Pawnless chess endgame

src: lebanesechess.com

A pawnless chess endgame is a chess endgame in which only a few pieces remain and none of them is a pawn. The basic checkmates are types of pawnless endgames. Endgames without pawns do not occur very often in practice except for the basic checkmates of king and queen versus king, king and rook versus king, and queen versus rook (Hooper 1970:4). Other cases that occur occasionally are (1) a rook and minor piece versus a rook and (2) a rook versus a minor piece, especially if the minor piece is a bishop (Nunn 2007:156-65).

The study of some pawnless endgames goes back centuries by players such as François-André Danican Philidor (1726-1795) and Domenico Lorenzo Ponziani (1719-1796). On the other hand, many of the details and recent results are due to the construction of endgame tablebases. Grandmaster John Nunn wrote a book (Secrets of Pawnless Endings) summarizing the research of endgame tablebases for several types of pawnless endings.

The assessment of endgame positions assumes optimal play by both sides. In some cases, one side of these endgames can force a win; in other cases, the game is a draw (i.e. a book draw).

Video Pawnless chess endgame

Terminology

major pieces are queens and rooks
minor pieces are knights and bishops
a rank is a row of squares on the chessboard
a file is a column of squares on the board
If a player has two bishops, they are assumed to be on opposite colors unless stated otherwise.

When the number of moves to win is specified, optimal play by both sides is assumed. The number of moves given to win is until either checkmate or the position is converted to a simpler position that is known to be a win. For example, with a queen versus a rook, that would be until either checkmate or the rook is captured, resulting in a position that leads to an elementary checkmate.

Maps Pawnless chess endgame

Basic checkmates

Checkmate can be forced against a lone king with a king plus (1) a queen, (2) a rook, (3) two bishops, or (4) a bishop and a knight (see Bishop and knight checkmate). See checkmate for more details. Checkmate is possible with two knights, but it cannot be forced. (See Two knights endgame.)

src: upload.wikimedia.org

Queen versus rook

A queen wins against a lone rook, unless there is an immediate draw by stalemate or due to perpetual check (Nunn 2002a:49) (or if the rook or king can immediately capture the queen). Normally the winning process involves the queen first winning the rook by a fork and then checkmating with the king and queen, but forced checkmates with the rook still on the board are possible in some positions or against incorrect defense. With perfect play, in the worst winning position, the queen can win the rook or checkmate within 31 moves (Müller & Lamprecht 2001:400).

The "third rank defense" by the rook is difficult for a human to crack. The "third rank defense" is when the rook is on the third rank or file from the edge of the board, his king is closer to the edge and the enemy king is on the other side (see the diagram). For example, the winning move in the position shown is the counterintuitive withdrawal of the queen from the seventh rank to a more central location, 1. Qf4, so the queen can make checking maneuvers to win the rook with a fork if it moves along the third rank. If the black king emerges from the back rank, 1... Kd7, then 2. Qa4+ Kc7; 3. Qa7+ forces Black into a second-rank defense (defending king on an edge of the board and the rook on the adjacent rank or file) after 3... Rb7. This position is a standard win, with White heading for the Philidor position with a queen versus rook (Müller & Lamprecht 2001:331-33). In 1895 Edward Freeborough edited an entire 130-page book of analysis of this endgame, The Chess Ending, King & Queen against King & Rook. A possible continuation: 4. Qc5+ Kb8 5. Kd6 Rg7 6. Qe5 Rc7 7. Qf4 Kc8 8. Qf5+ Kb8 9. Qe5 Rb7 10. Kc6+ Ka8 11. Qd5 Kb8 12. Qa5 [Philidor--mate in 7].

Example from game

In this 2001 game between Boris Gelfand and Peter Svidler, Black should win but the game was a draw because of the fifty-move rule. Black can win in several ways, for instance:

1... Qc8

2. Kf7 Qd8

3. Rg7+ Kf5

4. Rh7 Qd7+

5. Kg8 Qe8+

6. Kg7 Kg5, and wins.

The same position but with colors reversed occurred in a 2006 game between Alexander Morozevich and Dmitry Jakovenko - it was also drawn (Makarov 2007:170). At the end of that game the rook became a desperado and the game ended in stalemate after the rook was captured (otherwise the game would have eventually been a draw because of perpetual check, i.e. threefold repetition).

