The earliest type of bent-board chess is a game described by the Arab historian al-Masudi (died Y956) in his Muruj adh-dhahab (Fields of Gold) Y947 and described in more detail in several later manuscripts.
The board and opening position in this game, usually called Byzantine Round Chess (though it was played at other times and places, e.g. by Ala'addin, chess adviser to Timur in Samarkand circa Y1400) can be regarded as formed from the usual opening position by splitting the board down the middle (between the king and queen sides) and joining the short ends of the two halves, thus making a board of 16 radial 'ranks' and 4 circular 'files'.
The pieces denoted here as queen (Q) and bishop (B) are of course the Fers and Alfil in mediaeval chess. The K and F(Q) of one colour were usually interchanged so that the two fers moved on the same set of cells. There was no pawn promotion; if two pawns met head-on the opponent removed them. (It is not strictly correct to call this game Byzantine Chess as this would be the normal form of chess played in Byzantium, which would have been Shatranj, known locally under the Greek name of Zatrikion.)
Numerous forms of Modern Round Chess have been proposed, played with the modern forces. I suggest the new rules that a blocked pawn may make a diagonal move (like a phantom capture), and that a pawn promotes on making a complete circuit of the board. The pawns would then need to be marked with eyes to show which way they are travelling! In the above diagram it is assumed that pawns move head-first (in our other diagrams White pawns move head-first and Black pawns feet-first).
In 1983 David Reynolds came across a picture in a book (probably Joseph Strutt's Sports and Pastimes of the People of England 1801) of a circular chess board in use in medieval times. He was sufficiently intrigued by this to make his own board out of cardboard and to take it along to his local pub, the Tap and Spile in Hungate, Lincoln. The book gave an opening layout but did not say how to play, so David and his friends adapted modern rules in a natural way and found that it made a playable game. They now play on attractive wooden boards about 55 cm in diameter with a red cross and gold fleur-de-lys in the middle. The 2005 event in Lincoln will be the tenth Circular Chess World Championship.
In their rules the pawn can move two steps at its first move but there is neither en passant capture nor castling. Pawns promote on reaching the nearer of the opponent's piece rows. Null moves (taking a rook or queen all the way round and putting it back where it started, are banned.
Here is a bit of endgame theory by John Beasley from the above article. The lack of corners means that king and rook against king is only a draw, though king and queen still win, and king and any defended pawn (even a side pawn) win against a bare king because there is no stalemate defence. Rook and pawn against rook might seem to be drawn because the rook can sacrifice itself for the pawn, but it isn't as simple as this: put the rook behind the pawn and the king nearby, and if Black attacks the pawn on the file White swaps off rooks round the board and winsthe pawn ending, while if BBlack attacks from the side the pawn simply advances. Queen and pawn against queen? The king can hide from Black's checks, as in the position below: 1.Kc4 Qa6+ 2.Kb4 Qb7+ 3.Kc3. But whether White can use the breathing space to advance his pawn is another matter.
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The second diagram is an endgame study by John from VC37 p.85. White to play and win. Solution: 1.Kb3 (If 1.Kxa3? Kxc4 and Black draws because White is in zugzwang and must cede his pawn.) a2 (1. ... Kc5? 2.a5 etc wins) 2. Kxa2 Kxc4 3. Ka3 and this time it is Black who is in zugzwang.
The lack of a method of printing true circular diagrams of the board is one of the main restraints on the development of this variant. In his articles John Beasley has adopted the convention of showing the board as two 4-by-8 half-boards, which you have to imagine bent round until their top and bottom edges meet, but note that a1 is a light square. On page 48 of VC31 John gave three knight's tours of the circular board, one being from Twiss's Chess 1789, which has an account of the round game (Vol.2, pp.9-11).
Here are two games from the championships. To aid visualisation I follow the convention of giving both the start and finish cells of a move when they are in opposite halves of the board, but otherwise just the destination cell. The commentaries are by John Beasley.
The first is the deciding game between the three-times champion 1997-98-99 Francis Bowers and year 2000 winner Herman Kok. 1. a4 d5 2. b4 Bd7 3. a5 Na6 4. Ba3 b6 5. axb6 [diagram]. The aggressive long-term option for Black is now ... cxb6, giving each player a half-open ring on which to deploy his major pieces. Which would be the stronger: White's possession of the a/h ring, on which his rooks are already doubled and Black's pawn is likely to become weak, or Black's of the more central c/f ring with its potentially more immediate threat to the opponent's king?
