80-Square Chess 81 80-SQUARE CHESS Ed Trice1 Jamison, Pennsylvania, USA ABSTRACT Chess has evolved over a span of about 2300 years. The game was not always the “packaged game” that it is today. Its origins are traceable to Indian ashtapada boards commonly used among dice- playing games of that time. This game spread from culture to culture via several different means, periodically undergoing revision. Most chess players are aware of the radical reforms that shaped the game of chess during the Medieval Era, but probably only a few players are aware of the changes proposed by former World Champion José Raoul Capablanca in the 1920s. This paper focuses on the 80-square version of chess recommended by Capablanca, identifies some of the anomalies associated with Capablanca’s chess, and proposes a slightly modified version of his setup as a plausible enhancement, called Gothic chess. New values for the pieces on the 10 × 8 board are derived. Subsequently, five Gothic-chess computer programs are introduced. Finally conclusions and perspectives are given. 1. INTRODUCTION: A VERY BRIEF HISTORY OF CHESS Chess literature often ascribes the game’s origin to a man named Sissa, a Brahman Indian in the court of Rajah Balhait (Lasker, 1959). Sissa called the game chaturanga meaning “army composed of four members”. When Alexander the Great invaded India in 326 B.C., the Indian Army featured the same four components (Liddell, 1937) that had already appeared in the game of chaturanga, viz. chariots, foot soldiers, horses, and elephants. These early incarnations of the game (with two-player and four-player variations, each with or without dice) bore little resemblance to the 64-square board of recent times. It was in the Middle Ages, believed during the 15th century, that the rules of chess started to resemble the present configuration. Eventually the castling rule was added to help protect the King from the pieces that were given more power (the Queen and Bishop); the en passant rule entered the game as an option to evoke a special form of a pawn capture (Murray, 1913). The game of chess has not evolved since, but in the 1920s, World Champion José Raoul Capablanca was seriously considering altering the game. His proposed changes were aimed at making the contests more lively, cutting the average game length in half, and drastically reducing the high frequency of draws that were growing more commonplace among tournament players of the highest caliber (cf. Pritchard, 1994). 1.1 The Course of the Article Chess, by definition, is a variant of the game of chaturanga. Many cultures have their own preferred board game, which can trace its origins back to either chataranga or one of its very early offshoots. In essence, what becomes ‘the variant’ or what remains the mainstream game over the course of time is highly subjective and varies from culture to culture. In recent times, we have seen additional interest in at least asking the question: “Is the current configuration of the chessboard the one that results in the most satisfying game?” (Van Haeringen and Van den Herik, 2003). Throughout the course of this article we look into the mind of the great José Capablanca and reveal his quest to make the game more entertaining. Fifty years before Capablanca, we find Henry Bird on a similar trek to identify an acceptable variation with 80 squares. We note in passing that Pietro Carrera, in 1617, also sought an 80-square implementation, but for the sake of brevity we only examine the more contemporary variants. 1.2 Why Does Chess Evolve? It is important to reflect on what drove the changes that shaped the game of chess, viz. the desire to make the play more appealing. Older versions of the game contained pieces with much less power. For example, our present-day Queen, originally designated “the counselor” (Firz in Arabic), was weaker than the Knight. The 1 2190 Sunrise Way, Jamison, PA 18929. E-mail: GothicChessInfo@aol.com. http://www.GothicChess.Org. 82 ICGA Journal June 2004 Firz was only permitted to make one diagonal step at a time (Lasker, 1959, p. 30). Bishops were permitted to make two such diagonal steps on any given move. It is easy for us to see that the recalcitrant nature of such a game would produce longer contests wherein a great many resulted in draws. It required the “restless spirit of Europe” (Lasker, 1959, p. 30) to infuse new life into the game. According to Lasker, by the end of the 11th century, the earlier form of chess had spread throughout all of Europe. Within the next 400 years, the game had evolved almost completely into the game as it is played today. The rapid adoption of the rule changes (in just 400 years the European community had revamped and improved the game more than the previous 1400 years of combined efforts) that swept through Europe surely was a sign that players preferred battles with attacking flare that ended with one side emerging victorious. When you consider how slow communication was in Medieval times, and the arduousness of travel and life in general, the fact that chess evolved with a universal consensus is a minor miracle. This was accomplished without mass-produced chess sets, advertising campaigns, email, and telephones. It is clear that chess evolved in order to give the game greater complexity and mystery, to help foster the creation of interesting games that require fewer moves before an inevitable concession was reached, and to decrease the frequency of observed draws. 