Author: blass uri
Date: 15:01:29 07/12/99
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On July 12, 1999 at 15:50:08, KarinsDad wrote: <snipped> >Uri, > >I have looked at what you wrote and your equations. I even tried several >promotion examples for your conditional equation. I sort of understand that the >conditional equation works (and always will for all cases), however, I still do >not understand good/bad pawns. Could you please rephrase it? It could just be a >language thing (either that or I am just being dense). Please be specific (i.e. >verbose) since the previous explanation was not enough. > >Thanks, > >KarinsDad :) Declarations: passed pawns,pawns on the same file and promoted pawns are declared to be good pawns for example if white has pawns at h2,h3 then they are good pawns if white has a pawn at h4 and black has not a pawn at h5, h6 or h7 then this pawn is a good pawn. if white has a knight that was in the past a pawn then the knight is also a good pawn. The other pawns are declared to be bad pawns. Observation 1:white can get a new good pawn in a file only after a capture. Observation 2:white can get a new good pawn in a file only if the difference between white pawns and black pawns in the file is increasing after the capture(I substract black from white) Explanation:white gets a passed pawn only if the black pawn leave the file because of a capture and white gets more than one pawn in a file only after a capture of white that increase the number of white pawns in the file. observation 3:If the number of white pawns in a file does not increase after a capture then white can get at most one good pawn in this file observation 4:white can get at most 2 new good pawns after a capture Explanation:by observation 3 white can get more than one only if the number of white pawns in the file does increase after a capture. If it increased from 0 to 1 then white gets only one new good pawn. If it increased from 1 to 2 then white gets 2 new good pawns If there were at least 2 white pawns in the file before the capture then the old pawns are not *new* good pawns and only the new pawn in the file can be a new good pawn. When white does a capture then there are some options: a)white captured a piece with a piece and in this case for bith sides the number of good pawns was not changed(see observation 2). b)white captured a pawn with a piece and in this case there is only one file that white can get new good pawns in it and white can get only one good pawn in this file(see observation 3) black cannot get a new good pawn in this case (see observation 2) c)white captured a piece with a pawn(without loss of generality axb). In this case white can get good pawns only in the b file(observation 2) and at most 2 new good pawns(observation 4) black can get good pawns only at the a file(observation 2) black can get at most one new good pawn in this file(observation 3) d)White captured a pawn with a pawn(for example without loss of generality exf) then white can get good pawns only in the f file(observation 2) and can get at most 2 good pawns(observation 4) black can get good pawns only at the e file(observation 2) black can get at most one new good pawn in this file(observation 3) After a capture of white we see that white can get at most 2 good pawns and After a capture of black we see that white can get at most 1 good pawn(the same as black can get after a capture of white). Consequence: The number of good white pawns is at most the sum of twice the number of captures of white and the number of capture of black. In other words 2*(15-y1-y2-y3-y4-y5)+(15-x1-x2-x3-x4-x5) is the relevant upper bound. i is a lower bound to the number of white promoted pawns The number of promoted pawns is smaller or equal to the number of good pawns by definition of good pawns so we get: i<=2*(15-y1-y2-y3-y4-y5)+(15-x1-x2-x3-x4-x5 We get the second inequality in a similiar way. Uri
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