To the memory of Stan Goldovski
A first survey of Losing Chess endgame material published up to the end of 1999
John Beasley, 7 St James Road, Harpenden, Herts AL5 4NX, England; October 2000
"The endgame is the most appealing stage of the game, a garden of surprises," wrote David Pritchard about Losing Chess in The encyclopedia of chess variants (1994). However, the literature of Losing Chess is sadly fragmented (a typical mainstream chess magazine carries an article perhaps once every twenty years), and all too many composers and writers, myself included, have spent time and effort rediscovering what had already been published. This document is therefore a first attempt to list published material relating to endgame theory and endgame studies. I think it is reasonably complete as regards what has been published in England (I have excluded articles that merely quote earlier work), but it is less complete in respect of foreign material and its coverage of Russia and Eastern Europe is almost nil. Even so, it seems to me more useful to publish the document as it now is and to let others build on it than to hold it back while yet further researches are made. Given the recent computer-inspired explosion of activity in Losing Chess, the cut-off date is by no means ideal, but it is numerically convenient and I have imposed it in order to crystallize matters.
The document comprises a list of articles and studies in approximate date order (items within the same year being arranged in alphabetical order of author), a list of definitive analyses by computer, and an index by material covering positions with up to five men (on pages 30-32). My thanks are due to Ralf Binnewirtz, Paul Byway, Chris Feather, George Jelliss, Jörg Kuhlmann, Fabrice Liardet, Cedric Lytton, David Pritchard, Ken Whyld, Peter Wood, the library of the British Chess Problem Society (BCPS), and the Kokinklijke Bibliotheek in Den Haag for material (the contributions of Ralf Binnewirtz, Chris Feather, and Fabrice Liardet have been particularly valuable), and to Chris Feather and a singing friend known to me only as Ursula for translations. Round brackets (...) mean that I have relied on a quotation or transcription, square brackets [...] that I am merely reporting a reference and have not seen the item at all. Obvious and unimportant misprints have been ignored or silently corrected, and notation has been standardized even within quotations. Readers should be aware that I can read only English, French, Czech, and Slovak, and that interpretations of material in other languages may be no more than deductions from diagrams, moves, and isolated words looked up in dictionaries.
The distinction between "problems" and "endgames" in Losing Chess is not always clear, but I am treating anything with five men or less as an endgame, and some compositions with more men appear also to deserve inclusion. Positions from play are included only if they have some particular interest. Compositions requiring retrograde analysis are normally excluded, as are positions using other than the normal board and men, but there are a few exceptions in each class. Unless otherwise stated, I assume the rule that stalemate is a win for the player stalemated. Alternative solutions known to me are reported, except where uniqueness of solution is not an issue, but I have made no attempt to verify the correctness of every example. The advent of Stan Goldovskis program Giveaway Wizard should mean that most material published from mid-1998 onwards can be taken as correct (running on a 450MHz Pentium III, the program once took less than three seconds to give a definitive verdict on an ending with which I had struggled unsuccessfully for several weeks), but I have not systematically tested earlier material.
I have no authority to waive the rights of others, but in so far as anything in this document is original with myself it may be copied without payment or formality; I ask only that there be due acknowledgement. And if anyone wishes to incorporate what follows into a more extensive survey, he may do so with my good will.
Articles and studies
1885 Verney, G. H. Chess eccentricities, 1885, article " Take me Chess, invented by Walter Campbell, and played at Boyton Lodge, Wilts, in 1876" within a section "Social Chess" (p 191). The earliest description of the game known to me. The rules are broadly as at present, but the result in the event of stalemate is not defined, promotion is to "any Piece ... which has already been lost", and only active sacrifices are specified as compulsory ("If a player places one of his Pieces in such a position that his opponent can take it, he can insist on his antagonist taking it by saying the words Take me; and the antagonist is bound to take the Piece in the manner the player desires").
There is no specimen game, but the final paragraph is worth quoting in full:
"A curious feature of this game is that not until almost the last move can it be guessed which player will win; for it often occurs that when a player has only one Piece left on the board, his antagonist may by careful play cause this one Piece to take all his own Pieces which may be left."
So the subtlety of Losing Chess endings was recognized even at this early stage.
1901 Schellenberg, P. Dresdner Schach-Kalender (a booklet produced to celebrate the 25th anniversary of the Dresden Chess Association), 1901. A game-like ending: White Bg2, Pf7/a6/c5/d4/e4 (6), Black Ra8, Bg3, Pa7/h5 (4), White to play and win. I have seen only (a) a copy of the front and back covers, the position quoted appearing on the back cover under the heading "Schlagschach - Studie" with the stipulation "Weiß am Zuge gewinnt", (b) a copy of the contents page, which lists nothing apparently relevant, and (c) a copy of the section "Bemerkungen zu dem Titelbildnis" (pp 39-41 according to the contents page, pp 43-45 in reality) which I thought might throw further light but in fact talks only about an ordinary problem on the title page and does not mention the back cover at all. It therefore appears that the study appeared without solution. Klüver, assuming passive captures to be compulsory, gives 1 e5 Bxe5 2 Bxa8 Bxd4 3 f8Q Bxc5 4 Qxc5 h4 5 Qxa7 h3 6 Bg2 hxg2 7 Qf7! in his 1924 Deutsches Wochenshach article (see below), and the manuscript copy in a private collection of problems compiled by T. R. Dawson spells out the details: 7...g1K 8 Qf2 Kxf2 9-10 a8R, or 7...g1N 8 Qf3 Nxf3 9-10 a8N (a8R would also do). Fabrice Liardet points out that 6 Be4 h2 7 Bd3 also works, for example 7...h1B 8 Be4 Bxe4 9 Qa8 Bxa8 10 a7. The name of the editor and presumed composer of the study does not appear in the material I have seen, and I am relying on Klüver for it.
(1914) Markwick, F. W. Stratford Express, 19.xii.1914. A position from a game, after 1 e4 f5 2 exf5 e6 3 fxe6 dxe6 4 Qg4 Qxd2 5 Nxd2 Ba3 6 Qxe6 Bxb2 7 7 Qxe8 Bxc1 8 Qxg8 Rxg8 9 Rxc1 Bh3 10 Nxh3. Black now wins by successive sacrifices: 10...b5 11 Bxb5 Re8 12 Bxe8 Nd7 13 Bxd7 Re8 14 Bxe8 g5 15 Nxg5 a5 16 Nxh7 c6 17 Bxc6 a5 18 Bxa5. This is the earliest complete game known to me, and again we notice that passive captures (4...Qxd2, 6...Bxb2, etc) were treated as compulsory. Supplied to me from the Dawson collection.
1923 Klüver, H. Chess Amateur, iv.1923. Six studies.
(1923) Roese, W. Source not known to me (quoted in Boyers Les jeux déchecs non orthodoxes, 1951). White Rd1, Nh2, Pc2 (3), Black Bf6, Pb4 (2), White to play and win. 1 Ra1 Bxa1 2 c4 bxc3 3 Nf3.
1923 Watney, G. C. Chess Amateur, iv.1923 (from play). White Pe7 (1), Black Nd7, Pf7/h7/h6 (4), White to play and draw. 1 e8K! ("any other move loses") Nf6! ("again any other move loses") 2 Kxf7 Ng8! 3 Kxg8 h5 4 Kxh7 h4 5-6 Kh5 h2 7 Kg4! h1K! ("any other promotion losing"), with Dawsons exclamation marks throughout.
1924 Dawson, T. R. Deutsches Wochenschach, 31.v.1924 (dated from Dawsons notebooks in the BCPS Library). White Rb3, Bf2, Na2/b1, Pe4/g4/d3/g3/c2/g2 (10), Black Bf7 (1), can White win? The answer is Yes, despite the inability of Whites dark-squared bishop to sacrifice itself to Blacks light-squared: 1 Rb6 Bxa2 2 Rg6 Bxb1 3 g5 Bxc2 4 g4 Bxd3 5 Bh4 Bxe4 6 g3 Bxg6 and White is stalemated. There is a reference to a famous orthodox chess endgame by J. G. Campbell in which White draws similarly (no details are given, but I have seen such an ending quoted elsewhere as Chess Players Chronicle, iii.1855, White Kb5, Bg5, Pb6/a4/d4/b2 (6), Black Kf1, Pb7/d6/d5/g3/h3 (6), draw by 1-2 Ba5 and 3 b4). See also Dittmann 1987 and Nagorko 1999.
1924 Klüver, H. Deutsches Wochenschach, 31.v.1924 (no date on the extract seen by me, but dated thus in Dawsons notebooks). Article "Schlagschach" (pp 89-91) in a series "Feenspiele". A general article covering all aspects of the game. Items relating to the endgame are as follows.
(1925) Dawson, T. R. Deutsches Wochenschach, 15.iii.1925. Two items.
Taken from the Dawson notebooks (problems 2211 and 2240).
(1929) Törngren, P. H. Tidskrift, xi-xii.1929. White Ph2 (1), Black Pa6 (1), White to play and win. Supplied to me from the Dawson collection, where the solution is given as 1 h3 a5 2-5 h7 a1R/N 6 h8B; the companion line 5...a1K 6 h8R is not given, but it is a routine win whereas the win with Bh8 v Ra1 is an exception to the normal rule. The diagram in the Dawson collection has a note "cf. V. Onitiu", but I do not know to what this refers.
(1930) Sunyer, J. Els Escacs a Catalunya, viii.1930. White Nc5/h4/c3 (3), Black Pe3 (1), White to play and win (yes, three White knights). Supplied to me from the Dawson collection. Dawson gives no solution and Ken Whyld tells me that there appears to be none in Els Escacs a Catalunya, but I play 1 Nb5 e2 2 Nd4, after which promotions to K, Q, or R lose at once, 2...e1B allows 3 Nc2 Bxh4 4 Ne1 Bxe1 5 Ne6, and 2...e1N allows 3 Ng2 Nxg2 4 Ne4 with two more sacrifices to follow. Fabrice Liardet sends me a second solution, discovered by computer: 1 N3e4 e2 2 Ne6 e1B (if 2...e1N then 3 Ng2 Nxg2 4 Nd4 transposing into my solution) 3 N4c5/Nd8 Bxh4 4 Nd8/Nc5 Bxd8 5 Nd3.
1934 Dawson, T. R. Problemist Fairy Supplement, xii.1934. On an n x n board, set up rows of n pawns one square apart; who wins? This was inspired by Klüvers two-against-two pawn study described below, and proves to be unexpectedly deep. Dawson originally thought he had found a systematic solution valid for all n, but by the time he came to write Caissas wild roses in 1935 he had realized his error; a complete solution is yet to be found, and may well not exist. Although atypical as a chess problem, the problem has attracted attention in the literature of mathematical games, and an excellent account can be found in Winning ways for your mathematical plays (Berlekamp, Conway, and Guy, 1982).
1934 Fabel, K. Problemist Fairy Supplement, viii.1934. White Rc1, Ba6, Pd4/a2/c2 (5), Black Pd2 (1), White to play and draw. Given as best is 1 a4 forcing 1...dxc1B, any other Black move losing. 1 d5 and 1...Bc4 are also given as drawing, "but with fewer winning chances". This is described as "from play, versus H. Klüver", but a note on the manuscript diagram in the Dawson collection describes it as "based on" actual play.
1934 Klüver, H. Problemist Fairy Supplement, x.1934. Two studies.
1935 Slater, E. (presumably E. T. O.) Problemist Fairy Supplement, ii.1935. White Bf8, Nf6 (2), Black Kf7, Pd7/h7 (3), White to play and win. 1 Nxh7 Kxf8 2 Nxf8 d5 3 Ng6 d4 4 Nh4 d3 5 Ng2 d2 6 Ne1, or 2...d6 3-7 Ne4 d1B 8 Nd6. The point is that 1 Nxd7 Kxf8 2 Nxf8 doesnt work because Black can play 2...h6! and meet 3-7 Ng4 h1B 8 Nh6 by 8...Bc6. This is described as "the one exceptional win" with N v B, but Na2 v Bc1 (not reachable from this position) provides a second.
(1938) Dawson, T. R. Stratford Express, 23.xii.1938. A forced-capture problem with eleven men against two (wKg2, Qf5, Bh1, Ne4/h3, Pa5/a4/d4/h4/e2/h2, bBa3, Pb7) ending in a win by stalemate (1 Nd6, 2 Kf3, 3 Ng1, 4 Bg2, 5 Qf6, 6 Bf1, 7 Kf2, 8 e3, 9 Ba6). Quoted in The encyclopedia of chess variants (Pritchard 1994), and dated from the Dawson notebooks in the BCPS Library. In 1977, Panteleit showed an alternative win not depending on stalemate (1 a6, 2 Nd6, 3 Kf3, 4 Kg3, 5 Qc2, 6 Ng5, 7 e3, 8 Bc6, 9 Qc3, 10 a5, 11 Bb5).