Browne versus BELLE

Queen versus rook was one of the first endgames completely solved by computers constructing an endgame tablebase. A challenge was issued to Grandmaster Walter Browne in 1978 where Browne would have the queen in a difficult position, defended by BELLE using the queen versus rook tablebase. Browne could have won the rook or checkmated in 31 moves with perfect play. After 45 moves, Browne realized that he would not be able to win within 50 moves, according to the fifty-move rule. Browne studied the position, and later in the month played another match, from a different starting position. This time he won by capturing the rook on the 50th move (Nunn 2002a:49).

Garry Kasparov still has the magic! | ChessBase

src: en.chessbase.com

Queen versus two minor pieces

Defensive fortresses exist for any of the two minor pieces versus the queen. However, except in the case of two knights, the fortress usually cannot be reached against optimal play. (See fortress for more details about these endings.)

Queen versus bishop and knight: A queen normally wins against a bishop and knight, but there is one drawing fortress position forming a barrier against the enemy king's approach (Müller & Lamprecht 2001:339-41). Another position given by Ponziani in 1782 is more artificial: the queen's king is confined in a corner by the bishop and knight which are protected by their king (Hooper & Whyld 1992:46).
Queen versus two bishops: A queen has a theoretical forced win against two bishops in most positions, but the win may require up to seventy-one moves (a draw can be claimed after fifty moves under the rules of competition, see fifty-move rule); there is one drawing fortress position for the two bishops (Müller & Lamprecht 2001:339-41).
Queen versus two knights: Two knights can generally draw against a queen if the king is near its knights and they are in a reasonable position by setting up a fortress. (Müller & Lamprecht 2001:339-41).

src: upload.wikimedia.org

Common pawnless endings (rook and minor pieces)

John Nunn lists these types of pawnless endgames as being common: (1) a rook versus a minor piece and (2) a rook and a minor piece versus a rook (Nunn 2007:156-65).

Rook versus a bishop: this is usually a draw. The main exception is when the defending king is trapped in a corner that is of the same color square as his bishop (Nunn 2002a:31) (see Wrong bishop#Rook versus bishop). If the defending king is trapped in a corner that is the opposite color as his bishop, he draws (see Fortress (chess)#Fortress in a corner). See the game of Veselin Topalov versus Judit Polgar, where Topalov defended and drew the game to clinch a win of their 2008 Dos Hermanas match.
Rook versus a knight: this is usually a draw. There are two main exceptions: the knight is separated from the king and may be trapped and won or the king and knight are poorly placed (Nunn 2002a:9).Kamsky vs Bacrot, 2006 is an example of a rook vs knight ending which resulted in a win. In this game, Black underpromoted a pawn to a knight to avoid a checkmate and eventually lost the game after allowing his knight to be separated from the king.
Rook and a bishop versus a rook: this is one of the most common pawnless endgames and is usually a theoretical draw. However, the rook and bishop have good winning chances in practice because the defence is difficult. There are some winning positions such as the Philidor position, which occurs relatively often. There are two main defensive methods: the Cochrane Defense and the "second rank defense" (Nunn 2007:161-65). Forced wins require up to 59 moves. As a result, FIDE extended the fifty-move rule to 100 moves and then to 75 moves for this endgame, before returning to fifty moves (Speelman, Tisdall & Wade 1993:382). See rook and bishop versus rook endgame for more information.
Rook and a knight versus a rook: This is usually a simple draw with few winning positions. The winning positions require the defending king to be badly placed near a corner; this can not be forced in general (Nunn 2007:159-61). The Cochrane Defense can be used.

src: en.chessbase.com

Miscellaneous pawnless endings

Other types of pawnless endings have been studied (Nunn 2002a). Of course, there are positions that are exceptions to these general rules stated below.

The fifty-move rule is not taken into account, and it would often be applicable in practice. When one side has two bishops, they are assumed to be on opposite colored squares, unless otherwise stated. When each side has one bishop, the result often depends on whether or not the bishops are on the same color, so their colors will always be stated.

