Under a rather thin narrative coating, in which Sherlock Holmes matches wits yet again with the fiendish Professor Moriarty, this book is a rigorous treatise on and exercise in logical deduction -- a very Holmeslan sort of activity except that the "language" used in setting and solving the problems is the game of chess.

From a passing reference to this activity in "The Adventure of the Retired Colourman," ("Amberley excelled at chess -- one mark, Watson, of a scheming mind"), it has been generally assumed that Holmes despised the game. But new evidence has been coming to light in recent years (outside of the Conan Doyle canon, to be sure) to cast doubt on this easy assumption.

There is, for example, "The Moriarty Gambit," a short story by Fritz Leiber (available in "Chess in Literature," Avon/Equinox paperback, $4.95) in which Holmes administers to Moriarty a particularly elegant epaulette mate duringd the 1883 London International Tournament.

Leiber makes a persuasive argument in a statement by Holmes to Watson: "You knew I reveled in crypotograms, problems, and analysis of all sorts. Moreover, it ought to have occurred to you that no strongly intellectual man who used the anodynes of morphine and cocaine could possibly have failed to try the supreme mental anodyne of chess."

Now, Raymond Smullyan, a professor of mathematical logic and philosophy at the City University of New York, has provided an abundance of that anodyne -- 50 problems in retro-grade chess analysis that are considerably more powerful than cocaine or morphine and also less expensive (under 10 cents per fix in the paperback edition).

For confirmed addicts, there is also a hint of a promise that more may be in store. A key element in the book's plot is a manuscript of yet another book, "entitled 'Arabian Knights,' by an unfamiliar author, Nayllums Dnomyar." It is a collection of problems in retrograde chess analysis, and Watson remarks at the end of "The Chess Mysteries of Sherlock Holmes" that "much mystery surrounds its origin":

"The author's name appears to be a nom de plume, though we are not sure of this. The handwriting is in spots illegible, and we can only conjecture the meaning of some of the passages. Either the author never gave solutions to the problems, or the solutions were somehow separated from the rest of the manuscript and lost, so Holmes and I have to work out most of them. This and much other work remains to be done, but eventually we intend to publish this manuscript."

In comparison, "The Chess Mysteries of Sherlock Holmes" is light reading. Evidently Smullyan is a more considerate author than Nayllums what's-his-name. He gives solutions to all the problems he poses, and he wraps his exercises in a thin but engaging narrative. It should be noted, however, that Holmes and Watson, in this book, keep running into chess situations as frequently, vividly and improbably as the people in mystery novels run into corpses.

A final question may remain in the minds of one or two readers: just what is "retrograde chess analysis?" It is a kind of problem in which the principles of deductive logic are applied to a chess situation -- not usually to find a winning move, as in the more common and simple-minded chess problems, but rather to answer one or more questions about what must have happened in the game to bring it to the present situation. I will close with a sample. It is not taken from the book (it is from the October issue of "Capital M," the newsletter of Metropolitan Washington Mensa), and it poses as a regular chess problem while it is actually a problem in retrograde analysis.

Without retrograde analysis, this problem is baffling -- here seems to be no single move with which white can administer checkmate. But to a retro-grade analyst, the question is not how to win but what was black's last move. Clearly, that move could not have been made by the bishop or by the pawns on the a, d or h files. Less obviously but also clearly, it could not have been made by the immobilized black king, so it must have been the c-pawn, and it could not have been from the c6 square because then the white king would have been in check. Once it is determined that black's last move was from c7 to c5, white's winning move becomes obvious: pawn takes pawn en passant, mate.

The problems posed by Smullyan are considerably more abstruse and complex (and would take more space to examine than is available). But addicts of puzzles will turn to them as avidly as Holmes to his needle.