2. CAPABLANCA’S 80-SQUARE BOARD Former World Champion José Raoul Capablanca lost only 26 games of chess in his tournament career (not counting match games) spanning 29 years of active play (Reinfeld, 1942). His table of tournament records contains 267 wins, 26 losses, and 178 draws, in a total of 471 games. (Please note that Capablanca had four losses in a single tournament, the AVRO 1938 tournament, in which he suffered a stroke! Clearly his grave physical condition near the end of his life affected him in that event.) Even with this incredible performance record, Capablanca still felt that there were too many draws occurring at the highest level of tournament and match play. Chess was suffering from its own popularity. Many games were published and annotated at great length, allowing master play to “trickle down” and be imitated by players of lesser strength. Those who were already master-caliber players became even more informed regarding the latest issues in opening theory, middle-game strategies, and endgame tactics. The result was that the drawing frequency among the chess elite was sharply on the rise. Capablanca experimented with many different variants of chess, and some of them were wild and obtuse. From April 22 to 24, 1929, Geza Maroczy played Capablanca a two-game match on a board 16 columns wide by 12 rows in height. This game featured two complete sets of pieces sitting side-by-side horizontally. There were, therefore, two Kings per each side, both of which had to be checkmated! On this strange taller board, Pawns could leap up to 4 squares on their first move. Capablanca won the first game in 94 moves, and drew the second game in 82 moves (Winter 1989, pp. 184-185). It should be noted that this larger board made the game last longer since it was much harder to win. Players generally prefer a quick game (in terms of number of moves) that has a low occurrence of draws. So, the result of these long games did nothing to convince anyone that this 192 square board should be taken as a serious contender to replace the 64-square setup. So what setup did Capablanca prefer? We have the answer from Lasker (1959, pp. 38-39). He chose an 80-square board where he added two new piece types, called the Chancellor and Archbishop. “The pieces he [José Capablanca] added were both about as strong as a Queen. As counterparts of the latter, which combined the powers of Rook and Bishop, he had a Chancellor, moving like a Rook or a Knight, and an Archbishop, moving like a Bishop or a Knight.… Capablanca placed the Chancellor between the Bishop and Knight on the King’s wing, and the Archbishop on the corresponding square on the Queen’s wing, and, of course, added a Pawn in front of each….I played many a test game with Capablanca, and they rarely lasted more than twenty or twenty-five moves. We tried boards of 10x10 squares and 10x8 squares, and we concluded that the latter was preferable because hand-to-hand fights start earlier on it.”(Lasker, 1959, pp. 38-39). Capablanca’s preferred configuration of the chessboard is shown in Figure 1. A similar 80-square arrangement advocated by the English master Henry Bird fifty years earlier is shown in Figure 2. 80-Square Chess 83 Figure 1: Capablanca’s board, circa 1925. Figure 2: Bird’s board, circa 1875. 2.1 Undesired Features of the 80-Square Board After a brief inspection of the two boards, one might conclude that Bird’s setup appears to be more logically motivated. For example, Bird has his Bishops at the same relative distance from the edges of the board as in regular 8 × 8 chess, while Capablanca has pushed them inward by one file each. The light Bishop on d1/g8 and the dark Bishops on g1/d8 look out of place in Figure 1, although they are on the opposite colours as their 8 × 8 chess counterparts. More crucially, there is no way to fianchetto Bishops on the Capablanca board. Another curiosity on the Capablanca chessboard is the unprotected Knight’s Pawn on the i-file. A direct consequence of the Bishop being pushed inward, this Pawn is under immediate attack after White plays the natural 1. d4 pawn push. It is important to note that when the King would execute a castle on this wider board, he would be permitted to travel a third square horizontally, coming to rest on the Knight’s file. This is another reason why the Capablanca configuration is not entirely desirable: you could castle directly into an attack as a result of the built-in weaknesses of the starting setup. Below we discuss a subset of undesired positions (in 2.1.1). Then we provide an in-depth analysis of Trice’s mate (in 2.1.2). In 2.1.3 we provide some background about why Capablanca improved on Bird’s idea. 2.1.1 A Subset of Undesired Positions There are some other negative consequences associated with Capablanca’s inward displacement of the Bishops. As shown in Figure 3, playing a natural developing move with the King’s Knight will hamper the deployment range of your own King’s Bishop. Notice how the h-Pawn hems in the Bishop on the g1 square, allowing it only to move to the left should the King’s Pawn be put into play. Attempting to fix this by playing the Pawn to h3 instead of Nh3 does not solve the problem. White’s King’s Bishop would be free to head to the right after the h-Pawn is pushed once, but where can the King’s Knight land on its first move? Playing Nj3 then Nh2 after Bi3 costs a critical tempo, and Black’s e-Pawn can make one move to threaten the Bishop on i3, a Figure 3: Playing the natural 1. Nh3 in potential loss of another tempo. Pushing the g-Pawn Capablanca’s chess constrains the deployment then playing Ng2 looks more promising, but