story identification - Alien communication via a Chess Knight piece

I'm looking for a short story, most likely written before 1985 about an alien being able to communicate with a man only because he was playing chess (by himself?) and thinking about his Knight move.

I vaguely recall:

1. The chess game was at night.



2. The man was sleepy?


3. That it was the Knight piece is important, something to do with the alien physiology


4. It was definitely a short story not a whole novel. Part of an anthology perhaps.

Answer

Might be "The Exploration of Space" (1972) from the collection The Knights of the Limits by Barrington J. Bayley. Summary here:

The first person narrator, under opium-induced haze, is visited upon by a knight on a chess board. Evidently, it has arrived in a space-ship from a "locational-transitional" space. The ensuing conversation ranges over a few theories the author has about other types of possible spaces of existence. Einstein's field equation of General Relativity makes an appearance in tensor form. There is a minor discussion about the abstraction of numbers and a cursory paragraph about Cantor, transfinite numbers and transfinite spaces.

The first few pages are viewable on google books here. You can read the opening section where he is contemplating the possibility of universes with discretely organized space while looking at a chessboard under the influence of opium, then the pieces suddenly start to move around rapidly on their own, and the knight establishes communication within him, from which he learns the following:

My sense of excitement returned, however, when he went on to explain that he was not a space explorer such as our imagination might conjure by the phrase, but that he was an explorer of alternative types of spatial framework of which, he assured me, there were a good number in the universe. What we are pleased to call the sidereal universe, that is, the whole system of space-time observable by us on Earth, is merely one among a vast range of various systems. Even more astounding, in the circumstances, was the revelation that the Knight hailed from a system of space identical to that which I had a moment before been contemplating! One analogous to a game of chess, where space, instead of being continuous and homogeneous as we know it, was made up of discrete locations, infinite or at any rate indefinite in number, and to which entities can address themselves instantaneously and in any order, There is no extended spatial framework in which these locations are ordered or arrayed and all locations are equally available from any starting point (provided they are not already occupied). An entity may, however, occupy only one location at a time and therein lies the principle of order in this well-nigh incomprehensible world. Structures, systems and events consist of convoluted arabesque patterns of successive occupations, and of the game-like relationships these manoeuvres hold to one another. The chess-people's analogy of a long distance takes the form of a particularly difficult sequence of locations; alternatively the sequence could correspond to a particularly clever construct or device — the chess-people make little distinction between these two interpretations.

As do the occupants of a chessboard, the entities of this space (which I shall term locational-transitional space) vary in the range and ingenuity of their movements. Primitive organisms can do no more than transfer themselves slowly from one location to another, without pattern or direction, like pawns, while the most evolved intelligent species, like my friend the Knight, had advanced to dizzying achievements as laid down by the possibilities of such a realm. Their most staggering achievement was that of travel to other spaces; this was accomplished by a hazardous, almost infinitely long series of locations executed at colossal speed and comprising a pattern of such subtlety and complexity that my mind could not hope to comprehend it. Indeed, few even in the Knight's spatial realm comprehended it and for their science it constituted a triumph comparable to our release of atomic energy from matter.

The discerning reader who has followed me this far might justly wonder at the coincidence which brought these bizarre travellers to my presence at the very moment when I had been theoretically contemplating something resembling their home space. This question was uppermost in my mind, also, but there was, the Knight told me, no coincidence involved at all. On entering our continuum (which the Knight and the companions under his command did indirectly, via other realms less weird to them) the space explorers had become confused and lost their bearings, seeming to wander in a sea of primeval choas where no laws they could hypothesise, not even those garnered in their wide experience of spatial systems, seemed to obtain. Then, like a faint beacon of light in the uncognisable limbo, they had sighted a tiny oasis of ordered space, and with great expertise and luck had managed to steer their ship towards it.

That oasis was my chessboard. Not the board alone, of course — tens of thousands of chess games in progress at the same moment failed to catch their attention — but the fact that it had been illumined and made real by the thoughts I had entertained while gazing upon it, imbuing it with conceptions that approached, however haltingly, the conditions of their home world. Hence I owed the visitation to a lesser, more credible coincidence: chess and opium. At any rate, having landed their ship upon the board and thus bringing it under the influence of that vessel's internally maintained alien laws, they had carried out simple manipulations of the pieces in order to signal their presence and establish communication — the real ship and its occupants not being visible or even conceivable to me, since they did not have contiguous spatial extension.