Chess with Chinese Pieces
Daniel W. VanArsdale, 2004, 8/2023

" . . . its only a Rock group that split up, its nothing important."   John Lennon

Hundreds of different "Fairy" chess pieces have been used in chess problems over the last one hundred years. And so many variant forms of chess have been announced in recent years that it is hard to think of an appropriate name that has not already been used. I do not introduce any new fairy pieces or chess variants here. However I do propose that:
(1)  Special attention be given to the use of the Cannon, Vao and Leo in chess compositions.
(2) A new version of chess be developed which (i) allows some "free placement" of units, (ii) abandons castling
      and double pawn moves, and (iii) incorporates the Cannon, and Vao into the game.
The Cannon (Pao), Vao and Leo (Chinese line pieces) are fairy chess pieces that move exactly like the orthodox Rook, Bishop and Queen respectively, but to capture must hop over one piece of either color. Instructional examples and composed problems are presented below to demonstrate their tactical potential. I have tried to make these problems entertaining in themselves. Four criticisms of orthodox (European) chess are discussed, and it is argued that a free placement version of chess could overcome these criticisms and eventually sustain more interest than orthodox chess.

(8/7/2023)  In the almost 20 years since Chess with Chinese Pieces was first uploaded chess playing software has come to dominate the best human players. Merely adding new pieces would soon yield to renewed cyber domination. Perhaps going to a 10x10 board, adding 2 vaos and 2 cannons, and initiating some expansive form of free placement would temporarily challenge cyber domination and provide a better standard venue for problems. Great initial variety would result if player A can choose placement rules, within some boundaries, and player B can choose to place first or second.   

As for chess problems, it is noteworthy that a century before computers a prominent aesthetic for problems could be described as having a key that looks like a weak move, or a move that if a program disregarded any moves at all, the solution to the problem would be one of them. Note also that best move programs are very useful to problem composers by checking for cooks or unseen defenses. In a recent television documentary on the rise of AI, it was asserted that the recent AI defeat of the best human Go player greatly stimulated new interest in AI, particularly in East Asia. Perhaps humans and AI will "cooperate" for a sustained liberation of human creativity that promotes both the flourishing of our civilization and the ever expanding utility of AI.

CONTENTS:
     1. The Chinese Pieces: Cannon, Vao, Leo and Mao
    2. Problems and Endgame Studies with Chinese Line Pieces
    3. Critique of Orthodox Chess.
    4. Free Placement Chess.
    5. Contact, Links and References.

DIAGRAMS:
    INSTRUCTIONAL EXAMPLES:  1. The Cannon, Vao, Leo and Mao  2. Triple check

    PROBLEMS AND ENDGAME STUDIES: 
    3. Mate in three   4. Mate in three  5. White draws  6. White wins  7. Mate in three    8. Helpmate in two
    9. Series mate in four (2)  10. Mate in three (3)  11. Mate in two  12. Mate in two    13. Mate in three

    FREE PLACEMENT CHESS: A & B.  Burmese Pawn Placements.

1. The Chinese Pieces: Cannon, Vao, Leo and Mao.

I assume that the reader is familiar with the rules of orthodox chess and algebraic notation for describing moves. [Traditional English notation is also provided for solutions to problems.] The diagrams below use the convention of displaying the Chinese pieces by the symbols of the corresponding orthodox pieces rotated counterclockwise by 90 degrees. The following abbreviations are used: Cannon or Pao = C, Vao =V, Leo = L, Knight = S, check = +, checkmate = #.

The Cannon moves like an orthodox rook along a rank or file, but to capture must hop over one piece of either color (the "frame") and continue on the line beyond until encountering an opposing piece. This piece is then removed from the board and the Cannon occupies its square.

The Vao moves like a Bishop, but to capture must (like the Cannon) hop over a piece of either color.

The Leo combines the movement and capturing of the Cannon and Vao, just as the orthodox Queen combines the powers of the Rook and Bishop.

The Mao moves to and captures on squares a Knight's move away, but accomplishes this in two steps: (i) a one-step orthogonal move to an unoccupied square, and (ii) a one-step move on an outward diagonal from that square. Its move is not sufficiently distinct from that of the Knight to justify their both being present in a game. Further, the unblockable move of the orthodox Knight is to be preferred over the Mao, since it distinguishes the Knight in a useful and pleasing way from line pieces.

