Recently, ChessBase posted a challenge motivated by a 1908 study by Carl Behting, which has for many years defied computer analysis:
http://www.chessbase.com/newsdetail.asp?newsid=8388 The most recent challenge was not to solve the study by computer, but to determine whether the study is "cooked" by allowing a second solution. It is ironic that to resolve this challenge a computer is required, as we shall see.
The starting position of the study is:
Carl Behting, Baltische Schachblaetter 1908 
The main line of the solution runs:
1.Kc6!! g1Q 2.Nxh4! Qh1+ 3.Nhf3 And it is clear that the black king is imprisoned (there are of course several alternative variations, which are shown in the attached PGN.) Such prisons and fortresses are notoriously difficult for computers to handle, although they might stumble upon this draw by accident. I imagine a Monte Carlo analysis might be a fruitful way to proceed.
John Nunn proposes a possible cook, based on the idea that many Q v 2N endings are drawn. Here is his proposed potential cook:
1.Ng7+ Kg5 2.Nf3+ Kg4 3.Ke4 h3 4.Nf5 g1Q 5.Nxg1 h2 6.Nxh6 Kh5 7.Nf3 h1Q 8.Nf5 Kg4 9.Ne3+ Kg3 10.Nf5+ Kf2 11.N5d4
Challenge Position 
Can Black win?
John Nunn writes "Since in general positions with Q v 2N are drawn, if Black is to win he has to make progress without allowing an exchange of pawns."
The rub here is the "in general". In fact, if we remove the two pawns from the Challenge Position Black wins. The reason is that the knights supporting one another like that are actually quite vulnerable to the approach of the enemy king. The win consists of driving the king between the knights, so that neither knight can move since that would lose the other knight. Then drive the defending king to the edge of the board with the queen, and then approach with the king threatening mate, while simultaneously preventing the knights from getting back into the action without becoming en prise.
Let's see how this works in the Challenge Position without the two pawns:
C1 1...Qh7+ 2.Kf4 Qd3 3.Ke5 Ke3 The black king is now already attacking both knights. Next, drive away the white king:
4.Kd5 Qc3 5.Ke5 Qc8 6.Kd5 6...Qc7! 7.Ke6 Ke4 The white king has been separated from the knights. Now drive it to the edge of the board:
8.Kf6 Qd7 9.Kg6 Qe7 10.Kh5 Finally, drive the king into the corner and win by creating mating threats:
10...Qf6 11.Kg4 Qg6+ 12.Kh3 Qg8 13.Kh2 Qg4
14.Kh1 Ke3 15.Kh2 Kf2-+ This win is not completely straightforward, but understandable.
In the 18th century the members of the
Modenese School (Lolli, Ponziani, del Rio) thought that Q v 2N was a general win, precisely because they considered the starting position with the knights protecting each other the best defensive configuration. If that were the case, then the Challenge Position should turn out to be a straightforward win. However, things are a little more complicated.
To see this, consider the following position that Lolli thought to be a win:
Lolli, 1763 1.Kg3 Kd3? Black can draw with 1...Nd6 and several other moves.
2.Qd5+? Instead, White can win with 2.Kf4!
2...Kc3 3.Kf4 Nd3+ 4.Kf3 C2 This is an important turning point.
Lolli now mistakenly returns the knights to a mutually protecting configuration with 4...Nde5+? which loses to 5.Kf4 in a manner similar to that shown for position C1 above. Instead, Black has several drawing moves, including
4...Nb4 C3 Play might continue 5.Qd1 Nd2+ 6.Ke2 Nb3 7.Qe1+ Kc4 8..Qh4+ Kc3 9.Qf6+ Kc4 and White can't make progress.
As far as I know, this drawing move was first pointed out in the first edition of von Bilguer's
Handbuch des Schachspiels in 1843, with analysis most likely by von der Lasa. Volume IV of the ECE (Beograd 1989) instead attributes C3 to Lolli. John Nunn in his
Secrets of Pawnless Endings (2nd edition, 2002), refering to the ECE, coins the drawing method of having the knights adjacent to one another as the "Lolli defence". According to the
Handbuch, the general idea of positioning the two knights adjacent to one another to better keep away the attacking king seems to have first been published by Mendheim in 1832.
However, simply putting the knights adjacent to one another (with the king close by) is no guarantee of a draw. The
Handbuch considers the following position drawn:
However, 1.Qc7! wins (the
Handbuch only considers 1.Qe6? Kg7 when Black indeed draws).
What does all that have to do with our Challenge Position? One point is that Q v 2N, which very likely needs to be understood to solve it, is complicated. When the knights are not reasonably consolidated near their king, it can be almost impossible to gauge whether they can consolidate to achive the draw, or whether the queen wins. According to Noam Elkies "No human can reliably tell whether a given KQKNN position is won or drawn with best play or estimate how many moves a given won position requires..." (ICGA Journal, v. 24 2001, p. 96).
While this may sound discouraging, it suggests that endgame tablebases will be needed to resolve the challenge. Since 6-man tablebases are now readily available, for example at
http://www.k4it.de/index.php?topic=egtb&lang=en perhaps we can reduce the 7-man Challenge Position to a known 6-man tablebase position. This may sound a little dissatisfying as we could not do that in a regular game (although it would be possible in a correspondence game), but it is similar to a Mathematician solving a problem by reducing it to a previously solved problem. Furthermore, this material configuration almost never occurs in practice to begin with, so we are mostly on theoretical grounds anyway.
Returning to the Challenge Position:
we notice that White can attack the pawn on c4 with Kd5. Since it might be awkward to have to keep defending the pawn, let's check with the 6-man tablebases whether Black can win even without the c4 pawn. Unfortunately, the first attempt of just looking at the 6-man position with the white king on c4 and no black pawn returns a draw. However, trying various positions of the black king, we find that with the black king on g3 black wins:
C4 The tablebase considers this a mate in 50. This result is pretty convincing evidence that we should be able to force a transition to a won 6-man ending from the Challenge Position pretty quickly. It also makes sense that g3 is a better position for the black king than f2, since in order to penetrate the fortress the king should reach e4 via f4, since he cannot reach e3 because of the white pawn on d2.
The quickest winning move from the Challenge Position is:
1...Kg2! The idea is that if now 2.Kd5 Kg3 3.Kxc4 we have reached the winning position C4, while if 3.Ke4 Qh7+ the black king gets to f4 quickly. 1...Kg3 right away runs into 2.Ne2+, when Black best retraces his steps.
As the attached PGN demonstrates, Black can force a transition to a won 6-man position in at most 12 moves, which is also confirmed by a complete 7-man database, currently not publicly available. In the PGN we stop whenever a 6-man position is reached, since the result can then be looked up exactly.
We have thus proven that the potential cook is in fact not a cook, so the study is sound!
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A Behting study and ChessBase Challenge (Read 8900 times)