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This choice between side ring and inner ring is perhaps the most fundamental strategic question in the game, but it is not to be explored on this occasion because Black chooses: 5. ... axb6. In the long term, this might be thought of as tending to a draw; neither side can afford to concede control of the open ring, so exchanges will be inevitable, and once the rooks and queens are off the game would appear to favour the defence (the attacker has only a four-file front on which to operate, and the defender can switch his pieces from wing to wing much more quickly than the attacker can). However, this is in the long term, and in the short term Black has pressure on the b-pawn. 6. c3 Qe8-c8 7. Bb2 c5. Black clearly stands better, and a writer on development would point out that he has moved three pieces once each whereas White has moved one piece twice. But White's demise hardly seems imminent. 8. Ng1-c2 cxb4 9. Nxb4? This loses at least a pawn. 9. cxb4 was correct. 9. ... Nxb4 10. cxb4?? And now 10. Rxa8 was essential. 10. ... Rxa1 11. Rh1xa1 Rh8xa1 12. Bxa1 Ba4 mate [diagram].
The second game is between one of the main organisers of the tournament, Rob Stevens who won the event in its first year, and another regular player Eamonn Hunt. This is from the 2002 championship. Notes by John Beasley again. 1. b4 c6 2. a4 a5 3. b5 c5. I find this curious. I would automatically have played 3. ... cxb5 4. axb5 b6, when I have a protected passed pawn for the ending and my half-open c/f ring will surely be of mre use than his solidly blocked a/h ring. 4. c4 b6 5. g4 h6 6. h4 h5 [diagram]. 7. gxh5. Another move I fnd curious. It seems to leave White with a very weak h-pawn, and no obvious compensation. 7. ... Rxh5 8. f3 f5. And this is surely wrong. Black's natural play is to set a block of granite on the b/g ring by g6, and then to try and exploit the isolated h-pawn. 9. Nb1-f2 Rh7 10. Bg2 g6 11. Qf1 Bh6 12. Bc1-e1 Be3 13. Nh3 [diagram].
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13. ... Bxg1. One of the isues still unresolved in this game is whether B for N is usually a good swap or a bad one. Both pieces are appreciably weaker than in normal chess, whereas the rook is appreciably stronger, but how they continue to relate to each other is less clear. This said, during a pre-tournament practice with David Howell [the tournament winner], I offered him two pieces for a rook with an appropriate warning; he accepted them, and proceeded to win handsomely. 14. Rxg1 Nb8-f7 15. Nf4 Nf7-b8 16. Qf1-c3 Nf6 [diagram]. 17. Bh3. White unmasks his artillery on the b/g ring, and Black's men are so poorly placed that he breaks right through. 17. ... Rg7 18. Rb1. Here Bxf5 at once would seem to be playable; what have I missed? 18. ... Bc8xh7 19. Qb2 Qf7. Black lets it happen anyway. 20. Bxf5 e5 [diagram].
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21. Bxg6 Bxg6 22. Rxg6 Rxg6 23. Nxg6. White is two pawns ahead with full command of the b/g ring, and the rest is routine. 23 ... Qe6 24. f4 Qe6-h3+ 25. Kc2 Qh2+ 26. Be1-c1 Qe2 27. Nxe5 Qf1 28. Rxb8+ Rxb8 29. Qxb8+ Kd8-e8 30. Qb8-g6+ Ke8-d8 31. Qxf6 Qh1+ (a last hope, but perpetual check there isn't) 32. Kc3 Qh1-a1+ 33. Bb2 Qa1-f2+ 34. Kb3 d5 35. Nf7+ Kd7 36. Qe7+ Kc7 37. Qe7-b8+ Kd7 38. Qd8 mate.
Reports by John Beasley on the 1999 World Championship, held at Lincoln Castle, the year 2000 Championship, held at St Mary's Guildhall, Lincoln, and the 2001, 2002 and 2003 Championships, held at Edward King House, Lincoln were published in Variant Chess Vol.4, issue 32, p.66, Vol.5, issue 36 p.57, Vol.5, issue 38, pp 90-91, Vol.6, issue 41, pp.2-3 and Vol.6, issue 44, p.57. The following is a photograph taken at the 2004 Championship. Here is a photo of the excellent venue at Edward King House, taken by me at the 2004 Championships.
Following are 2 photos taken by George Jellis at the 2006 championships held in a marquee in the grounds of Lincoln Castle