Introduction
by George Jelliss
Numerous proposals have been put forward over the years for altering the shape of the chessboard. Here we examine some of the possibilities that modify the board slightly, perhaps requiring modification of the opening position, but do not enlarge the board to an extent that would need the addition of new forces to cover the space, nor reduce it so much that some pieces are made redundant.
The simplest such variants are those that omit or add ranks, changing the square to a rectangle. The minimal case is the 8×4 board where there are no vacant squares at all, but White has immediate mates by P×f3. Play on the less compressed 8×5, 8×6 or 8×7 boards however seems perfectly possible.

Also on lengthened boards 8×9, 8×10 and so on, though on larger boards it would seem advisable to increase the forward movepower of the pawns, and perhaps the knights.
On these longer boards (and other enlarged boards) I have peviously proposed the use of Opting Pawns (VC6, p.70) which are able to make the optional double forward move wherever they are on the board. The double move remains subject to enpassant capture if it passes over a square guarded by an opposing pawn. An opting pawn on the third fromlast rank can thus reach the promotion rank in one move, unhampered by e.p. capture since no pawns inhabit the back rank.
Probably the best known variant with a reshaped board is Morley`s Game, described in the delightfully written little book of reminiscences by F.V.Morley, My One Contribution to Chess (1947), in which he adds a ‘corridor’ of six squares to each side of the board.
Much of the book is about his father F. W. Morley, a mathematician who in 1899 (though this is not mentioned in the book) discovered ‘Morley`s Theorem’, an elegant and surprising result which shows that the trisectors of the angles of any triangle meet in three points that are the vertices of an equilateral triangle (For an account see H. S. M. Coxeter`s Introduction to Geometry, 1969).
F. W. Morley was also an occasional chess player and before emigrating to the USA in 1887 to become Professor of Mathematics at Johns Hopkins University, Baltimore, he stopped off at Simpson`s Divan on 17 September 1886 to play three games with the famous H. E. Bird, losing two and winning one, which went as follows (old style notation seems appropriate on this occasion): 1. P–K4 P–K4 2. Kt–KB3 Kt–QB3 3. P–Q4 P×P 4. Kt×P B–B4 5. B–K3 Q–R5? 6. Kt–QB3 B×Kt 7. B×B Kt×B 8. Q×Kt Kt–B3 9. KKtP–Kt3 Q–Kt5 10. B–K2 Q–K3 11. P–K5 Kt–Kt1 12. Castles (Q`s side) P–QR3? 13. B–Kt4 Q–QB3 14. P–K6 Kt–B3 15. P×QP ch. B×P 16. B×B ch. Kt×B 17. KR–K1 ch. K–Q1 18. Q×KtP R–K1 19. R×R ch. K×R 20. R–K1 ch. K–Q1 21. Q–Kt8 ch. and White wins.
Later in the book F. V. Morley transfers this game to the new board:

After White`s 9th move.
He remarks that “Between players who wished to stick to the old openings the first 8 moves might well have been the same.” ... “On the ancient board, Bird at his 9th move was in serious trouble.” But on the corridor board he could play 9...Q–KRC4 (KRC being ‘King`s Rook`s Corridor’). “Whether that is or is not a good move I must leave to more competent analysts.”
In chapter 6 he describes chess as a ‘dromenon’ which is “...a pattern of dynamic expression in which the performers express something larger than themselves, beyond their powers of speech to express and a therapeutic rhythm in which they find release and fulfilment...” (from Jane Harrison Ancient Art and Ritual). Could that be why we play Variant Chess?
In the last two chapters of My One Contribution to Chess F.V.M. makes a second contribution to the subject by adding further corridors to the other sides of the board, behind the kings, to make a ‘double corridor board’ which restores perfect square symmetry. He gives knight`s tours on the two new boards, constructed by Euler`s method.
In this he was anticipated by Ernest Bergholt who had used this 88cell board in 1918 to illustrate his method of constructing tours in ‘perfect quaternary symmetry’. (This is in fact a form of binary symmetry giving an impression of quaternary symmetry on boards where true quaternary symmetry is impossible. It is a mixture of rotary and axial quaternary symmetry.) This example is among the memoirs on the subject that he sent to H. J. R. Murray:
Troitzky Chess
by Paul Byway
Inspired by a remark of the famous endgame study analyst Troitzky that checkmate by two knights is possible if you add two extra squares to the board behind each back rank I expand this idea into a proposal for ‘Troitzky Chess’.
The following is a quote from the preface of Troitzky`s Collection of Chess Studies: “Tchigorin and Schiffers became my good friends and both were much interested in my experiments to devise a game of chess for four players, and a game complicated by the addition of four extra squares to the board.” Further detail is not given there. Does anyone know if he wrote more fully on these ideas elsewhere?
As well as adding the four squares proposed by Troitzky, I also remove the four corner cells, putting the rooks initially in the royal annexes instead. Then, for symmetry, I also add extra squares at the ends of the central ranks, to give a nearcircular board.

