The most interesting aspect of "The Royale" is its math-history footnote concerning Fermat's Last Theorem, still unproved in 1989 when the episode was made, and still allegedly unproved in the 24th century. Who would've guessed then that the theorem, after more than 350 years, would be proved in 1995?
I mention that footnote in an episode that otherwise exhibits almost no interest or merit. After the Enterprise crew finds a piece of NASA space debris from the mid-21st century and traces it to a nearby planet, Riker, Data, and Worf beam down and enter a mysterious building. Inside they find an alien representation of a 20th-century casino hotel as based on a "second-rate novel" owned by the NASA astronaut that had survived, and used as a template by aliens to build him an oasis in the middle of a barren, unlivable environment. It makes for a classic TNG Pointless Period Piece.
The first sentence of the novel was, "It was a dark and stormy night," and the episode makes much of the fact that the novel is a piece of trash with shallow characters and endless clichés. I suppose this is to cover the fact that "The Royale," as an episode of TNG, has shallow characters and endless clichés. Seriously, it must've been a hell of a writers' meeting: "Let's do an episode that's about bad clichés and lame dialog so we don't have to write something that's actually good!" (Apparently, they figured that by pointing out that the storyline is dreck, that somehow excuses it.) The away team becomes trapped in the hotel and can't escape, for no reason except that this is a Twilight Zone-style mystery that has arbitrary rules and no satisfactory explanations.
I suppose I could excuse a fantasy show if it were entertaining, but not this one. There's no mystery or wonder or suspense, but merely bad clichés, pointless guest characters, aimless dialog, and a plodding premise that never comes close to justifying its fantasy elements. (And, no, Data playing craps isn't enough.)