52 ( +1 | -1 ) endgame analysisi think i've mastered most endgame algorithms.. even some of the really complex ones, there's just one that's beyond my grasp.. the 2 rook + 2 bishop + 2 knight + queen + 8 pawn vs. 2 rook + 2 bishop + 2 knight + queen + 8 pawn.. for simplicity let's say none of the pawns have moved off the 2nd rank and that both sides can castle on both sides.
is this a forced win for white or a draw? i've been fairly successful in this endgame but can't say i've been forcing perfect moves from the start
42 ( +1 | -1 ) olympioAs the accepted definition of an endgame is a position where King safety isn't a major concern I do not think that the position in question would be classified as an endgame.
I personally think that the position in question is a draw but am not nearly strong enough to prove my assertion. It is a very difficult position and easy to misplay. If I thought it was a win for either side I would have to quit playing chess and concentrate on backgammon.
149 ( +1 | -1 ) More than one positionOlympio specified that no Pawns and no Rooks have been moved.
At first sight, and if my calculations are correct, this problem allows for 353,419 different positions:
(36 * 35 / 2) * (34 * 33 / 2) - 11
Eleven positions have to be substracted, since two or more knights cannot give check simultaneously.
From the position of the Knights it is easy to see which side has the move. If White has both Knights on the same color, and Black has Knights on different colors, it is Black to move, etc.
White Black ----- to move same different --- Black same same --- White different same --- Black different different --- White
From this consideration it follows that, when one of the sides is giving check, there exist certain illegal positions which were included in the above number. For instance, when White is giving check and his other Knight is on a dark square, the Black Knights must necessarily occupy squares of different color.
Thus the number of positions is somewhat less than 353,419. It would require a more detailed analysis to determine the exact number.
Many of these positions lose immediately, for instance positions where a Knight can be taken (and the opponent cannot do likewise), positions where two Knights can be forked, etc.
As to the initial position and all positions with 32 men, it is interesting to note that the 32-men tablebase requires less computer memory than the 31-men tablebase.
20 ( +1 | -1 ) not elevenOf course my statement about 11 positions having to be substracted, is incorrect: as soon as two Knights are giving check simultaneously, a whole class of positions has to be substracted, namely all the placements of the other two Knights, etc.
" .. the 2 rook + 2 bishop + 2 knight + queen + 8 pawn vs. 2 rook + 2 bishop + 2 knight + queen + 8 pawn.. for simplicity let's say none of the pawns have moved off the 2nd rank and that both sides can castle on both sides. "
21 ( +1 | -1 ) In the initial position you describe olympio, I think the winner is based on which King is in ' distant opposition', this is very typical in most endgames like this, I all depends on whos go it is.
6 ( +1 | -1 ) pebblesKnights cannot be everywere. They can be only on 3-6th files or b1, g1, b8 or g8.
102 ( +1 | -1 ) SoikinsI explained where the Knights can be in a previous post:
The first White Knight can be on b1, g1, b8, g8, and any of the 32 squares of the 3rd-6th row, a total of 36 possible squares; the second White Knight can be on any of the 35 remaining squares. Both White Knights together can be on 36 * 35 / 2 squares. Etc.
I also explained that if one Knight is giving check (for instance White Knight on d6), no other Knight can be in a position where it would give check; there are 11 such cases and each case corresponds with a certain number of positions which cannot be legally arrived at. Also, when one Knight gives check (and none of the others does), the colors of the squares where the other Knights stand on, cannot be chosen ad lib.
I also explained that, from the colors on which the Knights are, it can be deduced which side has the move.
I wonder if someone at GK can solve the easy problem of how many legal positions there exist exactly, where all the Pawns are on their initial squares and where both Kings can still castle short and long.
14 ( +1 | -1 ) ????if you have the bishops and no pawns have moved....the bishops are on their original squares and castling is impossible....or am i not reading this right?
80 ( +1 | -1 ) Castlingis STILL possible, is what I wrote; for instance, at the beginning of a game, castling is still possible.
The problem, as formulated in my previous post, doesn't say anything about the presence or absence of Bishops. It is not equivalent to the "Olympio's positions" problem (where all pieces are still on the board), which is only a subproblem of it. Bishops are either still standing on their original squares or they have been captured.
But if you prefer that castling should be possible immediately, here's a different problem:
How many legal positions do there exist exactly, where all the Pawns are on their initial squares and where both sides can castle on their next turn? This problem has some interesting pitfalls.
Obviously, this last problem is NOT a subproblem of the previous one.