1938 Niemann, J. Fairy Chess Review, xii.1938. A one-against-one nightrider study on a 10 x 10 cylindrical board. This is not really within the intended scope of this document (the "nightrider" is a fairy chess piece which moves along straight lines of squares a knights move apart), but like Dawson 1934 it is an interesting example of what is possible on a larger canvas. White is on f1, Black on a9. Black to play is dominated (for example, if he tries a five-step move around the cylinder, a9-c8-e7-f6-i5-a4, White plays a nine-step move to d10 or h10, say f1-h2-j3-b4-d5-f6-h7-j8-b9-d10, and Black must capture). White to move plays to f6, say f1-h2-j3-b4-d5-f6, and the equivalent nine-step move is not available to Black; he has nothing better than a9-c8-e7-g6 or the symmetrically equivalent move a9-i8-g7-e6, and Whites reply f6-h5-j4-b3 or f6-d5-b4-j3 again leaves him dominated.
1947 Fabel, K. Am Rande des Schachbretts, 1947, chapter "Schlagschach" (pp 28-33). A general article including studies, game positions, and observations. Chris Feather translates some passages on knight endings. Page 30: "The endgame of two knights against one is especially interesting. It was described as a win for the single knight by Klüver in his introductory essay in Deutsches Wochenschach but actually it is a draw, a point to which I shall return." Page 33: "Finally a further look at the endgame of two knights against one, which with best play is a draw. Let White have the single knight. There are three distinct cases in all.
"The ending of two knights against two is even more complicated, and furthermore is hardly ever seen in practice."
1947 Fabel, K. and Klüver, H. Fairy Chess Review, x.1947. Two studies.
[1947] Frey, R. L. The new complete Hoyle. This American publication is said by Boyer in Les jeux déchecs non orthodoxes (1951) to contain relevant material, but I have not seen it.
(1947) Niemann, J. Schachmatt, 28.xi.1947. White Qf3, Pa4 (2), Black Kb6, Na6 (2), White to play and win. 1 a5! (1 Qh3 Kc5, or 1 Qd1 Ka7 2 Qa1 Nc7) Kxa5 2 Qh1 (2 Qg2 Ka4 3 Qh1 Kb3) Ka4 3 Qg2 Ka5 4 Qf3. Taken from John Niemann / Eine Gedenkschrift (Büsing and Gruber, 1996).
1948 Charosh, M. Fairy Chess Review, vi.1948. (i) Given one knight each, who wins and how; (ii) given two knights each, none en prise, same question. The well-known answer is given for one knight each. For two knights each, Charosh claimed that if the total number of knights on one colour was odd, the first player would win, the winning plan being to give up a knight so as to bring the opposing knights to squares of the same colour, in such a way that the remaining knight could threaten to sacrifice itself to either. Example: White Nc1/f1, Black Nc8/g8, play 1 Ne3 Nge7 2 Nc2 Nd6 3 Nd3 Ne8 4 Nc5 Nd6 5 Nd7! and now 5...Nb7 6 Nc5 Nxc5 7 Nb4, or 5...Ndc8 6 Nb6 Nxb6 7 Nb4, or 5...Nf7 6 Ne5 Nxe5 7 Nb4, or 5...Nf5 6 Nd4 Nxd4 7 Ne5. However, this example was challenged by F. Hansson in the x.1948 issue (see below) and a number of exceptions given to the general rule.
1948 Hansson, F. Fairy Chess Review, x.1948. "When knights are bold": an article on knight endings. Hansson gives one exceptional study with two knights against one (see Fabel 1947 above), and eight with two knights against two mostly contradicting Charosh. In respect of two knights against one, Hansson says that the conclusion that the ending is generally drawn "must be qualified as the more probable result, as most exceptions are trivial," but he gives the following as "worth noting": White Na1/g5 (2), Black Nh6 (1), White to play and win (1 Ne4 N-- 2 N sacrifices and wins, but not 1 Nf7 when White loses). As regards Charoshs rule with two knights against two, he says, "It is as difficult to disprove that rule as to establish it. Although not completely convincing, the following examples may be sufficient." He then gives eight examples, the published solutions to some of them being reduced to "a dominant line of play out of many". Be it remembered that Charoshs rule amounts to saying that if one side has its knights on squares of the same colour and the other on squares of different colours - a "same-different" position, in the terminology used below - the player to move can win.
"With these exceptions to Mr Charoshs rule, it is difficult to believe that the four Kts ending can be reduced to any simple rule."
(1948) Klüver, H. Schachspiegel, ix.1948. White Kd3, Re4 (2), Black Kg2 (1), White to play and win. A demonstration of the winning process with KR v K, where it exists: 1 Ra4 Kh3 2 Ke4 Kh4 3 Ra6 Kh3 4 Ra5 Kh2 5 Ke3 (simple and systematic, though the computer shows 5 Rf5 to be quicker) Kh3 6 Rb5 Kh2 7 Rb4 Kh1 8 Ke2 Kh2 9 Ra4 Kg1 10 Ra3 Kh2 11 Rg3, or 1...Kg1 2 Ke4! Kf1 3 Rd4 Kg1 4 Kf4 (4 Rd3 is quicker) Kf1 5 Rd8 Kg1 6 Re8 etc. Supplied to me by Ralf Binnewirtz.
(1948) Niemann, J. Schachmatt, 7.iii.1948. White Pd5/a2/h2 (3), Black Pb7/d6 (2), White to play and win. 1 h4 (1 h3 b5 and Black will promote while wP is still on h6) b6 (if 1...b5 then 2 a4 straight away) 2 a3! (this time White must temporise) b5 3 a4 bxa4 and we have Törngren 1929: 4-6 h7 a1R 7 h8B. White must promote at just the right moment, not before Black and not more than one move after him. Supplied to me by Ralf Binnewirtz.
(1948) Niemann, J. Schachmatt, 23.v.1948. White Rc4, Nb4 (2), Black Bg5 (1), White to play and win (a) as set, (b) with WNb4 on b2. (a) 1 Rc1 Bxc1 2 Na2; (b) 1 Rh4 Bxh4 2 Nc4. Taken from John Niemann / Eine Gedenkschrift (Büsing and Gruber, 1996).
(1948) Schlensker, P. and Kniest, A. H. Schachmatt, 24.x.1948. White Pb4 (1), Black Bc8/g5, Nb8/g8, Pd7/e6 (6), can White to play win? No, after 1 b5 Bf4 2 b6 Bc7 3 bxc7 Bb7 4 cxb8B Black can draw by moving bNg8 to a dark square and then tempoing with bBb7, and other promotions lose (the source says that cxb8K also draws but in fact Black has a simple win). Supplied to me by Ralf Binnewirtz.
(1948) Schmidt, P. and Kniest, A. H. Source unknown to me, 18.xii.1948. White Kg8 (1), Black Bg5, Nb8, Pc5/e5 (4), can Black to play win? No, 1...c4 2 Kg7 Bf6 3 Kxf6 c3 4 Kxe5 c2 5 Kd4 c1K 6 Kc3 and it is White who wins. The computer thinks that 1...Bd8 holds out longer, but White wins in all lines. Supplied to me by Ralf Binnewirtz.
1949 "Kluever" (presumably Klüver), H. Fairy Chess Review, x.1949. White Kh2, Nh6, Pe2 (3), Black Kd4, Ne5 (2), White to play and win. 1 e4 Kxe4 2 Nf5 Kxf5 3 Kh3; not 1 Ng4 Nxg4 2 e3 (2 e4 Kxe4 and Black draws at least) Nxh2 3 exd4 Nf3 4-6 d7 Ne5 7 d8B Nd3.
1951 Boyer, J. (Joseph). Les jeux déchecs non orthodoxes, chapter "Les échecs battu-battant" (pp 49-52). A survey including three studies (Roese 1923, Fabel and Klüver 1947, Slater 1935) and listing several references.
1951 Kahl, P. Feenschach, ix.1951. Two studies.
("Its really Blacks move" is a problemists trick, and where the stipulation states or implies that it is White to play we shall normally content ourselves with examining the play forward from the diagram and shall not concern ourselves with whether the position can legally be reached in a game. However, in the first position here the stipulation is explicitly "Who wins?" and nothing is said or implied about whose move it is, so there is no real element of trickery, and the play is of genuine interest.)
1952 Hofmann, H. Feenschach, vi-vii.1952. White Kh6 (1), Black Pa5/g4 (2), White to play and win. 1 Kg6 a4 2 Kf6 (2 Kf5? a3 and 3-4...a1R wins) a3 3 Ke5 g3 4 Kd4 and wK will sacrifice himself to one bP or the other (but not 4 Ke4 g2 5 Kd3 g1R 6 Kd2 Re1 7 Kxe1 a2 and 8...a1K with a draw). Nor does running the g-pawn help: 2...g3 3 Ke5 (but not 3 Kf5 g2 4 Kf4 a3 5 Ke3 g1R and White must again settle for 6 Kd2 Re1 with a draw, or 4 Ke4? g1R followed by 5...Rb1 and Black will win) g2 4 Kd4 and wK will give himself to the a-pawn, or 1...g3 2 Kf5 g2 3 Kf4 (preventing 3...g1R) g1K 4 Ke4 a4 5 Kd3. The source gives additional analysis. It is a very instructive ending, both for the way White wins and for the way Black draws or even wins himself should White go wrong.
1955 Boyer, J. (Joseph). Le Courrier des Echecs ("Revue bi-mestrielle dEchecs par Correspondance") 53, ii.1955, article "Les échecs battu-battant" (pp 1-3). A general article including two studies (Klüver 1934 (2P v 2P), Fabel 1934) and "une excellente partie récente de nos tournois" played between E. T. O. Slater and H. Klüver which came down to an ending with White Pd7/a6 (2), Black Qg6, Pa7/e6 (3), White to play his 27th move. Play continued 27 d8K! (other promotions allow Black to sacrifice bQ and bPe6, winning by stalemate) Qb1! 28 Kc7 Qb8 29 Kxb8 e5 30 Kxa7 e4 31 Kb7 e3 32 a7 e2 33 a8R e1K! with a draw (all other promotions lose). Boyers exclamation marks throughout. This game was to attract further attention, as the next few entries demonstrate.
1955 Boyer, J. (Joseph). Engelhardts Schach-Taschen-Jahrbuch 1956, section "Schlagschach" (pp 40-43) within a chapter "Dreimal anders als sonst". An article including an exposition of the Slater-Klüver correspondence game mentioned above. The 4 v 4 pawn ending after Blacks 17th move is highlighted (White Pa3/e3/f2/g2 v Black Pa7/d7/e7/g6): 18 e4 g5 (18...d5 is shown to lose) 19 e5 (19 f4 gxf4 20 g3 fxg3 21 e5 transposes) e6 20 f4 gxf4 21 g3 fxg3 22 a4 g2 23 a5 g1Q 24 a6 d6 25 exd6 Qb1 26 d7 Qg6! and we have the position already seen. However, after 27 d8K Qb1 the move 28 Ke8 is suggested as an alternative to the actual Kc7, and a prize is offered for the best analysis. The game is referred to as having been played in July-November "1955", but if the dating of Le Courrier des Echecs is to be relied on this must be a misprint for 1954. The prize offer was repeated in the xii.1955 issue of Fairy Chess Review (p 54), but with wording implying that the position with wKe8 was actually reached in the game.
1955 Fabel, K. Rund um das Schachbrett, 1955, chapter "Wer verliert, gewinnt!" (pp 49-55). This is broadly similar to the chapter in Am Rande des Schachbretts (1947), but some of the examples are different and account has been taken of the 1948 Hansson article in Fairy Chess Review.