DIAGRAM 1.

The following lists all possible moves and captures by the Chinese Pieces in Diagram 1.
White Cannon a1. Moves to a2 through a8. Captures on c1.
White Vao b1. Moves to a2, d3. Captures on h7. It can not move to c2 because this would put the white King in check.
Black Cannon c1. Moves to c2, c3, c4, d1. Captures on a1.
Black Mao e1. Moves to c2+, d3+. Captures on f3. It can not move to g2 because it is blocked on f1.
Black Leo f1. Moves to e2, d3, c4+, b5, a6, f2, g1, g2, h3. Captures on f7.

I have designed this example so that it is a simple problem as well. In a "helpmate" Black cooperates with White to produce a checkmate of the black King as soon as possible. So, from a game perspective, Black makes the worst possible moves. The convention is for Black to play first in a helpmate. So for this position, can you "helpmate in one move"? This means Black makes a certain move that allows White to checkmate on the next move.

The "Cannon" was introduced into Chinese Chess (Xiangqi) in the ninth century after artillery had been used in wars (Li, p. 222). The "Chariot" in Xiangqi moves and captures like an orthodox Rook, and the Cannon also moves like a Rook when "deployed." But to capture (fire), the Cannon must employ a "frame" (a piece of either color), and the action then extends on the line beyond the frame. The Mao (Horse) was also present in early versions of Xiangqi. 

Apparently neither the Vao (called an "Arrow" by Fergus Duniho) nor the Leo were ever used in a chess variant, Asian or otherwise, until recently. The Chinese pieces were introduced into Western "Fairy Chess" problems by T. R. Dawson around 1914 (Dickins). In problem literature a Cannon is usually referred to as a "Pao," which is Dawson's rendering of the Chinese word for Cannon, and he made up the words "Vao" and "Leo." I prefer "Cannon" since the terms "Pao, Vao, Leo, Mao" are singsong and easy to confuse, and "Pao" starts with a "P" as does "Pawn."

If new pieces are to be added to orthodox chess, the Chinese line pieces (Cannon, Vao, Leo) are a logical choice. In earlier experiments I thought the "Lion" was preferable. This piece both moves and captures as the Leo captures. The Lion proved to be a terror in the opening but a tethered lamb in an endgame. By contrast the Leo is always a mobile piece, though much weaker than an orthodox Queen. The "Grasshopper" moves and captures like the Lion except only to the one square immediately beyond the frame piece. Thus it is a much weaker piece than even the Lion. It has been very popular in fairy chess problems, usually appearing in considerable numbers. This may be due more to fashion than function, but in any case it is too immobile to add much interest as a game piece. The "Nightrider" may make repeated Knight moves along a single direction. It too has been very popular in problems, but children have enough trouble learning how a Knight moves to ask them to master the Nightrider. Few other powerful fairy pieces seem suitable for use in a "near-orthodox" game, perhaps with the exception of the "Aldarider" (moves like a Queen except that it hops over every other square, occupied or not). Surely the Chinese line pieces are leading candidates to be added to orthodox chess, and to learn one is to learn all three. They do not capture as they move, but this is not a total breach of orthodoxy since the trait is shared by the Pawn. The apocryphal text, Dodgson's "Frogs Manuscript" Decoded, suggests Lewis Carroll planned to introduce Chinese line pieces into chess.

The Chinese line pieces (CVL = Cannon, Vao, Leo) have the following characteristics:

DIAGRAM 2.
With both a Cannon and orthodox Pawns on the board the oddity of a triple check (on three different lines) becomes possible. With an en passant capture the Cannon can check on the rank of the capturing pawn. In Diagram 2 Black's last move can be deduced ("retrograde analysis"). It could not have been with the King. The only possibility would be Kc6 - b6, but on c6 there is a double check by the Pawn on b5 and Vao on h1 that White could not have produced with one move. Likewise Black could not have played Pawn c6 - c5 since there would have been an impossible double check by the Bishop on d4 and Cannon on d6. Thus the position must have arisen by White checking with the Bishop on d4 and Black replying c7 - c5. White can now play b5 x c6, en passant, which produces a triple check and mate. With free placement we will argue that double pawn moves (and hence en passant capture) should be abandoned.

With two Maos, a Bishop and a Rook, a quadruple check can be easily constructed without requiring en passant capture.
 








