 other pieces.

[QUOTE=gd_barnes;259031]There are several requirements for the king to be able to castle and you've only shown one of them. They are:
1. As you stated, both must be in their original position.
2. Neither king nor rook can have previously moved during the game. Even if they are in their original position, if they have moved and then moved back to their original square, then castling is not allowed.
3. You cannot castle OUT of check.
4. You cannot castle INTO check.
5. You cannot castle OVER check.[/QUOTE]
He will of course be well aware of this. The issue being discussed at the time was what features of a position determine whether in a given position either side can legally castle at some point later in the game, not necessarily on the move now. That is one important feature of a position which is not reflected in the "configuration" of pieces on the board.

[QUOTE]One question : When determining "different" chess positions, is it considered a different position if it is white to move than if it is black to move even if all of the pieces are in the same place? I would think so because it makes a big difference in whether it is a legal position.[/QUOTE]
Yes, and that is an even more important feature as also discussed earlier in this thread.
I'd like to mention yet another one though which I don't think anyone has referred to yet. It was brought to light as a result of a game, I think involving Bronstein, in which the players reached a very closed position (blocked by pawns) and spent a long time manoeuvering their pieces, and in the course of this manoeuvering the two white rooks swapped positions with each other. The question was then asked: was the position with the two white rooks swapped the same as the one before? It was important because of the possibility of claiming a draw if the position repeated for a third time. FIDE, the governing body of international chess, ruled later on that it WAS the same position: so the two white rooks, the two black knights, and so on are not considered to have separate identities. :smile:

Well, when the position graph in the game has a loop in it, it is the same position -- outcomes from that point (except for possibly draws) are all the same.

OK, I now have an "official" number, though not double-checked, for three pieces on the board:
2 125 534

Attached is the somewhat cleaned-up proof. The original is in Open Office Writer format. I'm going to try to count the pawn positions in a different order to see if there can be fewer special cases.

[QUOTE=Christenson;259153]OK, I now have an "official" number, though not double-checked, for three pieces on the board:
2 125 534

Attached is the somewhat cleaned-up proof.[/QUOTE]
Congratulations!

Verifying the last section with its newly modified handling of K+R vs K and K+B vs K is firmly on my list of things to do. Those here who, like you, have a good eye for detecting possibly illegal set-ups might well want to do the same.

I have a very minor quibble for now. You have included in your analysis mentioning the under-promotions the remark:
[QUOTE]though no serious chess competitor would promote a pawn "only" to a rook or bishop[/QUOTE]Well, I agree it's pretty rare, but "serious chess competitors" certainly know about the risk of giving stalemate if you promote to a queen. Here's a nice [URL="http://www.xs4all.nl/%7Etimkr/chess2/minor.htm"]webpage[/URL] by Tim Krabbé listing a large number of serious underpromotions to rook or bishop which have occurred in top international tournaments and matches in recent decades.

Ok, so it sometimes *does* happen...but I'm betting that 3-piece positions are rare, and that stalemate issue doesn't happen in the 3-piece position.

Could you post a position in which promotion to a rook or bishop is better than promotion to a queen for us?

[QUOTE=Christenson;259212]Ok, so it sometimes *does* happen...but I'm betting that 3-piece positions are rare, and that stalemate issue doesn't happen in the 3-piece position.

Could you post a position in which promotion to a rook or bishop is better than promotion to a queen for us?[/QUOTE]

Promotion to bishop is a draw by insufficient mating material. You can't do worse than that, so it's not better than anything else.

An example for the rook case would be wKh5, bKh7, wPf7. Here, f8=R is the only winning move; f8=Q is stalemate.

[QUOTE=ckdo;259246]An example for the rook case would be wKh5, bKh7, wPf7. Here, f8=R is the only winning move; f8=Q is stalemate.[/QUOTE]
And just to vary it slightly, if we place the king on f6 instead of h5 then f8=R is no longer the only winning move, but it is the quickest win (mate next move). And again, f8=Q?? would give stalemate.

Now the stalemates make sense...one side has nowhere to move. Interesting that it's not a loss for the side that can't move. What a way to learn the subtleties of the rules!