1956 Hofmann, H. Feenschach, vi.1956. White Kh3 (1), Black Rf8, Pb7 (2), White to play and draw. A brilliant and most instructive ending, which apparently found no successful solver. 1 Kg3/2? Rc8! 2 Kf1/2/3/4 b5 3 Ke1/2/3/4/5 b4 4 Ke1/2/3/4 (4 Kd6 Rh8 5 Kd5 b3 6 Kd4 b2 7 Kc4 Rh7) b3 5 Ke-/f- (5 Kd5 Rh8 etc) b2 6 Ke2!? Rc7! and promotion to R or B next move. 1 Kg4! Rc8 (1...Ra8? 2 Kf3 b5 3 Ke2 Rc8 4 Kd1 Rh8 5 Kd2 and White wins) 2 Kf5 b5 3 Ke6 (threatening 4 Kd7 with a draw) Rc3 (3...Rc2/Rc1 see below) 4 Kd7!! Rh3 (4...Rf3 5 Ke6 Rf7 is only a draw) 5 Kd6 (now bR has been driven from the c-file, wK can go for bP) b4 6 Kd5 b3 7 Ke4 Rh1 8 Kd3 b2 9 Kd2 Re1 10 Kxe1. 3...Rc2 4 Kd7 Rh2 5 Kd6 b4 6 Kd5 b3 7 Kd4 b2 8 Ke3 Re2/Rf2 9 KxR draw (8...Rh8? 9 Kd2 wins); 3...Rc1 4 Kd7 Rh1 5-7 Kd4 b2 8 Kd3 Rf1 (8...b1R? 9 Kd2 wins) 9 Kd2 (not 9 Ke2 b1N 10 Kxf1 Nd2). The source gives further detail in minor lines. This and Hofmann 1952 should be studied by everyone who wishes to master Losing Chess, because endings like this frequently arise in practice.
1956 Hofmann, H. Die Schwalbe vi.1956. White Rh8, Ph2 (2), Black Rc1, Pb5 (2), White to play and win. 1 h4 b4 (1...Rc8 2 Rxc8 and 3-5 h7 will win) 2 h5 b3 3 h6 b2 (3...Rb1 4 h7 Rb2 5 Rc8! and either 5...Rc2 6 Rxc2 bxc2 7 h8R or 5...Rb1 6 Rc1 Rxc1 7 h8R) 4 h7 Rb1 (now bR must hide) 5 Ra8! Ra1 6 Rxa1 bxa1R 7 h8B.
1957 Klüver, H. Engelhardts Schach-Taschen-Jahrbuch 1958, article "Eine Schlagschachstudie" (pp 35-38). An analysis of the competition position set two years previously: White Ke8, Pa6 (2), Black Qb1, Pa7/e6 (3), Black to play his 28th move.
The analysis starts with three preliminary remarks.
There follows an exposition of the further course of the game ("der weitere Partieverlauf"): 28...Qh1 29 Kf8 e5 (now 30 Kg7? is met by 30...Qh8! as in the second bullet above) 30 Kf7! (but by reaching a position with Kf7/Pe5 from the original Ke8/Pe6, White has effectively gained a tempo) e4 (30...Qb7 still loses) 31 Kf6 e3 (the threat of Kg5 and Kg4 gives Black no other option, for if say 31...Q-- 32 Kg5 Qh6 33 Kxh6 e3 then White is one tempo ahead of the line in the second bullet and Black cannot hold the draw) 32 Ke5 (now an offer of the queen is hopeless, and if bQ removes herself to a square from which she cannot capture wK then he enters the field of the e-pawn from the left; Black chooses the third and most interesting way to die) Qb7 (this takes us into the first bullet) 33 axb7 a5 34 b8R a4 35 Kd4! (the quickest and most elegant, with 35...e2 36 Kd3 a3 37 Kxe2 a2 38 Kd2 a1K 39 Rb1 Kxb1 40 Kc1 or 35...a3 36 Kxe2 a2 37 Kd2 and the same). So Black resigned ("Schwarz gab daher auf").
All this is incompatible with the game as reported in Le Courrier des Echecs (see above), but it would seem that 28 Kc7 was the move actually played in the tournament and that the players then wondered what would have happened after 28 Ke8 and went back and replayed. The analysis was reproduced in translation in Fairy Chess Review (see below), and I have relied heavily on that translation here.
1958 Slater, E. T. O. Fairy Chess Review, iv.1958, p 175. A translation of the analysis above.
1960 Mortensen, J. Feenschach, ii-iii.1960. White Pa7 (1), Black Nb4/c4 (2), White to play and win. 1 a8K? Nb6 2 K-- Na8 3 Nxa8 Nd5! and the knight dominates the king; 1 a8N? Nb6 etc; 1 a8B! and even though Black can sacrifice one knight, White can sacrifice himself to the other.
(1967) Dornieden, M. Deutsche Schachzeitung, xii.1967, pp 406-7. An article "Silvesterschach" including one Losing Chess ending: White Nb5, Ph4 (2), Black Pd6/a3 (2), White to play and win. 1 Nxa3 (1 Nxd6 loses) d5 2-4 h7 d2 and now given is 5 h8Q which is indeed quickest, but 5 Nc2 is almost as simple and 5 h8R also wins. Supplied to me by Ralf Binnewirtz.
1973 Hoffmann, F. feenschach, vii.1973. White Ke5, Rh1, Bf1, Ng3, Pc4/g2/h2 (7), Black Bc8 (1), White to play and win. 1 Ke6 Bxe6 2 Ne2! Bxc4 3 Ng1 etc.
1975 Panteleit, U. feenschach, iv.1975. Two studies.
1977 Panteleit, U. feenschach, iv-vi.1977. Corrections of the two items above. In the case of the first study the correction merely involved giving a correct solution, but the second study was reset as wNa8 v bPe7 with stipulation "Black to play, White to win" and solution 1...e5 2-5 Nd1, 1...e6 2-5 Nd4/Nf4 and either 5...e1B 6 Ne6 or 5...e1- 6 Ne2. Also included was the non-stalemate solution to Dawson 1938.
1978 Carfora, A. Eteroscacco 2, vii-ix.1978. White Nd4, Pa2 (2), Black Nb2, Pb3/a4 (3), White to win in 4 moves. 1 axb3 axb3 2 Nxb3 Nd1 (best) 3 Nd2 and a sacrifice next move.
1978 Magari, R. Eteroscacco 1, iv-vi.1978 (pp 11-12), 2, vii-ix.1978 (pp 20-22), 3, x-xii.1978 (pp not known by me). The first part of this article discusses the general theory of "colour-change pieces" (the squares on the board are denoted by various colours and we consider pieces wich change colour systematically with each move) and has no particular relevance to Losing Chess. The remainder considers Losing Chess using three different types of man: (a) the ordinary knight; (b) the ordinary draughtsman, which moves one square diagonally but captures by jumping over an adjacent man on to an empty square immediately beyond, chain jumps being permitted; (c) an "elementary two-colour piece" which can move only one step horizontally or vertically. In endings with pieces of a single kind:
The author gives an instructive illustration of three elementary two-colour pieces against one on a 3x8 board: White on a2, a1, b1, Black on c3, Black to play. The natural line might seem to be 1...c3-c4 2 a2-a3 c4-c5 3 b1-b2 c5-b5 4 a1-a2 b5-c5 5 b2-b3 c5-c6 6 a3-a4 c6-b6 7 a2-a3 b6-c6 8 b3-b4 c6-c7, where White is clearly winning, but if Black concedes ground voluntarily instead of waiting until he is forced he can keep White at bay: 3...c5-c6! 4 b2-b3 c6-b6 5 a1-a2 b6-a6 6 a2-b2 a6-b6 7 b2-c2 b6-c6 8 c2-c3 c6-b6 9 b3-b2 b6-b5 10 b2-a2 b5-b6.
My interpretation of this item in the provisional versions of this document was badly wide of the mark, and I am grateful to Chris Feather for providing me with a proper translation.
1978 Salvadori, R. Eteroscacco 2, vii-ix.1978. An article "La promotion" (pp 11-12) dealing with a simple one-against-six study (White Pc7, Black Kf7, Rg8, Nf8/e5, Ph7/g5) where a promotion to king is necessary in order to win.
1979 Gik, E. 64, 1979, issues 2 and 26 (11.ii-17.ii and 28.vi-4.vii). Two articles. The first, which I know only through a translation into Italian by Marco Bonavoglia in Eteroscacco 7 (x-xii.1979), quotes two studies described earlier (Törngren 1929, Fabel and Klüver 1947). The second, which is in the BCPS Library, contains the refutations of 1 d4, 1 e4, and 1 d3 given in the Encyclopedia of chess variants (Pritchard, 1994), two problems by Dawson described above, and a twelve-against-one Russian draughts problem (White has a game array, Black a single man at h8).
1979 Magari, R. Eteroscacco 5, iv-vi.1979. An article "Il finale de C/P" (p 12) which deals with various cases of the ending N v P. The case Nh8 v Pd7 (Black to play) is given as lost, Nh8 v Pb7 as drawn by stalemate (1...b6 etc), and Nh8 v Pa7 as won (Black plays 1...a5 and White arrives too late). Also considered is Nb8 v Pd7, given as won for Black because the knight cannot get back to e1 or c1, Nd8 v Pd7 with the same result (the text actually says "Lo stesso si verifica col P nero in d7 e il C B in c8" but "c8" appears be a misprint), and Nf8 v Pd7, where Black wins because the knight can get back to g2. The win against a Black c-pawn, meeting ...c1B by Na2, is overlooked.
(Knight against pawn is the most complicated of the one-against-one endings, and several published statements have been incorrect or incomplete. Suppose for the moment that being stalemated wins, and that the pawn has just moved. If the knight is now on the square of the opposite colour, it can hope to win by sacrificing itself to the pawn before the latter promotes; if it is unable to do this, the pawn wins by promoting to knight. If the knight is on a square of the same colour, there can be no sacrifice to an unpromoted pawn, and a rooks or knights pawn can always win by promoting to bishop; but a knight may be able to defeat a bishops or centre pawn by forcing a promotion to bishop and then taking up one of the exceptional winning positions (Na2 v Bc1, Nd6 v Bd1). If stalemate is a draw, the "opposite colour" case is unchanged, but the knight may be able to draw a lost "same colour" ending by giving stalemate. Everything else can be deduced from this.)
(1979) Minieri, -. Telescacco, 1979. White Be4, Nd8/e1 (3), Black Pd7/e7 (2), White to play and win. Quoted by Leoncini and Magari in Manuale di scacchi eterodossi (see below) with solution 1 Nc6 dxc6 2 Bxc6 and now either 2...e6 3 Bd5 exd5 4 Ng2 or 2...e5 3 Ba4! e4 4 Nd3 exd3 5 Bc2, but there are alternatives just as quick.
1979 Salvadori, R. Eteroscacco 4, i-iii.1979. An elementary article "Chi muove per primo" (pp 10-11) discussing the endings P v P where both pawns are on the seventh rank.
1980 Kuhlmann, J. Die Schwalbe, xii.1980. White Pa7/g5 (2), Black Kd8, Ph4 (2), White to play, shortest win? This is one of the most complex pre-computer studies. Black threatens 1-3...h1R drawing at least, 1 a8B h3 2 Bg2 hxg2 is again only a draw, and 1 a8Q loses to 1...Kc8. So 1 g6 h3 2 g7 h2 3 g8B! (3 g8K also prevents 3...h1R, White having 4 Kh7 and as below, but it offers no hope of winning). Now 3...h1R is met by 4 Bh7 Rxh7 5 a8N! Ke7 6 Nb6 Kf7 7 Nc8 Kg7 8 Nd6, a lovely line, while 3...h1K loses to 4 a8R, and 3...h1N allows 4 a8N followed in due course by a win with BN v N. What about bK? If it moves off the eighth rank White can win by an immediate 4 a8R, and if 3...Ke8 then 4 Bf7 Kxf7 5 a8R. Hence 3...Kc8, and now 4 Ba2! is quickest (4 Bh7 Kd8 5 Bb1 Kc8 6 Ba2 is slower, and nothing else wins). Blacks rook promotion is again prevented (4...h1R 5 Bb1 Rxb1 6 a8K!), so he has nothing better than 4...Kd8, and now comes the coup de grace: 5 a8Q!! This finally kills Blacks hope of a rook promotion (5...h1 6 Qxh1 and White will win) and leads to a win in all variations, one of the lines holding out longest being 5...Kc8 6 Qxc8 h1K 7 Bc4 Kh2 8 Qf8 Kh3 9 Bd3 Kh2 10 Be2 Kh1 11 Qf5. A full analysis appeared in Die Schwalbe in 1999 (see Gruber 1999). The ending BN v N had already been explored (Panteleit 1975), but this appears to have been the first study to examine QB v K and to show the delightful winning line in the normally lost ending N v KR.
1980 Leoncini, M. and Magari, R. Manuale di scacchi eterodossi. There are two elementary P v P endgames with the pawns on the seventh rank on page 88 (also a problem with seven men against two) and two complete chapters on the endgame: "I finali" (pp 127-33) and "Considerazioni conclusive" (pp 134-5). "I finali" contains the following.
