“It is quite a three pipe problem, and I beg that you won’t speak to me for fifty minutes." If these three pipe-fulls were smoked within the stated 50 minutes and that 50 minutes was on-the-nose accurate, Holmes proves to be quite the prodigiously hyperactive ruminator. But is this an even possible achievement? Let's essay...
First, let's look at the pipe itself. His clay, unlike his long cherry-wood, bears no mention in canon as to its approximate length. Pipemakers of that period, however, seemed to have been cranking out 5-18" long offerings. Since we have no mention of neither long nor short as pertaining to Holmes' "disreputable" smoking device, let's split the difference by falling on the mean--at 11.5". This is still a tick on the long-side. A longer pipe means a cooler, slower smoke, but also a smaller bowl (less tobacco), usually. Finally, clay pipes are known to smoke hot. Read: fast.
Now, the tobacco. The cut of his shag then would be coarser and more irregular than today's of the same type. This would mean that it would tend toward burning slower. Adding to a guesstimate of pace could be the fact that he kept said shag in a Persian slipper. I outline why this seemingly odd container isn't necessarily a terrible idea in my article: "What Would Happen if You Were to Keep Your Cigars & Pipe Tobacco as Sherlock Holmes Kept His? (Sherlock Holmes Day 2021)" From there, this--
"Days would be how long pipe tobacco lasts in a cloth or leather tobacco pouch. Cloth or leather is what Persian slippers are made out of ... Let's go a step further in ideating. The pointed toe of a Persian slipper really packs in the freshness, and jamming some new shag down atop old, adds moisture." Days upon days, then? Conceivably. Also, Holmes smokes a goodly bunch, to say the least. Not much would be left-over to further dry-out at any given time.
So perhaps his tobacco wasn't overly dry. Of important note, pipe tobacco does not require the exacting, often painstaking humidification of cigars. In fact, many pipe smokers today, hold their stashes indefinitely in Mason jars, which were patented in 1858 by Mr. John Landis Mason. So if the Persian was doing damage to Holmes' keep, I'm sure he'd have employed other existent options. Suffice to say, his pipes probably weren't burning at any problematically rapid rate of speed.
So it would seem now that we have the gettable data in-hand. Being a professional smoker of sorts, I can offer up the hypothesis, again, all things considered (including my own experience as an almost exclusive clay pipe smoker), that it would take barely more than 25 minutes to smoke one pipe within these confines. Speaking of confines, Holmes was indoors, not outside where wind or mist could sway pacing one way or the other. I'll concede as much as 30 minutes per loaded pipe for argument's sake. Perhaps his thoughts distracted him from his puffings.
That would make his three (clay) pipe problem offer-up a runtime of roughly 90 minutes. In fact, to smoke all three within 50 minutes would mean sitting with each for only around 16 and a half minutes. We then must look at why this fails to add up. Sure, we can blame Watson's sometimes flimsy chronicling. We cannot, however, say Doyle didn't know his way around tobacco. Also of note is that Holmes is portrayed as being calm, even to the point of Watson wondering if he "had dropped asleep," as he smoked this pipe in REDH. Perhaps the "appendage" of his body was simply supplying the necessary fuel of his amazing brain's needs?
Quite simply it doesn't add up. Until, in Holmes' famed words: "When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - The Sign of the Four. Well, this doesn't exactly fit but eliminating the impossible does work here. And it is, as luck would have it, eliminated by itself. "when he suddenly sprang out of his chair with the gesture of a man who has made up his mind and put his pipe down upon the mantelpiece." his unfinished pipe, I'd say. Most likely seeing its remnants eventually dried and smoked in the next morning's dottle-filled bowl.
So, smoking three pipes in 50 minutes is not possible, to answer my own question rhetorically asked up-top. Here's the thing, however, Holmes asked for an exact 50 minutes and mentioned an inexact three pipes; perhaps that's all the 'baccy he had on hand? Later in REDH it becomes apparent that he had the case solved and the rest was leg-work to prove his assumption correct. Maybe he just used 'three pipes' as a means of describing meditation on a problem, but really meant that he needed precisely ten minutes to get ready in order to leave at the top of the hour.
"Sarasate plays at the St. James’s Hall this afternoon,” he remarked. “What do you think, Watson? Could your patients spare you for a few hours?"
::: very :::