without opportunities of the King’s Bishop. White having pushed the e-Pawn to release the d1 Bishop, Black has ... Ci6 to hit on the weakened i2 84 ICGA Journal June 2004 square. Of course ... Ci6 could be answered with Ch2, but this cuts off the Bishop’s retreat path and invites either ... Ad6 or ... c6 and ... Bc7 to chase the Chancellor. All of these attempts to secure an equalizing position after 1. h3 are met with strong counterplay. Pushing the h-Pawn two squares in Capablanca chess, seemingly freeing the King’s Bishop, Knight, and Chancellor, allows a violent attack against h3 and i2 by the enemy Archbishop, Queen, and Bishop, after they line up on the c8-j1 diagonal. White can try to do the equivalent quick kingside castle in Capablanca chess, but there is a positional detriment imposed. After 1. h4 d6 2. Nh3 e5 3. Bi3 (protecting the h-Pawn since the black Bishop on d8 is now attacking it) 3. … Nh6 4. Ch2 looks to allow 5. 0-0 without breaking a sweat. See Figure 4 where Black to move has 4. … Nj5, attacking the h-Pawn twice and the Bishop on i3, detracting from the merit of the position for White. After 4. … Nj5 and 5. … Nxi3, White’s structure is ruined on the kingside as 6. jxi3 is needed to recapture the Knight that Figure 4: Playing the pawn push 1. h4 in removed White’s Bishop. Capablanca’s chess in order to castle quickly after 1. … d6 2. Nh3 e5 3. Bi3 Nh6 4. Ch2 (shown above) will be thwarted by 4. … Nj5! 2.1.2 Trice’s mate From this subset of undesired positions can spring forth many others. One of the more entertaining exploitations of the weakened i-Pawn is now known as Trice’s Mate in Capablanca chess. It has the same deserving stature as the Scholar’s Mate in contemporary chess, but it is still worthy of print. Trice’s Mate involves a direct assault on i7 from the start. If not met properly, a sacrifice of the Chancellor and Queen is crowned with an unexpected solo-checkmate with the Archbishop on move 6. Figure 5: 1. d3 Nh6 2. Ci3!? Cg6?! Figure 6: 3. Qd2 f5. A small group of players from Philadelphia were experimenting with 80-square variants, attempting to rediscover Capablanca’s chess. The game Ed Trice vs. Joel Gehen, October 13, 1998, is the source of the positions shown above. It should be noted that 1. d3 was being played instead of the more “chess-like” 1. d4, since flank checks of the King are possible. The g8 Bishop can reach c4 and deliver check if both d4 and e4 are pushed. The black Archbishop can likewise inflict a check via a pawn push to b6, then sliding to a6. Throughout our course of rediscovering Capablanca’s chess, some uncomfortable arrangements for the player with the white pieces would result if pushing to d4 without preparation. 80-Square Chess 85 After 1. d3 Black must react to the Archbishop’s hit on the weak i7, and 1. … Nh6 supplies adequate interference. The follow-up with 2. Ci3 cannot be recommended, since the Chancellor can be antagonized with a mere push of the e-Pawn to reveal the black Bishop on d8. Black’s 2. … Cg6 adds another layer of protection to i7 via the Chancellor holding this square with its Knight aura, but assigning a major piece such a task is not the best way to proceed. With 3. Qd2 White compounds the attack on i7 with an x-ray threat, and 3. … f5 does not look out of the ordinary. But here, White has the shocking 4. Cxi7?!, which does not lead to a forced mate, but just about everyone falls for the resulting trap once. Figure 7: Black to move after 4. Cxi7?! Figure 8: After 4. … Cxi7 5. Qxh6. From Figure 7, Black will recapture White’s Chancellor with 4. … Cxi7. The purpose of White’s “sacrifice” of the Chancellor was to undermine the blatant weaknesses in the vicinity of i7, and set up a solo checkmate with the Archbishop if Black continues to react the way that a “regular chess” player would. White adds fuel to the fire with the surprising 5. Qxh6, seemingly sacrificing the Queen after having already parted with a Chancellor. Black appears to be in a quandary, with the Chancellor and Rook in a skewer of sorts by the Queen and Archbishop aligned along the same diagonal. The way out is through simplification: 5…Ci8 6. Qxj8 Cxj8 7. Axj8 Qi4! 8. Ag5 Qxi2 9. Ah3 Qi6! and Black wins. The game featured the miscue 5. … gxh6?? taking the Queen but walking into the solo-checkmate of the Archbishop, 6. Axh6#, as shown in Figure 9. The Archbishop delivers check as would a Bishop while also denying the black King access to f7. The knight component of the Archbishop secures the checkmate. This game demonstrated a radical exploitation of the weak i7 Pawn. There are countless other less extreme ways to undermine this weakness that are built into the fabric of Capablanca’s chess. This raises an interesting question. Why would a talent like Capablanca choose this particular configuration for his 80-square version of chess? Figure 9: With 5. … gxh6?? 