(This is perhaps a convenient point at which to summarize one-against-one endings. Excluding as "trivial" positions where the side to move must make an immediate capture or can win by making an immediate sacrifice, the general results for piece against piece are as follows: Q/R/B against Q/R/B is trivial; Q or R wins against N or K; B or K wins against N, draws against K; N against N is a win for whoever is to move when the knights are on squares of the same colour. All this was known to Klüver, who noted some positions where the normal rule did not apply (see Klüver 1924). Definitive analysis by computer has listed the exceptional positions as follows: (a) a set of "attack and wait" wins typified by Qb8 v Bb1, Rb8 v Bb1, Bh8 v Ra1, Na1 v Rc2, Na2 v Bc1, and Na2 v Kc1, where the piece attacked is unable to sacrifice itself and the attacker then sacrifices on the square its opponent has just vacated; (b) two "domination" wins for a knight, Nd6 v Bd1 and Nd4 v Ka1; (c) three similar "domination" wins for a bishop against a king, Ba4 v Ka1, Bc4 v Kc1, and Bd4 v Kd1, though not Bb4 v Kb1 where the king can hold the draw by moving to a1. Every pre-computer writer who attempted to give a complete exposition, myself included, appears to have overlooked at least one of these exceptions. Knowledge of the piece-against-piece results enables the results for pawn-against piece and pawn-against-pawn to be worked out, and Leoncini and Magari are correct apart from the case N v P and the obvious diagram misprints.)
"Considerazioni conclusive" includes four studies already noted above: Fabel and Klüver 1947, Slater 1935, Roese 1923, and Minieri 1979.
(1983) Büsing, G. Jugendschach, 1983. White Be3, Pe6/h2 (3), Black Ne2, Ph3 (2), White to play and draw. 1 Bg1 (1 Bc1/Bf4 NxB 2 e7 Nd3 3 e8B Nf2 4 Bf7 Nh1 and 5...Ng3 followed by 6-7...h1R will win, 1 Bd4 Nxd4 2 e7 Nf5 3 e8B Ne3) Nxg1 2 e7 Ne2 3 e8Q! (other promotions lose) Ng3 4 hxg3 h2 5 Qe3 (other moves again lose) h1K 6 Qg1 and 7-11 g8K. Supplied to me by Ralf Binnewirtz. This appears to be the earliest published position in which P=Q is necessary to force a draw, and it remains the most economical setting of the task.
1983 Büsing, G. Die Schwalbe, iv.1983. Two studies.
1985 Bonavoglia, M. Eteroscacco problemi 1, vi.1985. An article "Problemi di Vinciperdi" (pp 20-21) which quotes some known work (Dawson 1924, 2 x Dawson 1925, Törngren 1929, Fabel and Klüver 1947) and announces the tourney described below.
1987 Dittmann, W. Eteroscacco 38, iv-vi.1987. White Rh8/d4, Bb1, Ph7/f6/h6/a3/c2/g2 (9), Black Ba1, Pg6/g5/c3/b2 (5), White to play and win in ten moves. 1 Rd6, 2 Rf8, 3 h8R, 4 h7, 5 Bb1, 6 Bxb1, 7-8 Bg8, 9 f7 and 9...Bxf8 gives stalemate. This would normally be regarded as "in nine moves", but in Eteroscacco the final move which White was unable to play was also counted. Issue 23 of Variant Chess (summer 1997) reported a second and longer solution which would be valid if the solution merely stipulated "White to play and win", but the problem as published specifically stipulates "in 10 mosse". This was the prizewinner in the composing tourney mentioned above; there were ten compositions in the tourney report, but only four were sufficiently orthodox to be listed here.
1987 Hernitz, Z. Eteroscacco 38, iv-vi.1987. An unhonoured study from the same tourney: White Rh4, Nd2, Pa2 (3), Black Re1, Nd1, Pf2 (3), White to play and win. 1 Nf1 Rxf1 2 Ra4, and wins in all lines. Given in reply to 2...Nb2/Nc3 is 3 a3 winning by stalemate, but Fabrice Liardet points out that White has alternative continuations in which he can sacrifice both men.
1987 Sekhar, R. and Shankar, R. Eteroscacco 38, iv-vi.1987. Two further studies from the tourney, the first commended, the second unhonoured.
1989 Beasley, J. D. International problemists meeting, Bournemouth, 1989. White Kh5, Pf7 (2), Black Kh7 (1), White to play and win (a) in ordinary chess, (b) in Losing Chess. Play (a) 1 f8R, as is well known; (b) 1 Kh6 Kxh6 2 f8R again. This was composed for a light-hearted tourney for twin studies (orthodox chess being implicitly assumed) in which it was announced that more attention would be paid to novelty of twinning mechanism than to depth of play. I submitted it as a joke entry, and the judge gave it the prize for sheer cheek.
1991 Liardet, F. Schweizer Schach-Magazin, viii-ix.1991. Article "Le Qui perd gagne " (pp 282-5). A general article paying particular attention to the ending, and including a table showing the general result for most two-man and three-man endings without pawns. He identifies 2N v Q, KN v Q, and QN v K as endings which appear not to permit the formulation of a general rule, and examines the endings that follow in greater detail.
My translations throughout. This appears to have been the first attempt to analyse three-piece endings by identifying "winning" and "drawing" regions, and it remains of interest even though computer analysis has shown that the simple rules given here are not entirely valid.
1992 Evseev, G. International problemists meeting, Bonn, 1992. White Nb1 (1), Black Nh8/a6 (2), White to play and win. This was subsequently reproduced in several magazines (for example, in the British Chess Magazine in xi.1992), and I think it best to quote the solution as given in the article by Evseev and Poisson in Rex Multiplex in iv.1993. Here the main line is given as 1 Nc3 Nb8 (if 1...Nf7 then 2 Nb5/Ne4 and if 1...Ng6 then 2 Nd5, in each case threatening to sacrifice to both Black knights and winning at once, because if Black sacrifices he loses the N v N ending and if he doesnt sacrifice White will) 2 Nd1! (the only move to win) Na6 (if 2...Nf7 then 3 Nf2 Nd8 4 Nd3 and the Black knights are dominated) 3 Nb2/Nf2 Nc7 4 Nd3 Nb5 5 Nc5 Na3 6 Ne6 Nc4 7 Nd4 Nb2/Nb6 8 Nc6 Nd1/Na4 9 Nb4 and soon wins (9...Nf7 10 Nd3 or 9...Ng6 10 Nd5). This had been found by computer as the longest win with one knight against two, and was set as a solving challenge; I think I am correct in saying that nobody was successful.
1992 Evseev, G. 2nd Prize, Phénix, 1992 (published xii.1992). White Nb8 (1), Black Pd5/c3/e3 (3), White to play and win. 1 Nc6 and now (a) 1...d4 2 Nxd4 and either 2...c2 3 Nxc2 e2 4 Nd4 e1B 5 Ne6 (not 4 Ne3 e1B 5 Ng2 Bh4!) or 2...e2 3 Nxe2 c2 4 Nc3 c1B 5 Na2 (not 4 Nd4 c1B 5 Nc6 Bh6!), or (b) 1...c2 2 Nd4 and either (b1) 2...e2 3 Nxc2 (3 Nxe2? d4 4 Nxd4 c1B) and now either 3...d4 4 Nxd4 e1B 5 Ne6 or 3...e1- 4 Nxe1 d4 5-7 Nc4/Ne4 d1B 5 Nd6, or (b2) 2...c1B 3 Ne2 d4 4 Nxc1 and now either 4...d3 5 Nxd3 e2 6 Nf4 (6 Nc5? e1K) e1B 7 Ne6 or 4...e2 5 Nxe2 d3 6 Nc3/Ng3 d2 7 Ne4 d1B 8 Nd6. Only 1...e2 2 Ne5 d4 3 Nc4 is straightforward. This is much the most intensive setting yet made of N v P; the line 1...d4 leads to accurate wins with N v cP and N v eP, and after 1...c2 2 Nc4 each of the subvariations 2...e2 and 2...c1B leads to further wins with N v dP and N v eP.
1993 Beasley, J. D. The Problemist, i.1993. White Pd2 (1), Black Nd1 (1), White to play and win (a) as set, (b) with WPd2 on b2. (a) 1 d4 and 2-5 d8N; (b) 1 b3 and 2-6 b8B. This adds nothing to Magari 1979, or even to Panteleit 1975, and I imagine that it would not have been accepted for publication had a survey equivalent to the present been available at the time.
1993 Evseev, G. and Poisson, C. Rex Multiplex 41, iv.1993. "Finales de cavaliers en qui perd gagne " (pp 2048-9). An article on endings with one or two knights against one, outlining the general theory and giving the 1992 Evseev study.
1993 Richardson, I. Variant Chess 10, iv-vi.1993. White Kh8, Rh5, Bh4, Pg7/f6/e5 (6), Black Ba1 (1), White to play and win. 1 Bg5 Bxe5 2 Bh6 Bxf6 3 g8N! and so on, a delightful trifle.
1993 Wood, P. C. Variant Chess 10, iv-vi.1993. "Vinciperdi" (pp 26-9). An excellent general article, largely based on Boyer 1951, Fabel 1955, Evseev and Poisson 1993, and material from Eteroscacco, including three pages on the endgame. It includes an extensive selection of existing studies, though its only original is that by Richardson (see above).
1994 Pritchard, D. B. The encyclopedia of chess variants, article "Losing chess" (pp 176-9). A general survey of the game, including the correspondence game Slater-Klüver 1954, two existing studies, and one original composition in which a complete White army sacrifices itself to a single pawn. The statements regarding one-against-one endings are not free from error; in the ending P v N where the P is unmoved, it is stated that the P wins by moving to the same-colour square as that on which the knight stands and ultimately promoting to B (this only works with a rooks or knights pawn), and the ending R v B is inadvertently given as drawn.
1994 Wood, P. C. Variant Chess 13, i-iii.1994. White Pe6/h3/g2 (3), Black Bb6 (1), White to play and draw. An instructive study of bishop against pawns. Black would like to manoeuvre his bishop round to g7 or h6, after which he can patrol these two squares, eat the g- and h-pawns as they advance, and sacrifice to Whites eventual e7, but if he takes his eye off d8 without threatening a sacrifice White can play e7 and e8B and draw at once. Thus 1 g4 fails: 1...Bc7 2 g5 Be5! 3 g6 Bg7 and Black has achieved his aim, or 2 h4 Bf4! 3 h5 Bh6! 4 g5 Bxg5 5 e7 (the only chance) Bxe7 6 h6 Bg5 7 h7 Bh6 8 h8N Bg7 and sacrifices next move. 1 g3 leads to similar play (1...Be3 2 g4 Bb6 etc). Hence 1 h4 Be3 ("if Black waits, White obtains an easy draw by promoting the h-pawn to a King") 2 h5 Bg5 (2...Bh6? 3 g4 B-- 4 h6 and White wins) 3 e7! Bxe7 4 g4 (an immediate 4 h6 is met by 4...Bf6 5 h7 Bg7 and a sacrifice to Whites eventual g5, so White must get his g-pawn to g6 before advancing the h-pawn) Bb4 5 g5 "and Black cannot win: e.g. 5...Bc3 6 g6 Bb4 7 h6 Bf8 8 h7 Bb4 9 h8K".
1995 Byway, P. V. Variant Chess 18, autumn 1995. White Kc6, Pa7/h2 (3), Black Qe1 (1), White to play and win. The given solution is 1 h4 Qxh4 2 a8N (2 a8B Qe1 3 Bb7 Qh4 4 Ba8 Qe1 with a draw by repetition, a charming line, or 2 Kb6/7 Qf4 3 Kc7 Qxc7 wins) Qe1 3 Nb6 (3 Kc7 Qf2 and 4...Qb6 wins, 3 Kb6 Qg3 similarly, 3 Kb7 Qd2 4 Ka7 Qg5 5 Kb7 Qd2 and at the very least Black can hold the draw by oscillating between g5 and d2) and the queen is dominated. However, 2 a8K also wins: 2...Qe1 (nothing else) 3 Kab7 Qh4 (again nothing else) 4 Kbb6 Qh8 (still nothing else) 5 Kbb5, and again the queen is dominated. The positions with White Kc6, Ba8 or b7, and Black Qe1 (h4 is equivalent) were subsequently found by computer to be the only drawn positions with this material.
1996 Beasley, J. D. British Endgame Study News, special number 4, xii.1996. "Elementary duels in the Losing Game" (pp 2-3). An attempt to survey one-against-one endings, but omitting the "attack and wait" wins with N v R and the "domination" wins with B or N v K. The treatment of knight against unmoved pawn appears correct.