Queens only

Queen versus queen: usually a draw, but the side to move first wins in 41.75% of the positions (Müller & Lamprecht 2001:400). There are some wins when one queen is in the corner, e.g. as a result of promoting a rook pawn or bishop pawn (Hooper 1970:17-19).
Two queens versus one queen: Almost always a win. A cross-check may be necessary, see cross-check#Two queens versus one for an example (Müller & Lamprecht 2001:400). A draw is possible in a few exceptional positions if the weaker side has an immediate perpetual check, e.g. with a White king on a1 and White queens on a2 and b1, the Black king on e8, and the Black queen giving check on d4. Black has as an unlimited supply of checks on d4, a4, and d1, and the White king cannot escape the corner.
Two queens versus two queens: The first to move wins in 83% of the positions (see the Comte vs. Le Roy diagram for an example). Wins require up to 44 moves (Nunn 2002a:329,379), (Stiller 1996:175).

Major pieces only

Queen versus two rooks: this is usually a draw, but either side may have a win (Nunn 2002a:311).
Queen versus three rooks: this is nearly always a win for the rooks
Queen and a rook versus a queen and a rook: Despite the equality of material, the player to move first wins in 83% of the positions (Stiller 1996:175).
Queen and rook versus a queen: this is a win (Nunn 2002a:317).
Two rooks versus a rook: this is usually a win because the attacking king can usually escape checks by the opposing rook (which is hard to judge in advance) (Nunn 2002a:320).
Rook versus rook: this is normally a draw, but a win is possible in some positions where one of the kings is in the corner or on the edge of the board and threatened with checkmate (Levenfish & Smyslov 1971:13).

A curious ending of two rooks against three rooks occurred in a game between Paul Lamford and Gile Andruet from a match Wales versus France in 1980. This proved an easy win for the three rooks. Andruet had earlier been forced to underpromote to a rook to avoid a stalemating defence for Lamford.

Queens and rooks with minor pieces

Queen versus a rook and a minor piece: this is usually a draw (Müller & Lamprecht 2001:402). The queen has good winning chances if the king and rook are near one edge and the minor piece is near the opposite edge. In the case of the knight, the queen can trap it on the edge; then the king assists in winning it. Against the bishop, the queen makes moves eventually forcing the bishop onto a square where it can be won (Mednis 1996:120-29).
Two rooks and a minor piece versus a queen: this is usually a win for the three pieces, but it can take more than fifty moves (Müller & Lamprecht 2001:406).
Queen and a minor piece versus a rook and minor piece: this is normally a win for the queen (Müller & Lamprecht 2001:403-4).
Rook and two minor pieces versus a queen: draw (Müller & Lamprecht 2001:405).

Queen and a minor piece versus two rooks: this is usually a draw for a knight and a win for a bishop, although the win takes up to eighty-five moves. The best method of defense is to double the rooks on the third rank with the opposing king on the other side and keep the king behind the rooks. This case with a bishop and queen versus rooks is unusual in that such a small material advantage forces a win. It was thought to be a draw by human analysis, but computer analysis revealed a long forced win (Müller & Lamprecht 2001:404), (Nunn 2002a:328-29,367,372).

Queens and minor pieces

Queen versus one minor piece: a win for the queen (Hooper 1970:4).
Queen versus three minor pieces: draw except for a queen versus three bishops all on the same color, which is a win for the queen (Nunn 2002a:328).
Four minor pieces versus a queen: a win for the pieces if they are the usual four minor pieces (see the position from Kling and Horowitz) (Fine & Benko 2003:583), (Horwitz & Kling 1986:207). Alexey Troitsky showed that four knights win against a queen (Roycroft 1972:209).
Queen and a minor piece versus a queen: this is usually a draw unless the stronger side can quickly win (see Nyazova vs. Levant and Spassky vs. Karpov) (Speelman 1981:108). With a knight, however, the stronger side has good winning chances in practice because the knight can create non-linear threats to fork the opponent's pieces and very accurate play is required from the defender to hold the position. There are 38 positions of reciprocal zugzwang and the longest win takes 35 moves until the knight forks the queen and king (Nunn 2002a:70-122).

Examples from games

An endgame with queen and knight versus queen is usually drawn, but there are some exceptions where one side can quickly win material. In the game between Nyazova and Levant, White won:

1. Qe6+ Kh4 (1...Kxh5? Qg6+ Kh4 Qh6+ skewers the black queen)

2. Qf6+ Kh3

3. Qc3+ Kg2

4. Qd2+ Kg1

5. Qe3+ Kg2

6. Nf4+ 1-0 (6...Kf1 7. Qe2+ Kg1 8. Qe1+ Kh2 9. Qf2+ Qg2 10. Qxg2#)

White could have won more quickly by 1. Qg8+ Kh4 2. Qg3+ Kxh5 3. Qg6+ Kh4 4. Qh6+ and White skewers the black queen (Speelman 1981:108).