2. Problems and Endgame Studies with Chinese Line Pieces

I will assume the reader is not familiar with chess problems and explain some basic terminology. It is conventional for orthodox chess compositions that they could have arisen by legal moves in a game, preferably without any promotions. This restricts the number of each type of piece used, thus allowing a solver to set up the problem with a conventional chess set. It may be useful to define additional orthodoxies to classify chess compositions. Define "CVL" (Cannon-Vao-Leo) rules as follows:
A. Legal Initial Positions
(1) Each side has one King and any number of Pawns, Rooks, Knights, Bishops, Queens, Cannons, Vaos, or Leos, as long as all the pieces fit on the board.
(2) All positions are allowed unless they imply a check has been ignored. Thus Pawns may be on the first or last rank. Multiple Pawns can be lined up on a single file without implying a capture. It is not required that the initial position be derivable from some standard position by a sequence of legal moves.
B. Legal Moves
(1) All the moves of orthodox chess and Chinese line pieces are allowed except for double Pawns moves (and hence en passant capture) and castling.
(2) A Pawn can "promote" to an immobile unpromotable Pawn.
C. Options
(1) Orthodox Pawns are stipulated.
(2) The only fairy pieces that are present are Cannons.
(3) The board is 10x10.
For all the compositions below one can proceed as if they were just orthodox positions with Chinese pieces added. Thus any Pawns present on the second rank can move two squares forward (Option #1). Allowing Pawns to appear on the first or last rank would be a useful convention for composing. A case can be made that promotion to a dummy Pawn should be allowed in orthodox chess. This would be a slight simplification of the rules, and would probably not effect the outcome of
any game ever played. But for chess compositions, allowing pawns on the first or eighth rank would facilitate construction of chess problems and tasks. For a remarkable problem composed by Sam Lloyd utilizing this option see EMPLOY THE DEAD! - Sam Loyd's Dummy Promotion.
 
If free placement ever gets established castling should be abandoned since at the very start of play a King can be at the side of the board, and Rooks can be connected. Likewise double Pawn moves should be abandoned since a Pawn can start play near the center of the board. Nor is there any reason to prohibit them from occupying the first rank. Also, free placement may as well allow one to place Bishops or Vaos on the same color squares. Granted, generally this is likely a poor choice if there are only two such pieces for a player. But rules should be as simple as possible as long as they do not modify the spirit of the play. It should not be the function of rules to guide one away from errors.

In a game, adding to each player's force four or five Chinese line pieces creates a crowded and complex playing field. Experiments suggest the use of a 10x10 board would improve the game. As noted above, a Chinese line piece pins two pieces at once on a line. In attempting compositions expressing pinning and "half pinning" themes one often runs out of space. Thus a 10x10 board would expand possible compositional themes. The resulting increase in legal moves would also greatly burden computerized "brute force" evaluations of all possible moves in a game.

As mentioned above, of the three line pieces (C,V,L), Chinese Chess employs only the Cannon. To better accommodate an 8x8 board, and to lessen deviation from the orthodox Chinese game, perhaps only two Cannons should be added in a free placement alternative to European chess (option 2 above). There is less reason for this restriction in problems.

The purpose of the compositions here is to illustrate tactical capabilities of the Chinese line pieces. No game-like compositions are presented below, this despite the fact that the Chinese line pieces are more effective in the opening and middle game than in positions with few pieces. This is because of their power to penetrate, and because with many pieces on the board there is no scarcity of a frame to enable a capture. Problem 13, added in 2017, illustrates some unusual tactical possibilities of the cannon. Problems have been composed using Chinese pieces for over 80 years. Some recent ones can be found on-line (PDB Server, Phénix 1999). I do not have access to the The Fairy Chess Review (1936 - 1958), but judging from the published problems I have examined likely most of the settings employed here have not been anticipated.

I have used Alybadix, a chess problem solving program designed by Ilkka Blom, to check direct mate, helpmate and series problems below. "Fairybadix" is the component of Alybadix that accommodates fairy pieces (over 220) and fairy conditions. The Windows interface for Alybadix (APwin, developed by P. H. Wiereyn) was used to produce the diagrams. The endgame studies were not checked by computer - please report any errors. Links to the solution of each problem appear next to the diagrams. The diagram is repeated on its solution page. You may use any of the text and diagrams. Please retain credits.