1996 Beasley, J. D. diagrammes 119, x-xii.1996. "Les promotions uniques dans les échecs à qui perd gagne " (pp 2917-9). A survey of simple positions in which a specific promotion is needed to win or draw. Winning promotions to R, B, N, and K, and drawing promotions to B and K, are shown in one-against-one positions; otherwise, there is White Pa7, Black Rb6, Pb7/a6, win by a8Q only; White Ba8, Pb7, Black Bb6, draw by b8N only, add Black Pa7, draw by b8R only, further add Black Nd8, draw by b8Q only (a curious triplet which is the articles only real claim to novelty); White Pg7, Black Pd7/f7/g6, draw by 1 g8K (only move not to lose, but now it is Black who is struggling to draw) d5 (only move) 2 Kxf7 d4 3 Kxg6 d3 4 Kf5 d2 5 Ke4 d1K and both sides have had to promote to K (after Watney, Chess Amateur 1923, see above). The Watney position is wrongly described as being from 1924.
(Pa7 v Rb6/Pb7/Pa6 is one of the positions where it is easiest to prove that only P=Q wins, since all other moves allow three immediate sacrifices, but while writing this document I noticed that several settings more economical in material were possible: for example, the two-against-one pawn position Pb7 v Pa3/Pb3, where 1 b8R loses to 1...a2 2 Rxb3 a1N and nothing else apart from 1 b8Q offers any hope. This position will have been disclosed by Laurent Bartholdis 1998 computer analysis, but I do not know whether specific public attention was drawn to it during the period under review. For an earlier and more economical setting of P=Q to draw, see Büsing 1983.)
1996 Marks, U. achová skladba 53, ix.1996. White Kb1 (1), Black Zebra g4 (1), whoever is to move wins. The zebra moves like a knight, but three squares and two instead of two and one (Ze4 can move to b6, c7, g7, h6, h2, g1, c1, b2). White to play, 1 Kb2 Ze7 2 Kc2 Zh5 3 Kc3 Zf8 4 Kd3 Zd5 5 Kd4 (a manoeuvre not possible against a knight) Z-- 6 Kd5; Black to play, 1...Zd6 2 Ka1 (2 Kc1 Zf3 3 K-- Zc1) Zf3 3 Ka2 Zc5 4 K-- Za2.
1997 Beasley, J. D. The Problemist, i.1997. White Nc2/g1 (2), Black Pd3 (1), White to play and win. 1 Ne2 and either 1...dxc2 2 Nc3 c1B 3 Na2 (if 2 Nd4 c1B 3 Nc6 then 3...Bh6) or 1...dxe2 2 Nd4 e1B 3 Ne6 (if 2 Ne3 e1B 3 Ng2 then 3...Bh4). This is perhaps the simplest setting yet of the alternative wins for a knight against a bishops or centre pawn, though everything after the first move appears in the line 1...d4 of Evseevs 1992 phénix study. 1 Nf3/Nh3 dxc2 and 2...c1B; 1 Na1/Na3/Nb4/Nd4/Ne3 d2 and 2...d1B, with a win for Black in every case; 1 Ne1 d2 2 N(e)-- d1B similarly, or 2 N(g)-- dxe1N.
1997 Beasley, J. D. Variant Chess 25, autumn 1997. White Nh3 (1), Black Kh8, Ng8 (2), White to play and win. A much lighter affair: 1 Nf4 (1 Ng5 loses immediately, and if 1 Nf2/Ng1 then 1...Kh7, after which Black will leave his king at h7, win the N v N battle without allowing wN to sacrifice itself to bK, and then win with K v N) Nh6 2 Ng6 N-- (if 2...K-- then 3 Nh8 wins immediately) 3 Nxh8 and wins with N v N.
1997 Beasley, J. D. British Endgame Study News, special number 8, xii.1997. "Promotion studies in the Losing Game" (pp 2-3). An article on simple promotion studies, containing nothing new.
1997 Beasley, J. D. Variant Chess 26, winter 1997. Two studies with N to win against two unmoved pawns, described as "theoretical studies: the positions are natural (or at least simple) and the solutions are strategically motivated, but the play is not unique. An unmoved knights or rooks pawn wins against a knight, so Black will hope to advance one pawn, force White to capture it, and then win with the other: moving it one square or two as appropriate and winning by promoting to bishop. It follows that White must force the second pawn to move and commit itself before he captures the first." In the first study, White Nd1 (1), Black Pb7/g7 (2), we have 1 Ne3 and now two lines: (a) 1...g5 2 Ng2 g4 3 Ne1 g3 4 Nd3 g2 5 Ne5 b6 6 Nf3! g1- (6...b5 7 Nd2) 7 Nxg1 b5 8-9 Nc3, and (b) 1...b6 2 Ng2 b5 (2...g6 3 Ne3 g5 4 Ng2 g4 5 Nf4 g3 6 Nd3) 3 Ne3 with the echo line 3...b4 4 Nd1 b3 5 Ne3 b2 6 Nd5 g6 7 Nc3 b1- 8 Nxb1 and 9-10 Nf3. In the second, White Nd3 (1), Black Pa7/h7 (2), we have 1 Ne5 h6 2 Nc4! a6 (2...a5 3 Nxa5 h5 4-6 Ng2) 3 Ne5 a5 4 Nd3 a4 5 Ne5 a3 6 Nd3 a2 7 Nc1! a1B (7...h5 8 Nxa2 h4 9-11 Ng1) 8 Na2 Bc3 9 Nxc3 h5 10-11 Ng3. However, although 1 Ne3 in the first study is Whites most natural move, since it meets two of Black four moves by immediate sacrifices, 1 Nb2 works just as well (see below) and 1 Nf2 actually forces the win one move sooner. We have 1...g5 2 Nd3 g4 3 Ne1 rejoining line (a), or 1...g6 2 Nd3 b5 3 Nb2 b4 4 Nc4 b3 5 Ne3, or 1...b5 2 Nd1 g5 (2...g6 3 Ne3) 3 Nb2 b4 4 Nd1 b3 5 Nc3 b2 6 Ne2 echoing the play after 1...g6, or 1...b6 2 Ne4! (avoiding the need to manoeuvre wN round to d5) g5 3 Nxg5 and 4-5 Nc3.
(If the echo in the first study is thought worth rescuing, it could be done by starting wN on b2/c2/f2/g2 instead of d1. Suppose wNg2, and play the natural move 1 Ne3; then, as before, we have 1...g5 and 2-4...g2 forcing 5 Ne5 and 1...b6 and 2-5...b2 forcing 6 Nd5. 1 Ne1 leads to the same thing (Blacks additional moves 1...b5 and 1...g6 lose more quickly). The precision can be further increased, at the cost of a less natural starting position, by starting wN on a4 or h4. Suppose wNa4; now 1 Nb2 is Whites only move, and leads once more to 1...g5 and 2-4...g2 forcing 5 Ne5 and to 1...b6 and 2-5...b2 forcing 6 Nd5, while 1...g6 is met by 2 Nd3 with a quicker win.)
1997 Beasley, J. D. diagrammes 123, x-xii.1997. Article "Quelques études en qui perd gagne ". A survey article, containing nothing new apart perhaps from the example used to show the absence of a simple rule for two knights against one (White Na3 to play against Black Nb8/Nd1 loses, move everything one file right and White to play wins).
(1997) van der Bilt, V. Internet (copy downloaded in 2000, dated from internal text). Six studies and problems which were clearly composed independently but in fact largely rediscovered manoeuvres already known.
(The precise dating of Internet material is a matter of difficulty because documents can be altered without the change being apparent, and perhaps pedantically I have put all Internet dates in brackets, but this merely acknowledges the nature of the subject and should not be construed as a comment on the integrity of any author or provider of material. For that matter, the date on the front of a printed magazine is not always a reliable indication of when it finally gets into the hands of its readers.)
(1997) Geerlings, V. Internet. White Pf7 (1), Black Nc3, Pa2 (2), White to play and win. I havent seen the solution as posted, and am relying a copy sent to me by Fabrice Liardet plus some computer analysis. 1 f8B (1 f8Q a1K and 1 f8K Ne4 2 Ke8/Kg8 a1R both win for Black, 1 f8R a1K is drawn) and now 1...a1B/K 2 Ba3 is won for White, hence either 1...Ne2 2 Ba3 Nf4 (2...Nc3/Nd4 3 Bb2, 2...Ng1 3 Bc1, 2...Ng3 3 Bc1 Nh1 4 Bb2 and either 4...Nf2 5 Bd4 or 4...Ng3 5 Be5) 3 Bb2 Ng2 4 Ba1 and wB will attack bN at the next move and force bP to advance, or 1...Nd1 2 Bd6 Nb2 (2...Ne3/Nf2 3 Be5, 2...Nc3/a1K 3 Ba3) 3 Bb4 a1K 4 Ba3.
1997 Liardet, F. Variant Chess 25, autumn 1997. White Nd7, Pe7 (2), Black Nh4 (1), White to play and win. A study which illustrates the winning plan with BN v N (see Panteleit 1975 and Liardet 1991). 1 Nf6 Nf5 (1...Ng6 2 e8B is worse and other moves allow 2 e8K) 2 e8B Nd4 (2...Nh4 3 Bb5 followed by a knight move and eventually a bishop sacrifice) 3 Nd7 ("the key move") Nb5 4 Bf7 Na7 5 Be6 Nb5 6 Bc4 and 7 Bb5 wins. All non-trivial refutations of alternative first moves are given: 1 e8K Nf5 (threatening 2...Ne7, and if 2 Ke7 then 2...Nxe7 and wins with N v N); 1 e8R Ng6 with similar play; 1 e8N Nf5 2 Nc7 Ng3 with a draw; 1 Nb8/Nb6 Ng6 2 e8B Nf8 threatening 3...Nd7/Ng6, and if 3 Bg6 then 3...Nxg6 again gives the N v N win to Black; 1 Nc5 Nf5 2 e8B Ne7 similarly.
1997 Liardet, F. Variant Chess 26, winter 1997. Two studies.
(1997) Liardet, F. Internet (I am relying on a copy sent to me by the composer). White Pa5/f6 (2), Black Pf4 (1), White to play and win. 1 f7 f3 2 f8N (2 f8K f2 3 a6 f1B draw) f2 3 Nd7 ("not 3 a6 f1K with a drawn ending RN v K"). Now queen and rook lose immediately and promotion to knight is also hopeless, leaving 3...f1K 4 Ne5 Ke1 5 a6 Kd1 (5...Kd2 6 Nd3) after which 6-7 a8R will give a winning RN v K ending thanks to the centrally posted wN, and 3...f1B 4 a6 Bxa6 5 Nb8 with an exceptional win with N v B.
1998 Beasley, J. D. (discovered by computer). Variant Chess 27, spring 1998. White Nd5/g2 (2), Black Ra8 (1), White to play and win. Black to move would lose at once; White to move plays 1 Ngf4 (Nge3 is equivalent by symmetry) Ra1 2 Ng6 Ra2 3 Nge5 Ra8 (3...Rh2 4 Nb6 Rh1 5 Nbc4 Rh8 6 Nb2 with an echo of the original position, 3...Ra1 4 Nf6 Ra2 5 Nfd7 and either 5...Ra1 6 Ng4 winning immediately or 5...Rh2 6 Nb6 with the same position as after 3...Rh2 4 Nb6) 4 Ng4 Ra1 5 Ngf6 Ra2 6 Ng8(!) Ra1 7 Nge7 Ra2 (7...Rh1 8 Nb4 with an echo of the position after 3...Ra1 and 5...Ra1 6 Ng4) 8 Ng6 Ra1 9 Ngf4 Ra8 10 Ng2 and we are back at the starting position with Black to move. This study has been reprinted several times, the use of a knight to transfer the move to the opponent being unusual in any form of chess, but only this first appearance is listed here.
(A few words on "computer discovery" may not be out of place. Provided that the programmer has access to a sufficiently powerful machine, the construction of a definitive table of results for any particular combination of material is straightforward, and all that is then necessary is to search it for interesting positions. Even this can be largely done by computer, for example by calling for the longest win, for the positions of reciprocal zugzwang, and for any positions where a player has a significantly shorter win if it is his opponents move. But while the work involved is vastly less than in conventional composition, not least because the "composer" does not spend time analysing positions which eventually prove to be unsound, it does not follow that the positions that result are less interesting. Paul Byway wrote in Variant Chess about the two positions in the next item: "These discoveries, dredged from the sea of possible positions, have a gem-like quality that seems to be missing from most of our more laboured, human constructions.")
1998 Beasley, J. D. (discovered by computer). Variant Chess 28, summer 1998. Two studies.
1998 Beasley, J. D. Variant Chess 30, winter 1998. An article "Losing Chess in Geneva" reporting the meeting held in September 1998 (see Liardet 1998).