The second position is from a 1982 game between former world champion Boris Spassky and then world champion Anatoly Karpov. The position is a theoretical draw but Karpov later blundered in time trouble and resigned on move 84.

Example from a study

In this 1967 study by Vitaly Halberstadt, White wins. The solution is 1. Be5+ Ka8 2. Qb5! (not 2. Qxf7?? stalemate.) Qa7+! 3. Ke2! Qb6! 4. Qd5+ Qb7 5. Qa5+ Qa7 6. Qb4! Qa6+ 7. Kd2! Qc8 8. Qa5+ Kb7 9. Qb5+ Ka8 10. Bd6! Qb7 11. Qe8+ Ka7 12. Bc5+ Ka6 13. Qa4# (Nunn 2002b:48,232).

Rooks and minor pieces

Two rooks versus two minor pieces: this is normally a win for the rooks (Müller & Lamprecht 2001:405). Henri Rinck discovered more than 100 positions that are exceptions (Roycroft 1972:203).
Two bishops and a knight versus a rook: this is usually a win for the three pieces but it takes up to sixty-eight moves (Müller & Lamprecht 2001:404). Howard Staunton analyzed a position of this type in 1847, and correctly concluded that the normal result of this ending is a win for the three minor pieces (Staunton 1848:439-40).

A bishop and two knights versus a rook: this is usually a draw, but there are some wins for the three pieces requiring up to forty-nine moves (Müller & Lamprecht 2001:403). Staunton in 1847 correctly concluded that the normal result of this endgame is a draw (Staunton 1848:439). Bernhard Horwitz and Josef Kling gave the same appraisal in 1851 (Horwitz & Kling 1986:142). During adjournment of the Karpov versus Kasparov game, Kasparov (initially unsure if it is a draw) analyzed that a successful defense is having the king near a corner that the bishop does not control, keeping the rook far away to prevent forks, and threatening to sacrifice it (for stalemate or for the bishop, which results in a draw, see two knights endgame). Tablebases show that it is usually a draw, no matter which corner the defending king is in (Kasparov 2010:303). (See the position from the Karpov versus Kasparov game for a drawn position, and see fifty-move rule for more discussion of this game.) Curiously, Grandmaster James Plaskett also had an adjournment of a London league game at the same time, versus David Okike; the last week of October 1991. After resumption it quickly resolved itself into the same pawnless ending. That game, too, was drawn.
Rook and a bishop versus two knights: this is usually a win for the rook and bishop but it takes up to 223 moves (Müller & Lamprecht 2001:404). The result of this endgame was unknown until computer analysis proved the forced win.
Rook and a knight versus two knights: this is usually a draw but there are some wins (for the rook and knight) that take up to 243 moves (Nunn 2002a:330).
Rook and a bishop versus a bishop and knight: this is usually a draw if the bishops are on the same color. It is usually a win (for the rook and bishop) if the bishops are on opposite colors, but takes up to ninety-eight moves (Müller & Lamprecht 2001:404).
Rook and a bishop versus two bishops: this is usually a draw, but there are some long wins if the defending bishops are on the same color (Müller & Lamprecht 2001:404).
Rook versus two minor pieces: this is normally a draw (Hooper 1970:4).
Two rooks versus three minor pieces: this is normally a draw (Hooper 1970:4).
Rook and two minor pieces versus a rook: a win for the three pieces (Hooper 1970:4). With two knights, White must not exchange rooks and avoid losing a knight, but the three pieces have great checkmating power (Roycroft 1972:195,203).

