@12442
"If are rare meant never happen that would have been worth writing."
++ Promotions to a piece not previously captured happen, but are rare.
Underpromotions happen but are rare.
The combination of both rare events: underpromotion to a piece not previously captured never happens with optimal play by both sides.
"a lot of the essential analysis needed to solve chess is of imperfect play"
++ No. Weakly solving chess means hopping from the initial position to other drawn positions until ending in a certain draw. All positions won for white are pitfalls for black. All positions won for black are pitfalls for white. For each position won for white there is one mirror image position won for black. None of these are relevant to weakly solving Chess.
"we don't KNOW it is imperfect until the analysis is done"
++ There is no analysis needed to tell 1 e4 e5 2 Ba6? is imperfect.
"your sample of 10,000 positions" ++ Which is the summary of 10^17 positions.
"Do tell me how many of those positions had multiple underpromotions?"
++ Underpromotions to pieces not previously captured? None. Zero.
"chess is much harder than checkers"
++ Yes, but at the same time it is more logical and thus easier to prune.
In checkers the men can only move forward, capture is compulsory, and it is compulsory to capture more material if given a choice.
In Chess the non-pawn pieces can move backwards, but 1 Nf3 d5 2 Ng1 can be rejected as illogical.
Capture or recapture are not legally compulsory, but often logically.
If I capture your queen you are not obliged to recapture my queen, but you need a very good reason not to.
If you have a choice between capturing my pawn or my queen, you are not legally forced to capture my queen, but you need a very good reason to capture the pawn instead.
If you promote a pawn, you can freely choose between a queen, a rook, a bishop, or a knight, but you need a very good reason not to chose the queen.
(3*10^37 * 2 / 2 * 10.9456 / 10,000)^0.5 = 1.8*10^17
positions are relevant to weakly solving Chess.
"Everything about your calculation is wrong."
++ Everything about this calculation is right and certainly more than your 10^30 without any calculation at all.
Reading through the recent discussion causes some things to be notable.
Martin had noted 1) e4 e5 white resigns and e4 e5 black resigns and e4 e5 game continuing with white on move if I remember correctly.
Three different 'positions'.
You could also have e4 e5 Draw Agreed.
So that's four.
Yes, also arguably "draw offer made". In the Basic Rules it refers to a draw agreement as a single event, but it's difficult to see how that is practicable without one player first making an offer and the other subsequently accepting.
But I realise I should have been more careful in my definition of "fp-position". The set of continuations from each of these n-positions is empty, so they would represent a single fp-position according to that definition. I've corrected it.
And add three more 'attributes' if that's the right word.
e4 e5 White Flag falls and e4 e5 Black Flag Falls and e4 e5 Both Flags Down.
Here I think we're at cross purposes. Flags don't fall under Basic Rules and Tromp was obviously not attempting to count competition rules positions.
Clocks are also not easy things to take into account in solving. The series of n-positions constituting a game becomes continuous because the clocks decrement continuously and it's difficult to specify what is regarded as possible continuations if, say, the clock of the player on the move is down to 1ms.
But also I don't think "attribute" would be the right word anyway. Under competition rules you would have specified three extra positions (n-positions or fp-positions) having the same diagram as an attribute. The diagram is an attribute of a position, not vice versa in the usual way of looking at it.
Now you've got seven different positions from the same arrangement of pieces with the same player on move.
But as I said not in Tromp's figures.
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I think 1) Nf3 Nf6 does a better job on some of this because you can have even more 'positions' from that.
Nf3 Nf6 Draw by Threefold Repetition. Yes!
Again, wrong game.
And you could have Nf3 Nf6 Draw by 50 Move Rule because the knights could have moved around the board not just bounced back to their stables.
Ditto.
So now you've got Ten 'positions' with the same arrangement on the table.
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But even Nf3 Nf6 doesn't allow for either player on move.
Black Cannot be on move there no matter how it got there.
You can triangulate a King you can't triangulate a knight to shift whose move it is.
This is true. Tromp's reduction for illegal positions is intended to take account of that.
Note that with a position of e4 e5 without knowing all the moves we can add more attributes.
The Kings could have moved and then moved back.
So then there's 4 positions with white pawn e4 black pawn e5
Castling Illegal Both Sides.
But no way there to have Castling Illegal White Only nor Black Only.
I think all castling options are possible and that is accounted for in Tromp's figure.
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Thought about it and realized you could switch whose move it is with white pawn E4 and black pawn E5.
Would that be good for a student?
Like this:
1) e4 e5 2) Ke2 Ke7 3) Kf3 Kf6 4) Ke3 Ke7 5) Ke2 Ke8 6) Ke1
Now its the same arrangement but Black's on move instead !!
Or just play 1.e3 first.
So now you've got another 'Attribute' possibility on that position.
Not with my understanding(s) of position. The n-positions are obviously different and so are the fp-positions, because all continuations start with a move by a different player. So you've got different positions with the same diagram attributes.
You could have done it with the Queens too or the f-bishops.
Without 'losing castling'.
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knight moving versus knight moving cannot triangulate ...
Queens moving can 'switch' and so can bishops or rooks. And Kings.
Only knights and pawns can't do it.
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Later in the game with more pieces moved from their originals all the attributes become possible with the same arrangement on the board.
How many potential 'positions' does that make possible with the same piece arrangement?
Could be large when you multiply some of the attribute possibilities by each other (some of them won't multiply properly)
The product could move up over 100 with the same arrangement on the board?
Beginning to look like it.
It depends what game you're talking about.
Under competition rules the number of legal n-positions or fp-positions with this diagram.
is larger than Tromp's figure for legal chess positions (definitely over 100).
Under basic rules the number of n-positions is countably infinite so long as you don't countenance an uncountable number of draw offers, but the number of fp-positions is 22 so long as you don't countenance simultaneous resignations, draw offers or draw acceptances. Or just 4 if you don't countenance the events at all and simplify art. 4 to define piece moves as individual events (which is the number of positions Tromp counts with that diagram).