6. Axh6# is Trice’s Mate. 86 ICGA Journal June 2004 2.1.3 Why Capablanca improved on Bird’s idea It turns out that Capablanca may have been merely correcting an unpleasant feature of Bird’s board when he subsequently proposed his setup. From Figure 1, if you switch the locations of the Bishop and Archbishop on the queenside, and Chancellor and Bishop on the kingside, you would have Bird’s proposed 80-square board in Figure 2. Figure 10: Playing 1. Ch3 d5?? on Bird’s board. Figure 11: The smothered mate 2. Cxh7# is possible. At first glance Bird’s configuration looks to be more logical, but the h-Pawn is undefended in this scenario. Notice that the h-Pawn is a Knight’s move away from the King. Bird’s Chancellor could make a leap on the first move, 1. Ch3, which would threaten the smothered mate next, 2. Cxh7#, if the h-Pawn was not defended immediately. It would be hard to imagine new players avoiding 1. Ch3 knowing that this Fool’s Mate exists. As such, Bird’s configuration had the potential to stifle opening creativity, rather than expound upon it. So, Capablanca improved upon Bird’s idea, but with a superficial treatment of the starting configuration, a more subtle, lasting imbalance was left in place as a result. The weak i-Pawn could be exploited, especially by the fact that three diagonal piece vectors are all aiming in its vicinity (Archbishop on c1, Bishop on d1, Queen on e1) from Capablanca’s starting position. From the examples shown, we can see that an exciting variation on the game of chess awaits, if it is possible to cure some of the anomalies on the wider board with the new pieces. It turns out that with a minor correction to the Capablanca setup, the result is an 80-square version of the game that is balanced, harmonious, and contains many themes and tabias already familiar to the modern chess player. 3. MODIFYING CAPABLANCA’S SETUP TO GOTHIC CHESS Figure 12: The familiar 8×8 chessboard. Figure 13: Gothic chess: a variation on Capablanca’s chess. 80-Square Chess 87 The contemporary 8×8 chess setup is shown in Figure 12, with a 10×8 chess variant called Gothic chess shown in Figure 13. The pieces from left to right at the bottom of the Gothic-chess board are: Rook, Knight, Bishop, Queen, Chancellor, King, Archbishop, Bishop, Knight, Rook. As shown above, by separating the King and Queen, it is possible to defend all of the Pawns in the starting configuration without perturbing the rest of the relationships between the chess pieces. Gothic chess is similar enough to chess that players of all chess strengths can adapt to it very quickly. The first Gothic-chess game ever played in July 2000 featured the moves: 1. Nh3 d5 2. i3 Nh6 3. Bi2 Nc6 4. e3 Af6 5. Nc3 e5 6. Ne2 Be6 7. Ng3 g6 8. Ae2 Qd7 9. 0-0 Cd6 10. b3 0-0- 0 11. Ba3 Ce8 leading to the position shown in Figure 14. This position contains a King’s Indian formation for White that is easy to identify. Black has castled to the opposite side of the board, which inevitably leads to wars being waged on opposing flanks. The game continued as follows. 12. Bb2 Bg7 13. Ad3 Ci8 14. Ac5 Qe8 15. d4 e4 16. Qe2 Bf8 17. Aa4 i5 18. c4 i4 19. cxd5!? ixh3 20. dxc6 hxi2 21. cxb7+ Kxb7?! 22. Axe8! ixh1=Q+ 23. Cxh1 Rxe8 24. Qb5+ Ka8 25. Qc6+ Kb8 26. Qb5+ Kc8 27. d5! Bd7?? 28. Qa6+ Kb8 29. Bxf6 j5 30. Bd4 c5 31. dxc6 e.p. Bxc6 32. Qxa7+ Kc8 33. Cc1 Re6 34. Bb6 1-0 which was an exciting fight every step of the way. Figure 14: White to move after 11. Ba3 Ce8. 4. DERIVING VALUES FOR PIECES ON THE 10×8 BOARD Before a variant such as Gothic chess would gain in popularity, some acceptable values for the new pieces would have to be determined. While such values for chess pieces on an 8×8 board have been well documented for decades now, it was not apparent how their strengths would translate once migrated onto a 10×8 board. Would a Queen become even stronger on this larger board? Will the rectangular area weaken the effectiveness of a Knight? Also, what are the values for the new pieces? Should they just be given some arbitrary merit in addition to the sum of their decompartmentalized values? Recall the contemporary Queen is given a one pawn bonus for having the combined powers of the Rook and Bishop exuding from just one square (instead of two). Should the Chancellor and the Archbishop be treated similarly? These questions will be addressed below. In 4.1 we introduce the concept of ‘safe check’. Then in 4.2 we compute the ‘safe check’ probabilities for a Rook on boards of arbitrary sizes. In 4.3 we compare the ‘safe check’ probabilities of 8×8 boards to those of a 10×8 board for all pieces. In 4.4 we introduce Reinhard Scharnagl’s computation of the values of the pieces. Finally, in 4.5 we provide two examples from the Gothic-chess practice. 4.1 The Concept of ‘safe check’ The mathematician Henry Taylor (1876) established the concept of a ‘safe check’ as a means for estimating the value of the chess pieces on a square board with n squares per side. Taylor’s ‘safe check’ measured the number of moves that a given piece could check an enemy King without the King being able to capture it (trivially) on the next turn. The more squares on which such ‘safe checks’ could be given the stronger the piece is. Taylor compared the probabilities, i.e., the ratios of the number of safe checking moves to the total number of arrangements, in order to establish exchange values for the pieces. Because his work centered primarily on computations for a square board, the formulas needed to be re-derived for a rectangular board. 88 ICGA Journal June 2004 Figure 15: ‘Safe check’ nomenclature for derivation of rook values for any sized board. To begin, examine the diagrams shown in Figure 15. An aura of rook moves is shown in (A), illustrating the horizontal and vertical components of its power. This allows an enemy King that is placed in check the potential to make one of three different types of capturing responses. If the King is in any one of the four corners of the board when it is in check (B), you can see that there are two squares marked with an “X” showing where it could make a trivial capture of the Rook. These are the unsafe checking squares, so what remain would be the squares of ‘safe check’. If the King is on the edge of the board (C) but not in one of the four corners, then it has three squares available to capture the Rook. Closer to the center of the board (D) the King has four capturing moves that can snare the Rook. One last remark: For the sake of simplicity we do not place a King onto the board with the same colour as the side delivering check. While this would make these ‘endgame positions’ legal, it does not offer any insight into the theoretical value of the pieces, and it complicates computation of the ‘safe check’ probabilities. 