1998 Beasley, J. D. Variant Chess 30, winter 1998. White Ne1 (1), Black Na8, Pe7 (2), White to play and win. Whites aim is to force Black to play ...e6 while bN is on a light square, after which he can expect to win (given as a typical line is 1 Nc2 e6 2 Ne3 e5 3 Nf5 e4 4 Nh4 e3 5 Nf3 e2 6 Nh2 e1B 7 Nf1 threatening Nd2/Ng3 sacrificing wN, with an N v N win if Black sacrifices bB first). Black can try and defend either by playing ...e5 while bN is on a dark square, when lines such as the above will lose, or by playing ...Nd6 while bP is still on e7, "e.g. 1 Nf3 Nc7 2 Nd4 Ne8 3 Ne2 Nd6 and White is running short of squares". Hence 1 Ng2, ready to meet 1...Nc7 with 2 Ne3 and if say 2...Na6 then 3 Ng4 e6 (3...e5 4 Nxe5 is a win for White) 4 Nf6 etc, or 1...Nb6 2 Nf4 Nc4 (say) 3 Nh5 and much the same. 1 Nc2 is met by 1...Nb6, 1...Nd3 by 2 Nc7, 1 Nf3 by both moves. "This is essentially a strategic ending and there are many alternatives later in the play, but Whites first two moves are unique." The computer has sharpened Blacks answers to incorrect White first moves, Black having a win in 11 moves at most.
1998 Beasley, J. D. British Endgame Study News, special number 13, xii.1998. "Computer discoveries in the Losing Game" (pp 2-3). An article containing various positions either discovered by the computer or shown by the computer to be unique: the win with Nb1 v Nh8/Na6 (Evseev 1992), the draws with Qh4 v Kc6/Ba8 and Kc6/Bb7 (Byway 1995), the draw with Bd1 v Qd6/Na8 (play 1 Ba4 Qf4 with a reflection of the position), the reciprocal zugzwangs with Bh6/Bc1 v Kc6 and Qb8/Qf8 v Kd3, and the win with Ng2/Nd5 v Ra8 (Beasley 1998).
1998 Byway, P. V. Variant Chess 27, spring 1998. Four studies, one a twin.
(1998) Kohli, L. Internet (I am relying a copy sent to me by Fabrice Liardet). White Ra2, Ng4, Pd2 (3), Black Pd4 (1), White to play and win. Not 1 Rc2 d3 and White has no good move, but 1 Rb2 d3 2 Rc2 dxc2 3 Ne5/Ne3/Nf2. The alternatives at move 3 could be eliminated by starting wN at g6.
1998 Liardet, F. Bulletin Genevois des Echecs 36, ix-xii.1998. White Pd5/b2/f2 (3), Black Rb5/f5 (2), White to play and win. Play 1 d6 Rxb2 (say) 2 d7 Rfxf2 3 d8B; Black can sacrifice one rook, but then loses with R v B. If 2...Rbxf2 then 3 d8N.
(1998) Liardet, F. Au coin du bois 1998 (I am relying on a transcription sent to me by the composer). White Ra4, Pf7 (2), Black Kg5/h3 (2), White to play and win. It is usually impossible to win against two kings, but here White can profit from their bad position. 1 Rh4! Khxh4 (if 1...Kgxh4 then simplest is 2 f8Q Kh2 3 Qa3, though 2 f8R also wins) 2 f8N! Kgg4/Kf4 (2...Kg6 3 Nxg6 with an N v K win) 3 Nh7! and the knight will sacrifice itself on g5.
1998 Liardet, F. Variant Chess 27, spring 1998. Two studies.
These were set as competition pieces, but only one solver cracked the second study and none at all cracked the first.
1998 Liardet, F. phénix 62, iv.1998. White Re8, Pc7 (2), Black Ka6, Nd7 (2), White to play and win. The solution is 1 Rb8 Nxb8 2 cxb8N with an exceptional win with N v K, familiar ground by 1998, but it is heightened by the try 1 Rf8 Nxf8 2 c8B, met only by 2...Nd7 3 Bxa6 Nb8 with an exceptional win for Black with N v B on the same two squares.
1998 Liardet, F. phénix 65, vii-viii.1998. White Rb6, Pf7 (2), Black Pe2 (1), White to play and win. 1 Rf6 e1B (1...e1N 2 Rf3 Nxf3 3 f8R, 1...e1K 2 Ra6 and 3 f8B) 2 Rf2 Bxf2 3 f8R with an exceptional position where a bishop to play loses against a rook. This is heightened by the try 1 f8R, defeated only by 1...e1B. Either wR can now sacrifice itself independently against bB, but in each case this sacrifice leaves bB attacking the other wR and this time it is the bishop that will win.
1998 Liardet, F. Schweizerische Schachzeiting, ix.1998. An article "Quelques finales en qui perd gagne " including three examples described elsewhere in this document.
1998 Liardet, F. Eteroscacco 83, ix-x.1998 (pp 10-14), and 84, xi-xii.1998 (pp 3-8). "Premières rencontres internationales déchecs à qui perd gagne", a report of an international meeting held in Geneva on 12-13 September 1998, with the games played. Such endings as were reached tended to be misplayed in time scrambles, but one was of interest in spite of this: White Re6, Ph4 (2), Black Pf4 (1). White played 29 h5 (it is normally correct in endings to push any remaining pawns as fast as possible), and play continued 29...f3 30 h6 f2 31 h7 f1R? (31...f1B would have drawn) 32 Rf6 Rxf6 and 33 h8B would have won had Whites flag not fallen before he could play it. But for once, pushing the pawn was wrong. The correct line was to leave wP on h4 and play wR to e5, say 29-30 Re7 f2 31 Re5. Now 31...f1Q/R/B lose off-hand, 31...f1N is met by 32 Rf5 N-- 33 Rf1 Nxf1 34-37 h8B, and 31...f1K by 32 Ra5 and promotion to R or B as appropriate. It is an interesting example of the way in which the ending K v RB can occur naturally in play. Another game from the meeting would have ended with the same material had not the defender made a slip at an earlier stage.
1998 Liardet, F. Variant Chess 30, winter 1998. Two studies, one a twin.
(1999) "Angrim" (Ben Nye). Internet (copy forwarded to me by Fabrice Liardet). Build notes on a two-man and three-man database, excluding positions without pawns. Each entry gives the White and Black men, the percentages of draws and longest wins (to end of game), a position leading to the longest win and the length of this win, and a sample drawing position. The data appear to have been calculated using the "FICS" (Free Internet Chess Server) rule that a stalemate is a won for the player who is left with fewer pieces. The longest wins overall appear to be "66 ply" (33 moves) in the endings QP v P (Qa7/Pa3 v Pg3) and PP v P (Pa2/Pb2 v Pf6).
(1999) "Angrim" (Ben Nye). Internet (annotated copy forwarded to me by Fabrice Liardet). Similar build notes on a four-man database. Again there is no specimen play, but specific positions are commented as follows (White always listed first and always to play, and the positions are longest wins unless stated otherwise).
It should perhaps be stressed that these are first comments on a massive set of data which did not become available until very late in the period under review. A similar survey made a year or two hence might show considerable amplification. See also Liardet below.
(1999) "Angrim" (Ben Nye). Internet (annotated copy forwarded to me by Fabrice Liardet). A listing of non-trivial three-man positions of loss-loss reciprocal zugzwang (whoever is to move loses) involving at least one pawn. Some positions in the endings N v BP, N v NP, N v PP, P v NN, P v NP, and P v PP are omitted. The "ply" counts are one higher than might be expected (for example, Qa8 v Bh6/Pg7, with typical play WTM 1 Qh1 Bc1 2 Qxc1 g5 3 Qxg5 and BTM 1...g6 2 Qf8 Bxf8, is given as "loss in 6/4" although only 5 and 3 moves are actually played). See also Liardet below.
(1999) "Angrim" (Ben Nye). Internet (annotated copy forwarded to me by Fabrice Liardet). A listing of all non-trivial four-man positions of loss-loss reciprocal zugzwang (whoever is to move loses) involving neither knights nor pawns. See also Liardet below.
(1999) "Angrim" (Ben Nye). Internet (annotated copy forwarded to me by Fabrice Liardet). A listing of all non-trivial four-man positions of loss-loss reciprocal zugzwang (whoever is to move loses) involving precisely one knight or pawn. The following specific position is commented:
See also Liardet below.
(1999) "Angrim" (Ben Nye). Internet (copy forwarded to me by Fabrice Liardet). A listing of non-trivial four-man positions of loss-loss reciprocal zugzwang (whoever is to move loses) involving two knights or pawns. If an ending contains not more than six such positions, all are given; otherwise, only six are given.
1999 "Angrim" (Ben Nye) (discovered by computer). Variant Chess 32, summer 1999. White Ka6/f2 (2), Black Qh8, Rc8 (2), White to play and draw. 1 Ka7 Rd8 2 Kf3 Rc8 3 Kf2 Rd8 4 Kf3 etc. If White relaxes the pressure, Black wins by taking one of the kings: 1 Ka5 Rc3 2 Kf1 Rc2 etc, or 1 Kg3 Qh2 2 Kxh2 Rc2. Black also wins if White deviates from the main line at a later stage: 2 Kb7 Rc8, or 3 Ke2 Rb8.
1999 Beasley, J. D. Three-man pawnless endings in Losing Chess (26pp, published as a self-standing pamphlet). An exposition based on definitive analysis by computer. It contains the following.
The document includes solutions to over fifty positions, mainly "longest wins" or other positions of particular interest, and the various positions credited here as "Beasley (discovered by computer)" come either from it or from a preliminary version which was circulating privately from January 1998.
(These expositions resulted from attempts to extract general rules from a mass of unstructured data, and two omissions should be noted. In K v QB, it is not made clear that the kings only hope is to attack the queen at once, and the list of draws becomes much more comprehensible once this is realised. If the pressure is relaxed, the pieces have a certain win. In K v QN, attention should have been drawn to two important positions of domination with Nd5 against a king on the first rank: Qf6/Nd5 v Kd1 and Qa6/Nd5 v Kc1. In the latter, 1 Kb1 allows 1...Qa2 2 Kxa2 Nb4 with an exceptional win with N v K, an interesting counterpart to the win with Ne4 v Ka1 exploited in one of the Liardet 1997 studies in Variant Chess 26. A computer-based exposition is more reliable than a non-computer in its identification of exceptional cases, but it may not be so good at identifying logical connections between positions and where a competent pre-computer exposition exists it should always be studied as well.)
1999 Beasley, J. D. (discovered by computer). Variant Chess 32, summer 1999. Two studies.
1999 Beasley, J. D. British Endgame Study News, special number 18, xii.1999. "Paradoxical play in the Losing Game" (pp 2-3). A selection of studies showing paradoxical manoeuvres of various kinds: Nb5/Pd7 v Ka8 (Goldovski 1999), Nd4/Ne4 v Kb8 (the priority of Goldowski 1999 is acknowledged), Rb6/Pf7 v Pe2 (Liardet 1998), and Na4/Pg7/Ph7 v Rf1/Pd2 (Liardet 1998).
1999 Goldovski, S. The Problemist, iii.1999. White Nd4, Pd7 (2), Black Ka7 (1), White to play and win in 5. 1 Nb5 Ka8 2 Nc7 (2 Na7 only draws) Ka7 3 Na8 (3 Na6 Kxa6 4 d8R also wins, but not in 5) Kxa8 4 d8B and wB dominates bK. Although not formally published until 1999, this had been circulating privately since mid-1997, and it anticipates the discovery of similar knight manoeuvres by computer.
1999 Gruber, H. Die Schwalbe, xii.1999. Article "(Un)Vergängliche Schwalben" recapitulating various compositions published in Die Schwalbe, including a complete analysis of Kuhlmann 1980 (pp 304-6).
(1999) Liardet, F. Internet (copies of files forwarded to me). A large amount of material covering all aspects of the game, some of it certainly predating 1999. Endgame material is highlighted in the entries below.
(1999) Liardet, F. Internet (copy of file forwarded to me). Page "Points faibles" (Weak points): a discussion of three kinds of weak point and how to exploit them. All are relevant to endgame play, though only one is illustrated with an endgame example.
My translations.
(1999) Liardet, F. Internet (copy of file forwarded to me). Page "Les finales de qui perd gagne ". A general introductory page, leading into what follows.
(1999) Liardet, F. Internet (copy of file forwarded to me). Page "Principes généraux sur les finales". A general discussion, giving the general rule of strength ("grosso modo") R > K = Q = B > N > P ("in the ending, the pieces are generally free from obstacles and can exercise their intrinsic power"), and stressing (a) the basic principle of advancing the pawns as quickly as possible and (b) the importance of zugzwang particularly once the last pawn has promoted. "If in ordinary chess zugzwang is fairly rare, although being able to master it is essential, in Losing Chess endings it is the rule. It is nearly always impossible to proceed by direct attack (there are no weak points!) and to win it is necessary to drive the opponent to suicide." A "particularly striking" example is given: White Kf1, Nc3 (2), Black Qh6 (1), whoever is to move loses.
