Recently in another topic I mentioned the “two bishops against two knights” chess endgame problem. I claimed it was investigated over two decades ago by a computer program and established that it is a win situation for the two bishops’ side. Then I was unable to Google a solid reference for my claim.
I believe that subject to the ambiguity in what is meant by “a win situation for the two bishops”, your recollection is correct.
The 6-piece pawnless endgames were were first analyzed systematically by Lewis Stiller starting in the late 1980s and reported in his papers in 1991 and 1992. The storage technologies available at this time meant that only summarized results could be saved, such as the longest win, and the total number of wins, draws and losses. I can’t find these papers online, but the results also appear in Stiller (1995) and there’s a summary of the state of the art in Thompson (1996).
For the KBBKNN ending Stiller only analyzed positions with the two bishops on opposite coloured squares (and I think with white to move), and reported that the longest win for white was 37 moves and the percentage of wins for white was 63%.You probably also want to note Stiller’s caveat:
The percent-win can be misleading because of the advantage of the first move in a random position—White can often capture a piece in one move—and because it includes positions in which Black is in check.
So I think if you said “mostly a win for the two bishops from a random position with bishops on opposite-coloured squares, with the player with the bishops to move” that would be a fair summary of the facts.
Modern tablebases usually also include positions with the two bishops on the same colour square, so that analyses of these databases will give different results to Stiller. For example, according to Kirill Kryukov, the KBBKNN positions split like this:
With white (bishops) to move: 28429 losses, 885809752 draws (76%), 282912378 wins (24%)
With black (knights) to move: 54327970 losses (4%), 1247006005 draws (96%), 154105 wins
How could you have found this using Google? Well, it always helps to know of specialized databases to search (because the results tend to be of higher quality). I used Google Scholar to search for academic papers relevant to the keywords “6-piece chess endgame” and that returned Thompson (1996) as the first hit, and reading Thompson’s summary of the state of the art led me to the Stiller papers. Of course, domain expertise is a big help too: I realised after discovering Stiller (1995) in the course of this search that I have a copy of this on my bookshelves.
References
Lewis Stiller (1991), “Some results from a massively parallel retrograde analysis”, ICCA Journal 14:3, pp. 129–134.
Lewis Stiller (1992). “KQNKRR”. ICCA Journal 15:1, pp. 16–18.
What can I say—that this is the answer one can only wish. Bravo!
The information about this KBBKNN situation I’ve read around 1987, must have been a little deformed by that magazine. I’ve took them too seriously.
Now, I am going to investigate another piece I recall and I couldn’t find it online until now. This time from the Science magazine sometimes during 1980′s. The title I remember was “Never Out of Sorts”.
I believe that subject to the ambiguity in what is meant by “a win situation for the two bishops”, your recollection is correct.
The 6-piece pawnless endgames were were first analyzed systematically by Lewis Stiller starting in the late 1980s and reported in his papers in 1991 and 1992. The storage technologies available at this time meant that only summarized results could be saved, such as the longest win, and the total number of wins, draws and losses. I can’t find these papers online, but the results also appear in Stiller (1995) and there’s a summary of the state of the art in Thompson (1996).
For the KBBKNN ending Stiller only analyzed positions with the two bishops on opposite coloured squares (and I think with white to move), and reported that the longest win for white was 37 moves and the percentage of wins for white was 63%.You probably also want to note Stiller’s caveat:
So I think if you said “mostly a win for the two bishops from a random position with bishops on opposite-coloured squares, with the player with the bishops to move” that would be a fair summary of the facts.
Modern tablebases usually also include positions with the two bishops on the same colour square, so that analyses of these databases will give different results to Stiller. For example, according to Kirill Kryukov, the KBBKNN positions split like this:
How could you have found this using Google? Well, it always helps to know of specialized databases to search (because the results tend to be of higher quality). I used Google Scholar to search for academic papers relevant to the keywords “6-piece chess endgame” and that returned Thompson (1996) as the first hit, and reading Thompson’s summary of the state of the art led me to the Stiller papers. Of course, domain expertise is a big help too: I realised after discovering Stiller (1995) in the course of this search that I have a copy of this on my bookshelves.
References
Lewis Stiller (1991), “Some results from a massively parallel retrograde analysis”, ICCA Journal 14:3, pp. 129–134.
Lewis Stiller (1992). “KQNKRR”. ICCA Journal 15:1, pp. 16–18.
Lewis Stiller (1995). “Multilinear algebra and chess endgames”, in Games of No Chance edited by Richard J. Nowakowski, MSRI Publications Volume 29.
Ken Thompson (1996). “6-piece endgames”, ICCA Journal 19:4 pp. 215–226.
What can I say—that this is the answer one can only wish. Bravo!
The information about this KBBKNN situation I’ve read around 1987, must have been a little deformed by that magazine. I’ve took them too seriously.
Now, I am going to investigate another piece I recall and I couldn’t find it online until now. This time from the Science magazine sometimes during 1980′s. The title I remember was “Never Out of Sorts”.
The tables of contents for Science magazine are online. Looking through these might jog your memory. But there are quite a lot of issues.
Inside Google Scholar it is easy to find:
Baryon number conservation law ended