4.2 The ‘safe check’ Probabilities for a Rook Figure 16: Preparing to perform ‘safe check’ summations for a Rook on any sized board. For each of the checking geometries, there will be a different number of king arrangements (see Figure 16). In scenario (B) there will always be 4 corners, no matter what the dimensions of the board are. In scenario (C), the number of edges on the board is given by 2(r-2) + 2(f-2) = 2r - 4 + 2f - 4 = 2r + 2f - 8 which will vary depending on the dimensions of the board. In scenario (D) there are (r-2)(f-2) = rf - 2r -2f + 4 center arrangements. Each of these will be used in determining the safe check probability for the Rook on a board f files wide and r ranks high. • Corner Squares: 4 • Edge Squares: 2r + 2f - 8 • Center Squares: rf - 2r -2f + 4 • Total Squares = (4) + (2r + 2f - 8) + (rf - 2r -2f + 4) = rf 80-Square Chess 89 First, place the King in any of the four corners of the board, and enumerate all of the safe checks that a Rook can issue. From the diagram of Figure 17 it is easy to see that (r - 2) safe checks can be delivered vertically and (f - 2) safe checks can be made horizontally. With four corner squares, the total number of safe checks is given by: 4(r - 2 + f - 2) = 4(r + f - 4) We note that there are rf squares available to place the King, and rf - 1 squares remaining for the Rook once the King has taken up one square on the board. Then, we express the probability for safe check as the quotient of the safe check square count and the total number of Figure 17: Deriving the ‘safe check’ formula for possible arrangements. a Rook vs. an enemy King in the corner. P(safe check, corner) = 4(r + f - 4) / [(rf)(rf - 1)] Next, place the King on each edge square and repeat the safe checks count that a Rook can issue. From the diagram above there are (r - 2) safe vertical checks but now only (f - 3) safe checks can be made horizontally. Since the King is out of the corner, it can make a capture to either side of the Rook when checked horizontally. If the King resided on a vertical edge, a similar observation is made. There would be (f - 2) safe checks along horizon and (r - 2) safe checks vertically. But now, there is no longer a “constant” number of these arrangements. There are 2(r - 2) instances where the King makes (r - 3) + (f - 2) captures and 2(f - 2) instances where the King makes (f - 3) + (r - 2) captures. The total number of safe checks is given by: 2(r - 2)[(r - 3 + f - 2)] + 2(f - 2)[(f - 3 + r - 2)] = 2(r - 2)(r + f - 5) + 2(f - 2)(r + f - 5) = (r + f - 5)[(2r - 4) + (2f - 4)] = (r + f - 5)(2r + 2f - 8) = 2(r + f - 4)(r + f - 5) Expressing the probability for safe check as the quotient of this count and the total number of possible arrangements: P(safe check, edge) = 2(r + f - 4)(r + f - 5) / [(rf)(rf - 1)] Finally, place the King on every center square. There are (r - 3 + f - 3) safe checks for each king arrangement that is inset one square from the outermost portion of the board. This rectangle is (r - 2) squares high and (f - 2) squares wide. The count for this scenario is just a straight multiplication, so the probability is given by: P(safe check, center) = (r - 2)(f - 2)(r + f - 6) / [(rf)(rf - 1)] Expressing the total probability for safely checking with the Rook is just the sum of all of these components: • P(safe check, corner) = 4(r + f - 4) / [(rf)(rf - 1)] • P(safe check, edge) = 2(r + f - 4)(r + f - 5) / [(rf)(rf - 1)] • P(safe check, center) = (r - 2)(f - 2)(r + f - 6) / Figure 18: Deriving the ‘safe check’ formula [(rf)(rf - 1)] for a Rook vs. an enemy King on an edge. • Total probability = (r + f - 6) / (rf - 1) + 2(r + f) / [(rf)(rf - 1)] 90 ICGA Journal June 2004 Now we can check our formula with the Taylor result from 1876. Taylor discovered a Rook could safely check a King one-sixth of the time on the 64-square chessboard. So, we set r = f = 8 and see what we get2: (r + f - 6)/(rf - 1) + 2(r + f)/[(rf)(rf - 1)] = (8 + 8 - 6)/(64 - 1) + 2(8 + 8)/[(64)(64 - 1)] = 672/4032 = 1/6. This formula therefore agrees with the Taylor result for 8×8 chess. As a next step, we compute the Rook safe check probability for Gothic chess using f =10 and r = 8. (r + f - 6)/(rf - 1) + 2(r + f)/[(rf)(rf - 1)] = (8 + 10 - 6)/(80 - 1) + 2(8 + 10)/[(80)(80 - 1)] = 996/6320 = 249/1580 = 0.15759493671 Comparing 1/6 = 0.16666666667 to 0.15759493671 indicates that a Rook on the 10×8 Gothic-chess board is about 94.5 per cent of the strength of the Rook on an 8×8 board. So, the Gothic-chess Rook is about 5 per cent weaker than its chess counterpart. 