(1999) Liardet, F. Internet (copy of file forwarded to me). Page "Finales de deux pièces (sans pions)". A comprehensive exposition of two-man pawnless endings, containing a general table of results, a complete list of exceptions, and a demonstration of the winning method with N v N, K v N, and R v K.
(1999) Liardet, F. Internet (copy of file forwarded to me). Page "Finales de trois pièces (sans pions)". A comprehensive practical exposition of three-man pawnless endings, containing a general table of results and a more detailed discussion of particular cases. The ending KN v Q is described as "inclassable" (impossible to classify): "all results are possible, the draw being a little less frequent than the respective wins".
The following endings are given a more detailed treatment.
(There are indeed exceptions, and it ought perhaps to be stressed that one of the knights must be within the region. It isnt sufficient just to put a knight into the region, it must be able to maintain itself there. For an extreme example, consider Nc7/Ne5 v Rh2. Here, one knight is not just within the region, it is on one of the four central squares, and even the other knight is away from the edge; yet 1...Rf2 drives the central knight away, and Black will win (2 Nc6 Rf3 etc). There are also exceptions the other way, perhaps the most extreme being Nd7/Nc8 v Rg4. White is very cramped and he will never be able to put a knight even temporarily into the region, but he can maintain one knight at d7 and move the other between c8 and a7, and Black will never be able to advance.)
There follows a detailed and leisurely exposition of the study Nd7/Pe7 v Nh4 (Variant Chess 1997, see above), and then two exceptional drawing positions with BN v N and KN v N respectively: Bd8/Nf6 v Nd3 (play 1 Ne8 Nc5 2 Bf6 Nd3 3 Bd8 with a loss if either side deviates) and Kb8/Ng3 v Nc4 (play 1 Nh5 Ne5 2 Ka7 Nc4 3 Kb8 Ne5 similarly). There is also a reference to two computer-discovered studies which appear elsewhere in this document.
"1) The bishop wins if the knight is on a square of the same colour as the bishop. The knight always draws if it is at least two squares away from the edge and the king can join it, and also by a little touch of trickery exploiting the exceptional positions with knight against bishop." Three illustrative examples are given, with full analysis: Kh4/Ne3 v Bc3 (Black to play wins, White to play would draw), Kb4/Ne3 v Bg3 (Black to play can only draw because wK can seek refuge on a6, and if ...Bc7? then Ka5 and Nf5 with an N v B win), Kf3/Nd2 v Bb2 (Black to play wins, being careful to keep wK away from h4).
"2) The king and knight can only hope to win if the knight succeeds in establishing itself on a square opposite to that of the bishop and within the region d2-e2-g4-g5-e7-d7-b5-b4-d2." Example: Ke1/Nd2 v Bc8, White to play wins by 1 Kd1 (not 1 Ke2?? Bh3 and Black wins, nor 1 Kf2 Bd7 2 Ke3?? Bc6) Bd7 2 Kc2 2 Bc8/Be8 3 Kd3! Bd7 4 Kd4 Be8 5 Ke5. "Surprising as it might seem," the position after 1 Kd1 is reciprocal zugzwang: White to play could not win (2 Kc2 Bd7 3 Kd3 Be8! and if 4 Ke4? then 4...Bh5 wins for Black, or 3 Kc3 Bg4! and draws).
"Exceptionally, the bishop can win if the knight is on a square of the right colour but very badly placed, as is shown by one of the exercises." This position appears elsewhere in this document.
My translations.
(1999) Liardet, F. Internet (copy of file forwarded to me). Page "Finales de quatres pièces". An introduction to the various "four-piece ending" files of "Angrim" (see above, also Liardets notes below), together with a more detailed treatment of certain endings involving kings and rooks only. There is a warning that Angrims results assume the FICS rule that stalemate is a win for the player having the smaller number of men remaining on the board.
The following endings are examined in detail.
A drawing example is given as an exercise (see the page of exercises below). "When the White king is pinned against the edge by his colleague, Black can counter all attempts to escape, even though sometimes a certain precision is necessary."
My translations.
(1999) Liardet, F. Internet (copy of file forwarded to me). Page "Finales avec des pions". An introduction to the various "three-piece anding" and "four-piece ending" files of "Angrim" (see above, also Liardets notes below), together with a more detailed treatment of certain three-man endings with one White pawn. There is a warning that Angrims results assume the FICS rule that stalemate is a win for the player having the smaller number of men remaining on the board.
The following endings are examined in detail.
"1) When the White king is not in the way and the pawn can reach a square two ranks ahead of the Black king, it draws by promoting to rook or king. But often the presence of the White king allows Black to gain one or two tempi." Example: Kb7/Pd4 v Kb3. "Here, 1 d5 puts the pawn two squares ahead of the king, but it is not enough: 1...Kb4 2 d6 Ka5 3 Ka6 (or 3 Kc8 Ka6) Kxa6 4 d7 Kb7 and the pawn is caught. If the White king had been on b8, the position would have been drawn (the king would have been sacrificed on a7, permitting a promotion to king). On the other hand, if the White king had been on c6, Black would have won even wasting a tempo: 1 d5 Ka3!? 2 d6 Ka4! 3 Kb5 Kxb5 4 d7 Kc6!
"2) Otherwise Black has an easy win, if necessary by manoeuvring around the pawn from the long side." Example: Kc4/Pc2 v Kc7, Black to play. "The fact that the pawn still has the possibility of a two-step move does not prevent the win. 1...Kd7! (aiming for e2) 2 Kd4 Ke7 3 Ke4 Kf6 4 Kd3 Kf5 5 Kc3 (or 5 Kc4 Kf4 6 Kc5 Kf3 7 Kc4 Ke2 and wins) Kf4 6 Kc4 Kf3 7 Kc5 (or 7 Kc3 Kf2 8 Kc4 Ke2) Ke2 8 c4 Kf3 9 Kc6 Ke4 and the king will be able to sacrifice itself on d5.
"Note that a bishop or a knight is likewise useless when accompanying a pawn, and that the endings BP v K and NP v K are similarly won for the king unless the pawn is sufficiently far advanced."
"1) The king always wins if it can place itself in front of the pawn; whatever the position of the rook, the king also wins if it can reach the diagonal opposition with respect to the pawn.
"2) In general, the rook and pawn win if the king is penned two squares below the pawn." Example: Ra3/Pf4 v Ke2. "1 Ra4! (the rook is awkwardly placed where it is, and this is the only good rank for it; for example, 1 Ra7? Kd3 2 f5 Kd4 3 f6 Kd5 4 f7 Kc6, or 1 Ra8? Kd3 2 f5 Kd4 3 f6 Kd5 4 f7 Kd6 5 Rh8 {5 f8R? Kd7 and it is Black who wins} Kd7 6 Rc8 Kxc8 7 f8K/B) Kd2 (if Black does not try to attack the rook, White can advance peacefully by 2 f5) 2 f5 Kd3 (2...Kc3 3 Rb4 followed by a rook promotion) 3 Ra1! Kc4 (after 3...Kc2 White has time for 4 Ra7 {4...Kb3 5 Ra2}) 4 f6 Kd5 (4...Kb5 5 Ra4) 5 f7 Ke5 6 Ra2! and after this waiting move Black is forced to allow a rook or bishop promotion (we recall that the ending RB v K is winning).
"3) The rook also wins if it can cut the king off from the pawn by occupying the file adjacent to the pawn, unless the king is three ranks ahead of the pawn." However, there is a drawing case where the king gets two ranks behind the pawn, drives the rook off the file, and then goes forward, and I understand that the text here has now been changed. Set Ra1/Pd3 v Kg5 and White appears to have a win by 1 Re1 Kg6 2 d4 Kg7 3 d5 Kg8 4 d6 Kh7 5 d7 Kg7 6 Re2 followed by promotion to R or B, but if Black plays 1...Kg4! 2 d4 Kg3! (threat 3...Kf2 drawing) 3 Re6/7/8 Kf2!! as in Hofmann 1956 he can hold the draw. Set everything one rank higher, Ra2/Pd4 v Kg6, and White does win.
My translations.
(1999) Liardet, F. Internet (copies of files forwarded to me). Pages "Exercices didactiques", "Etudes de mon cru", "Etudes dautres compositeurs", and "Positions trouvées par ordinateur". Four pages of endgame exercises and studies, including several not noted elsewhere in this document.
(1999) Liardet, F. Internet (copy of file forwarded to me). Notes inserted into Angrims file of build notes on his four-man database (see above). The added notes highlight particular positions, giving the essentials of the play and commenting on features of special interest.
Liardet comments that subsequent computer analysis by Ben Nye ("Angrim") has shown that there are precisely six such fortress positions for the knights: the one used here, and Nd3/Nf3/Nxx where xx runs through the five squares f5/e4/e2/f1/h1. See also "Positions trouvées par ordinateur" above, "where you will find out why Nc4, Nf3, Nh1 is not a fortress".
(1999) Liardet, F. Internet (copy of file forwarded to me). Notes inserted into Angrims list of non-trivial three-man positions of loss-loss reciprocal zugzwang (see above). Some notes merely highlight particular positions without further comment, and the comments on these are my own responsibility.
(1999) Liardet, F. Internet (copy of file forwarded to me). Notes inserted into Angrims list of four-man positions of loss-loss reciprocal zugzwang (whoever is to move loses) involving neither knights nor pawns. One note merely highlights a particular position without further comment, and the comment on this is my own responsibility.
(1999) Liardet, F. Internet (copy of file forwarded to me). Notes inserted into Angrims list four-man positions of loss-loss reciprocal zugzwang (whoever is to move loses) involving precisely one knight or pawn. Some notes merely highlight particular positions without further comment, and the comments on these are my own responsibility.
(1999) Nagorko, A. Internet. White Bc7/e5/g2, Nc6, Pb6/b4/c3 (7), Black Bg5/g1, Pf2 (3), White to play and win. 1 Bf6 Bxf6 2 Nb8 Bxc3 3 Ba8! Bxb4 4 Bd6 (4 Bh2 also works) Bxd6 5 b7 Bxb8 and stalemate. The purpose of bBg1/bPf2 is unclear, particularly as without them there would be no dual at move 4.
(Surprisingly, the win by stalemate against a bishop of the opposite colour seems to have inspired only three compositions, Dawson 1924, Dittmann 1987, and this one, and even Dittmann 1987 relies on "Win in n moves" to force the stalemate. While I was writing this document, the following curious position occurred to me: White Kb8, Pa7/b7/b6 (4), Black Be5 (1), win only by 1 a8B, winning against a lone bishop by promoting to a bishop of the opposite colour!)
"1999" Anybody carrying this survey forward is asked to note that issue 34 of Variant Chess, although nominally dated "Winter 1999", did not appear until early in 2000, and that material from it has not been included here.
Definitive analyses by computer
It is not clear to what extent these have been formally published, but their existence is known and I think they should be included.
1992 Evseev, G. Two knights against one. Data not made generally available so far as I know, but one position reported in the literature (see Evseev 1992 and Evseev and Poisson 1993 above).
1998 Beasley, J. D. All three-man pawnless endings. Data made available on disc, with an interrogation program; results for two knights against one confirm published details regarding Evseev 1992. Described in Three-man pawnless endings in Losing Chess (see Beasley 1999).
1998 Bartholdi, L. All three-man endings with or without pawns, calculated both assuming stalemate to be a win and assuming it to be a draw. The database is part of the Losing Chess playing program Iznogoud, but there is no separate interrogation program.
1998 Bartholdi, L. Three kings against rook. Data not made generally available so far as I know.
1999 "Angrim" (Ben Nye). All four-man endings, with and without pawns. The data were created for use with the Losing Chess playing program ASCP, and so far as I know only summaries and selected positions are as yet generally available. Moves are counted to end of game, and the FICS stalemate rule (stalemate is a win for the player with the smaller number of men) is assumed.
Index of positions with up to five men
Because some material combinations have been explored in several studies, sometimes with colours reversed, I have ignored colour and have always put the side with the smaller number of men first. I have indexed initial positions comprehensively but have been more selective with positions occurring in play, generally including the latter only when they appear to be significant. Multiple items within a year are indexed as /1, /2, etc, and bullets within an item as a, b, c. It should be stressed that in some cases the author cited is an editor or compiler and not the originator of the material.
