This is one of the so-called "classic" locked room mysteries, in which an appparently impossible murder takes place. In fact, lead sleuth Dr. Gideon Fell lectures at one point about the myriad locked room crime methods. The actual solution to the murder at hand is so convoluted it made my brain hurt; much more enjoyable is the interplay among the characters.
A wonderful book from a writer whose mysteries range from top notch (this one) to basically unreadable.
This is generally considered the zenith of locked room mysteries, and contains the famous "locked room lecture" in which Gideon Fell enumerates the various ways in which a murderer can do his deed and make it seem impossible.
In a poll conducted for his anthology "All But Impossible," Edward D. Hoch asked a number of mystery writers to name the best miracle-crime novels, and "The Three Coffins" topped the list (Second was Hake Talbot's "Rim of the Pit."