Is the Cluedo joke in Sherlock's "The Hounds of Baskerville" a random gag or is the game genuinely broken?

It is just a simple joke about Sherlock's stubborn refusal to consider that he might have missed the point, or to see things in anyone's terms but his own. You don't need to know the rules of Cluedo to get the joke ("Then the rules are wrong!" is enough), but I feel it adds a layer.

Watson is right: Sherlock's theory is simply comically impossible in the rules of Cluedo, and Cluedo is not "broken" - but it does have a problem with the kind of human errors that Sherlock hates to allow for.

Cluedo's gameplay hinges on players accurately responding to queries about the secret cards they are holding. If a player says they don't have a card which they have but absent-mindedly overlooked, it can break the game: other players may build logically-sound deductions on this that can cost them the game because they're built on a false premise.

This is a throwaway gag, but as someone who grew up playing quite a lot of Cluedo, it painted a very vivid picture for me:

• Sherlock had (maybe reluctantly) agreed to play a game with Watson (and probably others). Once he started playing, he got far too immersed in the murder-mystery premise
• He built elaborate deductions on every move another player did, as a good Cluedo player does. Unfortunately, someone at some point made a human error, so his logically-sound deductions took him to an impossible place.
• He obviously would not ever consider that his logic could have been at fault
• Normally at this point, the keen-eyed player would say "Hang on, that's not possible because [list of six observations from the last 10 rounds] means you can't have X card" then someone would say "But I do have X card. Oh, I must have made a mistake 10 rounds ago" and it gets very awkward, the game falls apart, and it possibly ends in an argument.
• However, Sherlock being Sherlock, it appears he got so absorbed in the murder-mystery premise of the game, he lost track that this was just a fun social activity between fallable humans, where possibilities are limited to which cards actually exist and "a player messed up" is a valid possibility.
• In keeping with "Once you have ruled out the impossible, whatever is left, nomatter how improbable, must be the truth", he constructed an elaborate narrative that made sense within the imaginary fictional universe of the game, but not in the real-life scenario of some not-quite friends awkwardly trying to play a board game with a high-functioning sociopath.
• Watson tried (and failed) to reason with him and save the game; but also had to be careful to avoid exposing whichever player had made the human error, because you can just imagine how Sherlock would have treated them.

It's a throwaway joke that conjures an amusing image of a long-suffering Watson trying to sustain a normal social event around Sherlock's peculiarities.

I'm not 100% sure it actually works though... Human errors in Cluedo tend to result in false certainty of an option. I can't see any way someone in a Cluedo game could logically rule out all options of a type, as this line implies.

Like a lot of details in Sherlock, it's best enjoyed if you think about it a little, but not too much.

From what we know about the character, I suspect Sherlock might not have bothered trying to follow the rules in the first place, or indeed to even learn what they were.

While most of us might think of Clue(do) as an entertaining game of deduction, for Sherlock, playing it the way it's intended to be played would likely be incredibly dull. So, rather than play along - going turn-by-turn, asking who has which cards and so on - it's quite plausible Sherlock would try to approach the case as he would any other: by observing the suspects, crime scene, "police report", etc... Basically, all the "flavor" that's largely meant to be ignored.

I like to imagine a group of folks gathered around the board, their initial enthusiasm fading fast, as over Watson's futile attempts to intervene Sherlock delivers an extended, rapid-fire monologue:

... Naturally the murder weapon couldn't have been the dagger. Even the police aren't so incompetent as to miss traces of blood that ...

... couldn't have tied the knot, which was clearly made in a hurry by someone right-handed, whereas from his tie it is obvious that Mr. Green is, in fact, left-handed ...

... in that short time Miss Scarlett couldn't have possibly made it to the library undetected, all the way from the lounge where she had been chatting up Professor Plum (rather pointless, as it's apparent any interest he may have in women is purely academic) ...

... you're asking yourself: "Then who set the dining table"? Wrong question. What you should be asking is why. Well ...

... Therefore, we must conclude that this was no murder at all, and that the perpetrator was, in fact, Dr. Black himself!

This does bring up one question: if Sherlock has already "cracked the case", why would he propose playing it again? But I'd say it's more than possible he paid so little attention to the rules ("unnecessary information" not worth storing) that, in his mind, the game could well deliver a new set of facts each time.

Cluedo (Clue in the US) was developed during WWII by a musician who was trying to pass the time during air raids. It was developed as a Murder Mystery Game in the vein of Agatha Christie novels. As such, it's impossible for the person who died (Dr. Black in the UK, Mr. Boddy in the US) to have committed suicide. To make this obvious to the players, the victim has no playable cards. I.E., Mr. Boddy can not have killed Mr. Boddy in the Study with a rope.

The gag allows for the conclusion that in an earlier Cluedo game, Sherlock made an error in that he somehow eliminated everyone, including the murderer, from being a potential suspect. Hence his statement that it was the only possible solution.

The UK rules explicitly state, "This evening Samuel Black was found murdered in his mansion!" The presumption by Sherlock's statement is; if it was a suicide, there is no murder and therefore the rules are wrong.