Pawns promote on the farthest square of whatever file they are on. Castling is by king swapping places with an adjacent rook. Since the rooks are initially in the annex, under this castlingbyinterchange rule you could have doubled centralised rooks after 1. 000!? It seems that a corner rook stops the wing unravelling, so in Troitzky chess devastating wing attacks may be normal. After From`s Gambit (1.PKB4? PK4!) Black threatens to win the rook`s pawn with check, and White is in a bad way.
The centre game: 1. PK4 PK4 2. PQ4 P×P 3. Q×P 000 4. 000 NQB3 is probably slightly better for Black. 3. Q×P threatens the QRP, but it might be poisoned, for if he takes it Black would reply NQB3 gaining time and the wing is secure.
On this board, as Troitzky showed, N + N can checkmate, with the aid of the K. Here is an illustrative endgame.

1. Nd7 Kg9 2. Ki7 Kh9 3.Kh7 Kg9 4. Ki8 Kf10 5. Kh8 Ke10 6. Kg8 Kf10 7. Kf8 Ke10 8. Nc9† Kf10 9. Nd9‡
Values of the pieces on this board: The rook is valued at 4½ pawns and it needs to be centralised, since its range is then greater. The mobility of the knight (measured by the average number of moves it can make from each square on the open board) is increased from 5.25 to 5.65 and of the bishop from 8.75 to 9.35 while that of the rook reduces from 14 to 13.76.
George Jelliss notes that alternative opening positions are possible in which all eight pawns are guarded initially, e.g. with K and Q in the annex and the other pieces moved inward one file, but castling, if permitted, would need to be redefined once again.
In this final idea I propose doing away with the board edges altogether! Without some restriction on moves, the kings could retreat indefinitely and no mate would be possible.
Rule (1): No group of n men may have more than √n cells between it and the nearest man. Where the root is taken to the nearest whole number.
This rule ensures the forces do not lose contact by drifting off into far separated groups. In the opening position the white and black groups of 16 are separated by √16 = 4 cells, which is thus at the required limit.
When n = 1 the rule implies that: No man may move to where it is more than two king moves from the nearest man to the arrival cell. This stops the mobile pieces riding off into the distance.
A further implication of the rule is that: A move that results in a group violating rule 1 is prohibited. Thus if piece A is the only piece two king moves from piece B, then piece B can only ‘orbit about’ piece A.
Initially the ‘board’, in the sense of all cells where a single piece may exist, is 12×12, but as the game progresses the outline may change. The cells to which actual single moves can be made will cover a smaller area than this, as illustrated here for the opening position.

Rule (2): A pawn promotes when it reaches a cell where it is not blocked, yet cannot move. This is because to move would take it more than two steps from any other man.
These concepts should also be extendable to Open Space Chess. Alternatives to rule 1 allowing capture by ‘stranding’ men, or groups of men, might be worth considering, but I`ve not found a workable formulation. With only two kings left whoever moves would win by moving away, which seems unsatisfactory.