DIAGRAM 3.


 

Diagram 3 presents a "direct mate" problem: The moves alternate with White playing first. White must find the move on his first turn (the "key") that results in a checkmate in the stipulated number of moves, regardless of the defenses chosen by black. So for this problem White must checkmate on his third move. An additional, unintended first move by White which also solves the problem is called a "cook." With the availability of problem solving computer programs, direct mate problems are rarely unsound now. A common way to designate that a diagram represents a direct mate in n moves is by the symbol "#n".
 
  SOLUTION
 
 
 
 
 
 
 
 
  

DIAGRAM 4.

 
Another direct mate in three moves.

 
  SOLUTION
 
 
 
 
 
 
 


 
 


DIAGRAM 5.

 

The usual specifications for an endgame study are for white to win or to draw, but not in a fixed number of moves. To "win" means White obtains a clear material advantage, perhaps checkmates in some lines, or arrives at a position known in the literature to be a win. The specification to draw means that there are sufficient exchanges that neither side can force a win, perhaps White or Black is stalemated in some lines, or a position can be forced to recur. The use of fairy pieces in endgame studies is much less frequent than their use in direct mate and helpmate problems.
 
SOLUTION
 
 
 
 




DIAGRAM 6.


 

 
 
 
 
Diagram 6 illustrates the use of a Chinese piece, here the white Vao on c2, to modify an existing composition. The pawns here are orthodox (option 1 above). Thus the black pawn on h7 can move to h5. The original study by Holzhausen and Sohege appears on the solution link.
In memory of Kate Wolf.
 
SOLUTION
 
 




 

DIAGRAM 7. 





     Mate in three.



   SOLUTION
DIAGRAM 8.

 
   

Many published problems with fairy pieces are "helpmates," in which Black and White cooperate to achieve a mating move by White. It is conventional for Black to move first, and for Black moves to be listed first in transcribing a solution. Thus in Diagram 8, Black moves first, the players alternate moving, and White's second move is a checkmate. Since there is no variational play in a helpmate, it has become conventional in two move helpmates for there to be two or more thematically related solutions.
 
SOLUTION
 
 
 
 
 
 
   

DIAGRAM 9.


 
In a "series mate" only white makes moves, without any checking until the final move which must be a checkmate. Here Problem 1 is the diagram. For Problem 2 replace the Cannon with a Rook. The mating position in Problem 2 is more elegant and may have been anticipated.
 
SOLUTION
 







DIAGRAM 10.
 

Here, as in many problems, the unsuccessful tries by White may be of interest. Thus for each of the three direct mate problems here, the solver may wish to examine why various choices for promotion do not succeed. These tries, and Black's responses, are presented on the solution page. It is conventional in transcribing tries to use an exclamation point after a Black move when it is the only Black move that defeats the try. If you only have time for one of these, make it the last (without a black pawn).

SOLUTION
DIAGRAM 11.



White checkmates in two moves.


SOLUTION
DIAGRAM 12.





This mate in two moves is an extension of the "Organ Pipe" theme, first presented by Sam Loyd in 1857.  See No. 452 in Sam Loyd and his Chess Problems by Alain C. White. This problem was published in The Problemist (March, 2006), a publication of The British Chess Problem Society.


SOLUTION
DIAGRAM 13.

Mate in three.  Cannons (paos) on a4, g1, h1.
Daniel W. VanArsdale,  8/2017.



       Solution

                            
3. Critique of Orthodox Chess

Orthodox chess is vulnerable to criticism on the following points.

Memorization of openings is too great a requirement.
To compete at the expert level and above in orthodox chess requires much study of opening analysis, whatever one's talent and creativity may be. Understandably, calls for reform encounter the hostility of many who have invested thousands of hours to this task. Memorization of some standard endgame procedures is also useful, but the effort required for this is much less than that for openings, and the settings are far more universal.

The rise of chess playing computer programs makes "memorization" a serious threat to orthodox chess as a game of wits. In the humiliating 1997 match victory of "Deep Blue" over then world champion Gary Kasparov, the huge opening repertoire of the computer was decisive (Chess Life, p. 45). Granted the best tactical chess players often excel at opening theory, and innovation in the opening demands a high order of creativity. But such praise can not apply to Deep Blue, which won the sixth and final game of the match by the automated retrieval of an opening variation. Does it speak well of chess that this capability is such an important factor in determining who wins?