4.3 A Comparison of Safe Check Probabilities for 8×8 and 8×10 Boards Knowing how one piece on the 10×8 board compares in strength to the 8×8 board is a great start. But in order to determine the values of the pieces when compared to one another, the formula computed above must be generalized for the Knight and the Bishop as well. To spare readers the math, I will present the Knight formula3 without all of the steps: Knight Safe Check Probabilities = 8/(rf - 1) + (16 - 12r - 12f) / [(rf)(rf - 1)] The equation for the Bishop is quite a disaster. This formula changes when your rank/file count goes from odd to even or even to odd. It is populated with conditional clauses that are not easy to represent in a concise mathematical form. It was easier to compute the 10×8 result directly using a graphical method specific to the 10×8 board. The result is given in Table 1. Safe Check Probabilities Piece 8×8 board 10×8 board % difference Knight 12/144 110/1580 -16.67% Bishop 13/144 133/1580 -6.77% Rook 24/144 249/1580 -5.50% Archbishop 25/144 243/1580 -11.41% Chancellor 36/144 359/1580 -9.11% Queen 37/144 382/1580 -5.95% Table 1: Safe check probabilities on 8×8 and 10×8 boards. There are a few things worthy to note at this point. These are not exact ratios universally accepted by the chess-playing public. The work of Taylor in 1876 provided a foundation upon which players could experiment and adjust the relative merits of the pieces over the years. In this respect, Taylor’s equations will compute semiconditional values for the pieces (Katsenelinboigen, 1997, p. 53). In this fashion, we differentiate these data from conditional, unconditional, partially conditional, and positional values that are computed in a variety of ways by the contemporary chess master at various stages of the game. For example, no strong chess player considers a Rook to be worth exactly two knights worth of material, as the 8×8 safe check entries indicate. A Rook is much stronger than two Knights in the ending, since it can force a 2 This is also possible via the probability formula for an r × r board being 2 (r – 2) / [r (r + 1)], which equals 1/6 if r = 8. 3 If r = f the formula reduces to 8(r – 2) / [r2(r + 1)], yielding 1/12 if r = 8. 80-Square Chess 91 checkmate of the enemy King, whereas a pair of Knights cannot force a checkmate (they can mate if the opponent commits a grave error.) A Rook also participates in castling so it has a special function to perform. To discourage trading a Rook for two minor pieces, most masters took the approach of making it less valuable than a pair of minors. The value attributed to a Queen was likewise adjusted by the masters. It is not merely the “sum of its parts” of Bishop + Rook. Since it can do on one square what these piece do on two squares, the modern master has given it an additional bonus. The new Gothic-chess pieces values were likewise adjusted according to some intuitive observations over the course of many games. The Archbishop is very deadly in closed positions, particularly because it can issue a checkmate unassisted. This piece tends to be worth much more than the sum of its minor piece values in the opening phase of the game. As Pawns come off of the board, the Archbishop’s power diminishes. It is very difficult to checkmate in the ending of Archbishop + King vs. King. The longest win for this endgame features a mate in 21 moves (Bourzutschky and Trice, 2004). In contrast, the Chancellor does not combine the powers of Rook and Knight very well in the opening phase, yet its longest win against a lone enemy King requires only 13 moves. Our experience so far has shown that the Archbishop is the most deadly piece in the opening, the Chancellor is the most deadly piece in the middle game, and the Queen is the most deadly piece in the endgame. 4.4 Scharnagl’s Piece Values Reinhard Scharnagl (Germany) has independently computed values for Gothic-chess pieces that differ slightly from the author’s calculations. Scharnagl (2004) is the author of the Strategiespiel-programm mit Intelligent Rückkoppelnden Funktionen algorithm (S.M.I.R.F.), which is capable of dynamically determining values for chess pieces on boards of any dimension. He applied this technique to the new Gothic-chess pieces as well (cf. Table 2). Piece Scharnagl Trice Pawn 100 100 Knight 306 275..250 Bishop 360 310 Rook 543 440..530 Archbishop 665 690..645 Chancellor 849 840..865 Queen 903 910..960 Table 2: Gothic-chess piece values calculated by Reinhard Scharnagl and Ed Trice. The values in the rightmost column indicate the range over which the piece weights will vary over the course of the game. For example, at the start of a game, the Knight value is set at 275, and it will decay to 250 by the time a pawnless endgame is reached. This is done to encourage a software program to prefer a Rook (530) to a pair of Knights (500) in the endgame. Also, those weights will keep the program from surrendering a Knight and Bishop (585) for a Rook and Pawn (540) in the early stages of a game. During the opening phase and early middle game, the two developed minor pieces are preferable to the Rook and Pawn. One of the most difficult piece exchanging scenarios concerns the Archbishop + Knight vs. Queen (and Pawn). In regular 8×8 chess, three minor pieces (on three squares) is a satisfying exchange for a Queen. In Gothic chess, the forces of Archbishop and Knight are equivalent to having the three minor pieces on two squares, a more powerful combination of material. But, on the 10×8 board, the Knight has lost some power. So the question remains, under what circumstances should the Archbishop and Knight be surrendered for a Queen or a Queen and a Pawn? In pawn-heavy clusters, the Archbishop and Knight (965) clearly outperform the Queen (910) so the pair of pieces should be retained. They should be exchanged if an opportunity presents itself to snare a Pawn in addition to the Queen (1010). However, towards the end of the game, on a board devoid of Pawns, the Archbishop and Knight (895) are considerably less valuable than the Queen (960). Note that the side with the Queen would still allow it to be traded if it could win a pawn in addition to the Archbishop and Knight (995). 