Minor pieces only

Two minor pieces versus one minor piece:
- Two bishops versus a knight: this is a win (except for a few trivial positions where Black can immediately force a draw), but it can take up to 66 moves. (Nunn 1995:267). See Effect of tablebases on endgame theory, fortress (chess)#A semi-fortress and see the example from the Botvinnik versus Tal game below.
- other cases: this is normally a draw in all other cases (Müller & Lamprecht 2001:402), (Hooper 1970:4). Edmar Mednis considered the difficulty of defending these positions:
  - Two bishops versus one bishop: The easiest for the defender to draw, unless the king is caught in a corner.
  - Two knights versus one bishop: any normal position is an easy draw.
  - Two knights versus one knight: an easy draw if the king is not trapped on the edge. However, if the king is trapped on the edge, there may be a win similar to the two knights versus a pawn endgame.
  - Bishop and knight versus a bishop on the same color: may be lost if the king is on the edge; otherwise an easy draw.
  - Bishop and knight versus a bishop on the opposite color: normally a draw but the defense may be difficult if the defending king is confined near a corner that the attacking bishop controls.
  - Bishop and knight versus a knight: best winning chances (other than two bishops versus knight). The difficulty of defense is not clear and the knight can be lost if it is separated from its king (Mednis 1996:36-40).
Three knights and a king can force checkmate against a lone king within 20 moves (unless the defending king can win one of the knights), but this can only happen if the attacking side has underpromoted a pawn to a knight. (Fine 1941:5-6).
Three minor pieces versus one minor piece: a win except in some unusual situations involving an underpromotion to a bishop on the same color as a player's existing bishop. More than fifty moves may be required to win (Müller & Lamprecht 2001:403,406). Three knights win against one knight (Dvoretsky 2011:283).
Two minor pieces:
- Two bishops is a basic checkmate
- A bishop and knight is a basic checkmate, see bishop and knight checkmate
- Two knights cannot force checkmate, see two knights endgame
Trivial cases: These are all trivial draws in general: bishop only, knight only, bishop versus knight, bishop versus bishop, knight versus knight.

Example from game

An ending with two bishops versus a knight occurred in the seventeenth game of the 1961 World Chess Championship match between Mikhail Botvinnik and Mikhail Tal. The position occurred after White captured a pawn on a6 on his 77th move, and White resigned on move 84.

77... Bf1+

78. Kb6 Kd6

79. Na5

White to move may draw in this position: 1. Nb7+ Kd5 2. Kc7 Bd2 3. Kb6 Bf4 4. Nd8 Be3+ 5. Kc7 (Hooper 1970:5). White gets his knight to b7 with his king next to it to form a long-term fortress.

79.... Bc5+

80. Kb7 Be2

81. Nb3 Be3

82. Na5 Kc5

83. Kc7 Bf4+

84. 0-1

The game might continue 84. Kd7 Kb6 85. Nb3 Be3, followed by ...Bd1 and ...Bd4 (Speelman 1981:109-10), for example 86. Kd6 Bd1 87. Na1 Bd4 88. Kd5 Bxa1 (Hooper 1970:5).

src: upload.wikimedia.org

Examples with an extra minor piece

An extra minor piece on one side with a queen versus queen endgame or rook versus rook endgame is normally a theoretical draw. An endgame with two minor pieces versus one is also drawn, except in the case of two bishops versus a knight. But a rook and two minor pieces versus a rook and one minor piece is different. In these two examples from games, the extra minor piece is enough to win.

In this position, if the bishops were on the same color, White might have a chance to exchange bishops and reach an easily drawn position. (Exchanging rooks would also result in a draw.) Black wins:

1... Re3

2. Bd4 Re2+

3. Kc1 Nb4

4. Bg7 Rc2+

5. Kd1 Be2+

6. resigns, because 6. Ke1 Nd3 is checkmate (Speelman 1981:108-9).

In this position, if White could exchange bishops (or rooks) he would reach a drawn position. However, Black has a winning attack:

1... Rb3+

2. Kh2 Bc6

3. Rb8 Rc3

4. Rb2 Kf5

5. Bg3 Be4

6. Re2 Bg5

7. Rb2 Be4

8. Rf2 Rc1

9. resigns, (Speelman 1981:109).