Computers will Dominate.
A recent tabulation lists 15 chess programs having a rating of over 2600. Shredder 7.04 was highest at 2726. There are only about 100 players in the world with a rating of over 2600.  So already the typical club player has virtually no chance against many chess playing programs. One source claims that the top 200 players in the world are holding their own against the robots in the last three years. Perhaps, but just another doubling of chip speed could reverse that. Computer domination of competitive chess is already a fact. And think of the temptation to cheat in human tournaments by using a concealed or remote computer.

There are too many draws at high levels of play.
Whatever game is being played, it is difficult to prevent the combatants from prematurely agreeing to a draw. But even without connivance, chess at the highest levels is subject to excessive draws. In the 23 world championship matches from 1951 to 2000 the draws numbered 321 out of 492 games (link), or 62 percent. In the 1986 match between Karpov and Kasparov 40 of the 48 games were drawn.

Certain rules are unnecessary.
It is reasonable to judge that one set of rules is more "simple" than another if it employs fewer words. Another criterion, more important but more difficult to measure, would be how easy it is to learn a set of rules. It is appealing to play a game under the simplest possible rules that enable typical play.

The most arbitrary and difficult to learn rules of orthodox chess describe (1) the initial placement of the pieces, (2) castling, (3) en passant capture. Of course an adult masters these rules after a few hours of study and playing. But they are not so easy for children, who if unsupervised may play for years and still place a Queen on the wrong color, or not know about en passant capture. All three of these rules are unnecessary if free placement is implemented.

Orthodox chess, with its fixed initial position, is unlikely to be independently invented, as by an extraterrestrial civilization. By contrast, the Asian game Weiqi (Go) has such simple rules that there may be many alien civilizations that play the same game. Further support for this conjecture derives from the Weiqi symbolism of flood control (Li, p. 140).


4. Free Placement.

Any chess variant good enough to be played extensively, and possibly even rival orthodox chess, would probably have to be the invention of several people, or even a community of players.

In H. J. R. Murray's A History of Chess (1913) we discover that in Burma (Myanmar) the pawns were lined up asymmetrically on advanced ranks (Diagrams A & B) and the remaining pieces placed freely behind them. A player could even replace one of the pawns on the front line with a major piece, placing the dispossessed pawn anywhere behind the pawn line. In India the pieces were set up in a fixed position, but then White, and next Black, made several moves in a row, restricted to their side of the board (Murray, p. 83). In an Abyssinian version of chess (1868) an indefinite number of preliminary adjustments were allowed, alternate moves only beginning after one side captured a pawn (Murray, p. 364).

              DIAGRAM A.                                                              DIAGRAM B.
             

Over the years there have been many suggestions to allow shuffling the positions of the major pieces in orthodox chess behind the line of Pawns on the second rank. One such proposal, involving randomization, was made by former world chess champion Bobby Fischer (Chess Life, p. 531). In this proposal castling is awkwardly maintained, as if it were an essential feature of chess. But if we allow all the pieces, including Pawns, to be placed freely within some "home territory" (such as the first three ranks), we at once make castling unnecessary since the King can be placed in a secured location on the side of the board. Free placement also removes the motivation for permitting a double first move by Pawns since they can start out on the third (or fourth) rank. Pawns might also be placed on the first rank for defensive purposes.

In recent years variants have been proposed that permit some freedom of placement. In "Free Programme Chess" the units are placed in turn anywhere in one's half of the board. The Kings are placed first. Pawns may be placed freely but can not be doubled or placed on the first rank. They retain the double move from the second rank. Bishops must be placed on opposite colors (Pritchard, p. 20). A master level tournament was held using these rules in Tbilisi, Georgia in 1995. Some free placement chess variants can be found on The Chess Variant Pages (for example, the variant called "Free Placement"). We suspect that the best game will result from free placement with restrictions. Just deciding what home territory should be is not easy. Murray (p. 454) claimed that early experiments "in which the pieces were rearranged so as to be more nearly in contact at the commencement of play," did not survive because with the advent of a much more powerful Queen and Bishop the forces were too close together. But if so, the forces can simply be moved apart one or more spaces, alternately employing straight pawn lines or an asymmetric Burmese arrangement (Diagrams A or B above).