92 ICGA Journal June 2004 4.5 An Example from Master Play Some of the exchange opportunities mentioned above present themselves from positions that appear to be tranquil in the middle-game phase of the game. Consider the examples in the Figures 19 and 20. They are taken from actual master class play. Look at the position shown in Figure 19. It does not appear to be out of the ordinary, but White has the opportunity to win either a Chancellor and two Pawns or a Queen and one Pawn for the Archbishop and Knight (and Black can choose which bundle to surrender). Figure 19: White to move from Ed Trice (2403) Figure 20: Black to move after 17. Nxd5! cxd5 vs. Uwe Kreuzer (2218, Germany). 18. Ng5 Ad8 19. Ab5! Notice both the Chancellor on e8 and the Queen on d6 are under attack by the White Archbishop on b5. Figure 20 shows the unusual skewering capability of the Archbishop, (sometimes referred to jokingly as a “skork” by Gothic-chess players, since it is a kind of skewer and a fork) as it hits the black Chancellor on e8 with its Bishop aura and the black Queen on d6 is now within the scope of its Knight component. If Black retreats the Queen, then Axe8 and Bxd5 will net White a Chancellor and two Pawns for the Archbishop and Knight. In the actual game, Black played 19…Bb7 to defend the d5 pawn, leading to 20. Axd6 Cxd6 where White won a Queen and Pawn for the Archbishop and Knight. There are countless middle game positions where dynamic engagements involving the Archbishop and Knight for Chancellor/Queen with/without Pawns await to be discovered. As interest in programming software to play this game continues to grow, and as the master class of tournament players increases in number, we will have a better understanding of the intrinsic exchange values of such combinations. 5. GOTHIC-CHESS SOFTWARE PROGRAMS Readers might be surprised to learn that there are currently five programs capable of playing Gothic chess as of June 2004. These programs are: GOTHIC VORTEX, TSCP GOTHIC, REINHARD’S GOTHIC, CAPA-GNU, and the ZILLIONS-OF-GAMES engine. Table 3 provides an overview of these programs. At the University of London there is another software project in the preliminary phases under the direction of Don Beal (2004). In the fall of 2004 a program-only tournament will be played via peer-to-peer connections to determine the “official” strongest Gothic-chess program in the world. Persons interested in registering for this event can see the details online at the GothicChess.org website. 80-Square Chess 93 Gothic-chess Programs/Engines Program Author(s) Country Description A project stemming from the CRAFTY chess program originally by Robert Hyatt and Bert Gower. It uses almost all of the algorithms and data structures found in the CRAFTY code, Gil Dodgen and ported to the 80-bit environment. GOTHIC GOTHIC VORTEX Ed Trice USA VORTEX is now a very fast searching bitboard engine with a highly-tuned evaluation function. It currently has only one loss in over 30 games played with the other programs (its sole loss to the impressive TSCP GOTHIC). A project that grew out of the TSCP chess program by Tom Kerrigan. Michel Langeveld corrected some bugs in the TSCP chess The program and added some additional features TSCP GOTHIC Michel Langeveld Netherlands (better hashing, Static Exchange Evaluation, etc). Michel’s Gothic-chess implementation appears to be one of the strongest programs currently developed. Reinhard Scharnagl implemented his own dynamic evaluation technology which formulates values for pieces on boards of any Reinhard dimension automatically. His values differed REINHARD’S GOTHIC Germany Scharnagl from those of GOTHIC VORTEX, so he asked the GOTHIC VORTEX team to compile a version using his values and play the engines against one another. This program is an old GNU port to play Capablanca’s chess. The setup feature allows the program to play Gothic chess. This is an older program (runs only under DOS) that CAPA-GNU Bill Angel USA uses relatively antiquated technology, and, to be fair, it is being configured differently from what it was intended to play. It is being included here for the sake of completeness. The Z.O.G. program is a generic games engine. It can be scripted to play any game, from tick-tack-toe to checkers to chess to Go to any chess variant. It is the program that most serious chess-variant enthusiasts are Jeff Mallett and ZILLIONS-OF-GAMES USA familiar with playing. It uses a generalized Mark Lefler search and evaluation function that is derived after parsing the rules file that programmers can script fairly quickly. It is impressive in its versatility and use of use, but did not fair well in our matches. Table 3: Existing Gothic-chess programs and their authors. 