Speelman gave these conclusions:

Rook and two bishops versus rook and bishop - thought to be a win
Rook, bishop, and knight versus rook and bishop - good winning chances, probably a win if the bishops are on opposite colors
Rook, bishop, and knight versus rook and knight - thought to be a win (Speelman 1981:170).

src: upload.wikimedia.org

Summary

Grandmaster Ian Rogers summarized several of these endgames (Rogers 2010:37-39).

src: upload.wikimedia.org

Fine's rule

In his landmark 1941 book Basic Chess Endings, Reuben Fine inaccurately stated, "Without pawns one must be at least a Rook ahead in order to be able to mate. The only exceptions to this that hold in all cases are that the double exchange wins and that a Queen cannot successfully defend against four minor pieces." (Fine 1941:572) Kenneth Harkness also stated this "rule" (Harkness 1967:49). Fine also stated "There is a basic rule that in endings without pawns one must be at least a rook ahead to be able to win in general." (Fine 1941:553) This inaccurate statement was repeated in the 2003 edition revised by Grandmaster Pal Benko (Fine & Benko 2003:585). However, Fine recognized elsewhere in his book that a queen wins against a rook (Fine 1941:561) and that a queen normally beats a knight and a bishop (with the exception of one drawing fortress) (Fine 1941:570-71). The advantage of a rook corresponds to a five-point material advantage using the traditional relative value of the pieces (pawn=1, knight=3, bishop=3, rook=5, queen=9). It turns out that there are several more exceptions, but they are endgames that rarely occur in actual games. Fine's statement has been superseded by computer analysis (Howell 1997:136).

A four-point material advantage is often enough to win in some endings without pawns. For example, a queen wins versus a rook (as mentioned above, but 31 moves may be required); as well as when there is matching additional material on both sides, i.e.: a queen and any minor piece versus a rook and any minor piece; a queen and a rook versus two rooks; and two queens versus a queen and a rook. Another type of win with a four-point material advantage is the double exchange - two rooks versus any two minor pieces. There are some other endgames with four-point material differences that are generally long theoretical wins. In practice, the fifty-move rule comes into play because more than fifty moves are often required to either checkmate or reduce the endgame to a simpler case: two bishops and a knight versus a rook (requires up to 68 moves); and two rooks and a minor piece versus a queen (requires up to 82 moves for the bishop, 101 moves for the knight).

A three-point material advantage can also result in a forced win, in some cases. For instance, some of the cases of a queen versus two minor piece are such positions (as mentioned above). In addition, the four minor pieces win against a queen. Two bishops win against a knight, but it takes up to 66 moves if a bishop is initially trapped in a corner (Nunn 1995:265ff).

There are some long general theoretical wins with only a two- or three-point material advantage but the fifty-move rule usually comes into play because of the number of moves required: two bishops versus a knight (66 moves); a queen and bishop versus two rooks (two-point material advantage, can require 84 moves); a rook and bishop versus a bishop on the opposite color and a knight (a two-point material advantage, requires up to 98 moves); and a rook and bishop versus two knights (two-point material advantage, but it requires up to 222 moves) (Müller & Lamprecht 2001:400-6) (Nunn 2002a:325-29).

Finally, there are some other unusual exceptions to Fine's rule involving underpromotions. Some of these are (1) a queen wins against three bishops of the same color (no difference in material points), up to 51 moves are required; (2) a rook and knight win against two bishops on the same color (two point difference), up to 140 moves are needed; and (3) three bishops (two on the same color) win against a rook (four point difference), requiring up to 69 moves, and (4) four knights win against a queen (85 moves). This was proved by computer in 2005 and was the first ending with seven pieces that was completely solved. (See endgame tablebase.)

src: lebanesechess.com

General remarks on these endings

Many of these endings are listed as a win in a certain number of moves. That assumes perfect play by both sides, which is rarely achieved if the number of moves is large. Also, finding the right moves may be exceedingly difficult for one or both sides. When a forced win is more than fifty moves long, some positions can be won within the fifty move limit (for a draw claim) and others cannot. Also, generally all of the combinations of pieces that are usually a theoretical draw have some non-trivial positions that are a win for one side. Similarly, combinations that are generally a win for one side often have non-trivial positions which result in draws.

Tables

This a table listing several pawnless endings, the number of moves in the longest win, and the winning percentage for the first player. The winning percentage can be misleading - it is the percentage of wins out of all possible positions, even if a piece can immediately be captured or won by a skewer, pin, or fork. The largest number of moves to a win is the number of moves until either checkmate or transformation to a simpler position due to winning a piece. Also, the fifty-move rule is not taken into account (Speelman, Tisdall & Wade 1993:7-8).

This table shows six-piece endgames (Stiller 1996).

This table shows seven-piece endgames (Lomonosov tablebases).

Note many of the combinations listed involves single piece capture and transposition into six-piece territory almost immediately from the starting position.

Pawnless chess endgame

Saturday, April 14, 2018