If one thinks the war analogy of chess is useful, free placement is more realistic and better training than starting play in one fixed arrangement as in orthodox chess. Recent wars involve considerable positioning of forces and supplies long before hostilities commence. And once they do, battle lines no longer conform to traditional arrangements of foot soldiers, cavalry, elephants, artillery, commanders, etc. as in centuries past.

Free placement provides the opportunity to introduce new pieces in the palette of Western chess. It has been proposed that the Chinese line pieces are the best choice. With an increased force it is reasonable to expand the playing field, thus the use of a 10 x10 board would probably make the best game if, say, two Cannons, two Vaos and a Leo were added to each player's orthodox pieces. Murray stated that "enlarged games of chess have rarely shown any vitality," because they were too taxing to be recreational (p. 454). But a larger board may decrease complications that arise in a cluttered position. And the 12 x 8 "Courier Game" was popular from 1200 to after 1650, about the duration of orthodox chess so far.

Free Placement ameliorates each of the four problems with orthodox chess listed above. With thousands of viable starting positions the possibility of defeating an opponent by using a previously analyzed opening is greatly lessened. The placement phase should also make matters much more difficult for computer programs.  Ultimately the computers would excel in this strategic effort also, but to do so will require a different and more interesting programming method than the current brute force evaluation of all possible moves. And during play, the use of a 10x10 board and additional pieces will increase the number of possible moves. If the number on each turn were increased by just 50%, after 12 "plies" (six moves) the number of variations to test increases by over 100 times. That should slow down the silicon competitors. There should be fewer draws also. An agreement to draw can be prohibited during placement, and the greater freedom of arrangement and loss of familiar guideposts should engender aggressive schemes and blunders. Also the penetrating power of the Chinese line pieces should make blocked positions more difficult to maintain. For example, the sacrifice of a Vao for two Pawns should be a frequent consideration since the Vao is a less valuable piece than either a Knight or Bishop. Finally, free placement can eliminate the most arbitrary rules of orthodox chess: the initial position, castling, double pawn moves and en passant capture. If we also allow promotion to a Pawn, only then will all the following simple descriptions apply to chess.

(1) The empty board is homogeneous. Only the edge of the board, and for Pawns their direction of movement, effect the power of a  piece. With free placement and in the course of a game, any piece may occupy any square.
(2) Only one piece moves at a time.
(3) A capturing piece always occupies the square of the piece captured.
(4) All positions are non-historical. The possible moves and captures in a position are determined solely by the location of the pieces on the board and who has the move, regardless of the particular sequence of moves that could have produced the position.
Elimination of en passant capture and castling excludes many problems that involve deducing what the last move was. But castling is an absurd option in the context of free placement of the major pieces. Determining the last move could still be a problem.

The concept of "free placement" can be carried a step beyond alternate placement to form the starting position to a liberation of the players to start the game however they may agree to. This could include an agreement on whether to use an 8x8 or 10x10 board, and which, if any, Chinese line pieces to employ. Agreement on how to place the pieces could produce the following options. Of course if there is no agreement between the players some default rules for placement must be in place. These should be simple, expedite placement, and produce a great variety of opening positions that are not so complex as to overwhelm players with early tactical dangers. Probably some combination of fixed positions (as for Pawns) and free placement (for major pieces) may work best. It may take much experimentation to find the right placements rules: here we offer only the following broad suggestions.

5. Contact, Links and References.

Comments, corrections or suggestions are welcomed.

Email: Daniel W. VanArsdale

REFERENCES AND LINKS

Alybadix. http://alybadix.bl.ee/
Chess Life, Special Summer Issue 1997, Vol. 52, No. 7.
The Chess Variants Pages. http://www.chessvariants.com/
Dickins, Anthony. A Guide to Fairy Chess. Dover Publications, New York. 1971.
Dodgson's "Frogs Manuscript" Decoded
Duniho, Fergus. Eurasian Chess. http://www.chessvariants.com/large.dir/eurasian.html
Li, David H. The Genealogy of Chess. Premier Publishing. 1998.
Murray, H. J. R. A History of Chess. Oxford University Press, Oxbow Books reprint. 1913.
Pritchard, D. B. Popular Chess Variants. B. T. Batsford Ltd, London. 2000.
Employ the Dead! - Sam Loyd's Dummy Promotion
White, Alain C. Sam Loyd and His Chess Problems. Dover  Publications. 1962.


 
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