6. CONCLUSIONS AND PERSPECTIVES There is a reason why the game of chess has not evolved in relatively recent history: it is very difficult to improve upon a game that has been play-tested so much over the centuries. Great minds such as Bird and Capablanca had good ideas as to how to “make the game even better”. It is apparent that they had not 94 ICGA Journal June 2004 performed a great deal of play-testing with their respective variants, and without this crucial element a different version of the game with mass appeal will never come to fruition. Below we give some facts and offer a few perspectives. Gothic chess has been around since the year 2000, and is currently played in 47 countries. There are boards and pieces available for this game, a free software program available for download at GothicChess.org, a website to play against others from around the world (http://www.BrainKing.com) and a bi-monthly periodical dedicated to Gothic chess. While there are those advocates in the chess world who insist that any chess variant will never become “mainstream”, it would be tough for them to argue with the 30,000 sets sold to date.4 Gothic chess is a fun, well thought out, and heavily play-tested chess variant. It incorporates ideas from some of the greatest chess minds who have played the game. The configuration of the board cuts the playing time in half before one side usually concedes (35 moves is a long game in Gothic chess.) The BrainKing.com website reports that there have been 1697 wins for White, 1750 wins for Black, and 71 draws out of the 3518 games played as of May 28, 2004. This is a 2.01% drawing percentage, orders of magnitude lower than what is observed in regular chess. The history of chess evolution has shown that changes to the game that have reduced the playing time and drawing percentages observed between the adversaries, as well as made the game more interesting, have had a strong impact on the subsequent shaping of chess. Only time will tell if Gothic chess will produce a similar level of impact that has been the recurring theme over centuries of evolution. ACKNOWLEDGEMENTS Dr. Aron Katsenelinboigen of The Wharton School of the University of Pennsylvania, as early as 1998, was instrumental in aiding my search in locating the papers of Henry Taylor. As I was struggling to “reinvent the wheel” in terms of deriving a mathematical model to precipitate new values for the pieces on an 80-square board, Dr. Katsenelinboigen merely had to state: “Look here”. 4 However, we are aware of the following facts. In Eastern Europe, the game of hexagonal chess (Glinsky) was rather popular; about 100,000 sets were sold and many tournaments have been played. After some tens of years and the death of its inventor, the game went down in popularity. The games of Fischer random chess and Janus chess are also gaining somewhat in popularity, but currently no numbers are available (Kok, 2004) 80-Square Chess 95 A decade before this, while I was in the throws of improving my own chess program (THE SNIPER), Norman Worthington, then the Chief Technical Officer for The Software Toolworks, was of invaluable assistance in helping me locate the bibliographical sources upon which the excellent software manual accompanying the CHESSMASTER 2100 chess program was based. This software manual sparked my interest in hunting down documentation of the means to play Capablanca’s chess, as it mentioned that he wanted to change the game by adding new pieces, but it did not indicate where they were to be placed. That quest lead to finding Edward Lasker’s book, The Adventure of Chess. I must highly recommend this book to chess players of any strength and interest level. The book is simply superb. To Edward Lasker, my deepest thanks. To Norman Worthington, you have my gratitude. Marc Bourzutschky was of tremendous assistance in reworking Eugene Nalimov’s endgame database generator to solve Gothic-chess endings. Thanks to Marc, GOTHIC VORTEX will be capable of announcing mate in 268. Some truly spectacular endings are shown at http://www.GothicChess.org/databases.html. Last, but certainly not least, I would like to recognize Gilbert Keddie Dodgen, bit-shuffler extraordinaire, who thought that rotated bitboards for a rectangular playing surface might only be “a little non-trivial”. Thanks for changing the correct three #define statements Gilbert. 7. REFERENCES Beal, D. (2004). Personal communication to Ed Trice. Bourzutschky, M. and Trice, E. (2004). Marc Bourzutschky and Ed Trice independently computed the set of all 5-Piece Gothic-chess endgame table bases. The longest win requires 268 moves (535 plies) in the ending of Queen + Pawn vs. Queen. This will be the topic of a separate paper. It should be noted that this is longer than the longest 6-piece win in chess (262 moves), computed by Ken Thompson in 2000. Katsenelinboigen, A (1997). The Concept of Indeterminism and its Applications: Economics, Social Systems, Ethics, Artificial Intelligence, and Aesthetics. Präger, Westport, CT. ISBN 0-275-95788-8. Kok, F. (2004). Personal communication to the Editor. Lasker, E(dward) (1959). The Adventure of Chess. Second revised edition, Dover, New York. ISBN 0-486- 20510-X. Liddell, D.M. (1937). Chessmen, Harcourt, Brace & Co. Murray, H.J.R. (1913). A History of Chess. Oxford University Press, Oxford. Pritchard, D.B. (1994). The Encyclopedia of Chess Variants. Games & Puzzles Publications, Godalming. Reinfeld, F. (1942). The Immortal Games of Capablanca. Horowitz & Harknes, New York. Reprinted in 1990, Dover, New York. ISBN 0-486-26333-9. Scharnagl, R. (2004). Refer to the website: http://www.chessbox.de/Compu/fullchess1_e.html for a complete description of the algorithm as well as his book on Fisher Random Chess (ISBN: 3-8334-1322-0). Taylor, H (1876). On the Relative Value of the Pieces in Chess, Philosophical Magazine, Vol. 5, March 1876, pp. 221-229. Van Haeringen, H. and Van den Herik, H.J. (2003). Superchess. ICGA Journal, Vol. 26, No. 4, pp. 239-250. Winter, E. (1989). Capablanca: A Compendium of Games, Notes, Articles, Correspondence, Illustrations, and Other Rare Archival Materials, McFarland & Company. ISBN 0-89950-455-8.