Two men
Many two-man endings are straightforward, and most articles cover a range of cases. Only explicit articles and compositions are indexed here, and it should be appreciated that a lot more must have been discovered at a very early stage. For example, the natural line in Schellenberg 1901 comes down to a win with R v K, and it is inconceivable that the winning nature of this elementary ending had not already been recognized.
Piece v piece: Klüver 1923, Klüver 1924, Niemann 1938 (nightriders), Leoncini and Magari 1980, Liardet 1991, Beasley 1996, Marks 1996 (king against zebra), Beasley 1999/1, Liardet 1999/5.
Piece v pawn: Klüver 1923, Klüver 1924, Panteleit 1975, Panteleit 1977, Magari 1979, Leoncini and Magari 1980, Beasley 1993, Beasley 1996/1.
Pawn v pawn: Törngren 1929, Salvadori 1979, Leoncini and Magari 1980.
Exceptional piece-against-piece winning positions are exploited in play as follows.
R v B: Liardet 1998/5.
B v K: Beasley 1999/2b, Goldovski 1999.
B v R: Törngren 1929, Hofmann 1956/2, Niemann 1948/1, van der Bilt 1997c, Liardet 1998/1, Liardet 1998/3b, Liardet 1998/5, game analysis in Liardet 1998/7.
N v K: Klüver 1923c, Hofmann 1956/1, Mortensen 1960, Liardet 1997/2a, Liardet 1998/2, Liardet 1998/3a, Liardet 1998/4, Liardet 1998/8b, Beasley 1999/2a, Liardet 1999/10a.
N v R: Liardet 1998/3ab, Liardet 1998/8b.
N v B: Sunyer 1930, Slater 1935, Niemann 1948/2, Klüver 1949, Magari 1979, Evseev 1992/2, Beasley 1997/1, van der Bilt 1997ab, Liardet 1997/3, Liardet 1998/4, Liardet 1999/6d, Liardet 1999/9i, Liardet 1999/10gi.
Three men
Three-man pawnless endings are examined in general terms in Beasley 1999/1 and Liardet 1999/6, and three-man endings with at least one pawn in Angrim 1999/1, Angrim 1999/3, Liardet 1999/8, and Liardet 1999/11. Individual endings are further examined or exploited as follows.
K v KQ: Klüver 1924b, Fabel and Klüver 1947a.
K v KR: Klüver 1924b, Klüver 1948, Leoncini and Magari 1980f, Liardet 1999/6e, Lairdet 1999/9a (in play).
K v KB: Klüver 1924b, Fabel and Klüver 1947b (in play), Byway 1998c (in play), Byway 1998d.
K v KN: Klüver 1924b, Liardet 1997/2b (in play).
K v KP: Beasley 1989, Liardet 1999/8a.
K v 2Q: Beasley 1998/5.
K v QB: Kuhlmann 1980 (in play), Beasley 1999/1.
K v QN: Klüver 1924b, Liardet 1991, Liardet 1997/2a (in play), Beasley 1999/1.
K v RB: Klüver 1934a, Leoncini and Magari 1980d, Liardet 1991e, game analysis in Liardet 1998/7, Liardet 1999/6g.
K v RN: Klüver 1923b, Klüver 1924b, Geerlings 1997 (in play), Liardet 1997/2a (in play), Liardet 1997/3 (in play), Liardet 1999/6e.
K v RP: Hofmann 1952 (in play), Hofmann 1956/1, Büsing 1983/a (in play), Liardet 1999/8b.
K v 2B: Leoncini and Magari 1980g, Liardet 1991d, Beasley 1998/5, Liardet 1999/6f.
K v BN: Beasley 1999/2b.
K v BP: Kahl 1951b (in play), Liardet 1999/8a.
K v 2N: Beasley 1999/2a, Beasley 1999/3.
K v NP: Goldovski 1999, Liardet 1999/8a.
K v 2P: Klüver 1923c, Hofmann 1952, Leoncini and Magari 1980e, Büsing 1983/2a.
Q v 2K: Byway 1995 (in play).
Q v KB: Byway 1995 (in play).
Q v KN: Niemann 1947 (in play), Liardet 1991, Byway 1995 (in play), Geerlings 1997 (in play), Beasley 1998/2b, Liardet 1999/4, Liardet 1999/6.
(continued)
Three men (continued)
Q v BP: Angrim 1999/3.
Q v 2N: Liardet 1991.
Q v 2P: Liardet 1999/11c.
R v KB: Byway 1998c (in play).
R v KN: Geerlings 1997 (in play), Liardet 1997/2b (in play).
R v BN: Sekhar and Shankar 1987a.
R v 2N: Liardet 1991f, Beasley 1998/1, Liardet 1999/6b.
B v KN: Liardet 1991c, Byway 1998a (in play), Liardet 1999/6d, Liardet 1999/9b, Liardet 1999/10k (in play).
B v KP: Liardet 1999/9c.
B v QN: Beasley 1998/5.
B v 2R: Liardet 1998/1 (in play), Liardet 1998/5 (in play).
B v RN: Niemann 1948/2.
B v BP: Beasley 1996/2.
B v NP: Geerlings 1997 (in play).
N v KQ: Byway 1998ab (in play).
N v KR: Kuhlmann 1980 (in play), Byway 1998ab (in play).
N v KN: Beasley 1997/2, Liardet 1998/3a (in play), Liardet 1999/6c, Liardet 1999/10n (in play).
N v KP: Liardet 1999/11a.
N v QN: Liardet 1999/6c, Liardet 1999/9f.
N v RN: Liardet 1998/3a (in play), Liardet 1999/6c.
N v RP: Liardet 1999/11b.
N v BN: Panteleit 1975a (in play), Kuhlmann 1980 (in play), Liardet 1991b, Liardet 1997/1 (in play), Beasley 1998/2a, Beasley 1998/4 (in play) , Liardet 1999/6c, Liardet 1999/10d (in play).
N v 2N: Klüver 1924b, Fabel 1947, Hansson 1948, Magari 1978, Leoncini and Magari 1980c, Evseev 1992/1, Evseev and Poisson 1993, Beasley 1997/5, Liardet 1999/6a.
N v NP: Panteleit 1975a, Liardet 1997/1, Beasley 1998/4.
N v 2P: Evseev 1992/2 (in play), Beasley 1997/4, van der Bilt 1997ab (the latter in play).
P v KQ: Byway 1998b.
P v KR: Byway 1998b.
P v KB: Fabel and Klüver 1947b, Byway 1998c.
P v KP: Byway 1998a, Liardet 1999/9a.
(continued)
Three men (continued)
P v QP: Angrim 1999/1.
P v 2R: Klüver 1923f.
P v RN: Liardet 1999/11d.
P v RP: Liardet 1998/5, Liardet 1998/7, Liardet 1998/8a, Liardet 1999/8c.
P v 2N: Klüver 1923e, Mortensen 1960, Beasley 1997/1.
P v NP: Geerlings 1997.
P v 2P: Liardet 1997/3, Angrim 1999/1.
Four men
Four-man endings are examined in general terms in Angrim 1999/2, Angrim 1999/4-6, Liardet 1999/7, Liardet 1999/10, and Liardet 1999/12-13. Individual endings are further examined or exploited as follows.
One man against three
K v 3K: Liardet 1999/7d.
K v KKR: Liardet 1999/7c.
K v KRR: Liardet 1999/7b, Liardet 1999/9d.
K v KBN: Angrim 1999/2l.
K v KBP: Angrim 1999/2l.
K v KNP: Angrim 1999/2l.
K v KPP: Angrim 1999/2l.
K v QBB: Liardet 1999/12bc.
K v QNP: Angrim 1999/2m.
K v RRB: Leoncini and Magari 1980d.
K v RBP: Kahl 1951b (in play), Liardet 1999/10e.
K v BNP: Angrim 1999/2l.
K v NPP: Liardet 1999/9e.
K v 3P: Liardet 1999/10f, Leoncini and Magari 1980e.
Q v KPP: Byway 1995.
R v 3K: Liardet 1999/7a, Liardet 1999/10uv, Liardet 1999/12d.
R v KKR: Liardet 1999/10w, Liardet 1999/12e.
R v KKN: Liardet 1999/13p.
R v KKP: Liardet 1999/13q.
R v KBP: Liardet 1999/10x.
B v QNN: Liardet 1999/10j.
B v RRP: Dawson 1925b, Liardet 1999/13n.
B v RNN: Liardet 1999/10i.
B v RNP: Liardet 1999/10kl.
B v 3N: Angrim 1999/2n, Liardet 1999/9i, Liardet 1999/10gh.
B v 3P: Wood 1994.
(continued)
One man against three (continued)
N v KKB: Liardet 1999/10m.
N v KKN: Liardet 1999/10n.
N v KKP: Liardet 1999/10o.
N v KRB: Liardet 1999/13o.
N v KRN: Liardet 1998/8b.
N v KBB: Liardet 1999/10pq.
N v RRB: Liardet 1999/10s.
N v RRN: Liardet 1999/10t.
N v RBB: Liardet 1999/10r.
N v RPP: van der Bilt 1997b.
N v 3N: Magari 1978, Leoncini and Magari 1980c.
N v 3P: Evseev 1992/2.
P v KBN: Angrim 1999/2l.
P v KBP: Angrim 1999/2l.
P v KNP: Angrim 1999/2l.
P v KPP: Angrim 1999/2o.
P v QNP: Angrim 1999/2p.
P v RNP: Kohli 1998.
P v RPP: Beasley 1996/2.
P v BNP: Angrim 1999/2l.
P v 3N: Sunyer 1930.
P v NPP: Angrim 1999/2l.
P v 3P: Beasley 1996/2.
Single-step movers: Magari 1978.
Two men against two
2K v QR: Angrim 1999/7.
2K v RP: Liardet 1998/2.
2K v BN: Angrim 1999/2a.
2K v 2N: Angrim 1999/2b, Liardet 1999/10a.
KQ v KB: Angrim 1999/2c.
KQ v KN: Angrim 1999/2f, Liardet 1999/13c.
KQ v RB: Liardet 1999/12a.
KQ v BN: Liardet 1999/13j.
KQ v 2N: Liardet 1999/10c.
KR v QR: Liardet 1999/10b.
KR v RN: Angrim 1999/2g, Liardet 1999/9g.
KR v BN: Liardet 1999/13i.
KB v QR: Angrim 1999/2d.
KB v RB: Sekhar and Shankar 1987b.
KB v RN: Liardet 1999/13a.
KB v RP: Liardet 1999/13b.
KB v 2P: Angrim 1999/2e.
KN v KP: Liardet 1999/2a, Liardet 1997/2b (in play).
(continued)
Two men against two (continued)
KN v 2Q: Liardet 1999/13h.
KN v QR: Liardet 1999/13fg.
KN v QP: Niemann 1947.
KN v RP: Liardet 1998/4.
KN v RB: Liardet 1999/13de.
KN v BP: Liardet 1997/2b (in play).
KN v 2P: Liardet 1997/2b.
KP v 2P: Kuhlmann 1980.
QR v BP: Angrim 1999/5a, Liardet 1999/13m.
QB v 2N: Angrim 1999/2h, Liardet 1999/9h.
QP v RP: Angrim 1999/2k.
2R v 2P: Angrim 1999/2j.
RB v RN: Liardet 1999/13kl.
RN v BN: Liardet 1999/10d.
RP v RP: Hofmann 1956/2.
BP v BP: Beasley 1996/2.
2N v 2N: Klüver 1924d, Fabel 1947, Charosh 1948, Hansson 1948, Magari 1978, Leoncini and Magari 1980c, Angrim 1999/2i.
NP v NP: Liardet 1997/2a.
NP v 2P: Dornieden 1967, Büsing 1983/1.
2P v 2P: Klüver 1934b.
Five men (one against four)
K v BNPP: Schmidt and Kniest 1948.
K v 4P: Leoncini and Magari 1980e.
R v KBPP: van der Bilt 1997c.
B v BNPP: Schlensker and Kniest 1948 (in play).
P v NPPP: Watney 1923.
Five men (two against three)
KN v KNP: Klüver 1949.
KP v QPP: Boyer 1955/2 (in play), Klüver 1957, Slater 1958.
2R v 3P: Liardet 1998/1.
RP v NPP: Liardet 1998/3b.
BN v KPP: Slater 1935.
BP v RNP: Roese 1923.
BP v BNP: Beasley 1996/2.
NP v RPP: van der Bilt 1997d.
NP v BPP: Büsing 1983/1.
NP v NPP: Carfora 1978.
2P v KPP: van der Bilt 1997e.
2P v QPP: Boyer 1955/1.
2P v RPP: Büsing 1983/2b.
2P v BNN: Minieri 1979.
2P v 3P: Niemann 1948/1.