My players stopped by a tavern near my setting's wizard college, and because I have a bunch of IRL college nerds at my table, I added a tiny detail I thought they'd like: a chalkboard sign advertising trivia night. I was a fool and forgot that most basic rule of DMing: if it isn't important, it will be all the players can think about. So...now I'm running a pub trivia session next week. Help.

First, I run an intrigue campaign, so the trivia night is a prime opportunity to make some NPC connections, overhear some rumors, and learn a little about the criminal underworld associated with their current quest. I don't intend the trivia portion to last all session, but my players are incredibly excited about it, so I want to do something substantive - and this is where I need help, because my setting is a bit tricky.

My campaign started a month ago and is set in a city whose residents have general amnesia, so the party doesn't have very much in-game knowledge. I do NOT want to ask a bunch of lore-dump questions and feed the answers to players with the best history checks, so I'm looking for ideas of functional minigames for the entire party to do in the context of a pub trivia competition.

Since the tavern is in the alchemy district and the party is an alchemist, a rogue/bard, and a scribes wizard, I thought maybe one of the pub challenges could be creating a new flavor of alchemical beer - make the trivia night a mashup of knowledge games, logic puzzles, and head-to-head practical challenges. Thoughts? Other ideas? Thanks!

Due to the location of the tavern being near the wizard college, it is reasonable that the proprietor expects to serve members of that college and will thus include several questions that pertain to the college's fields of study. This is an easy way to insert history/political sciences students and professors as NPCs. All the history professors I've met love retelling history to everyone they meet. Perhaps the party sits down next to one such professor who spends the entire night loudly trying to impress their friends by spouting history fact after history fact. 

As for minigames, its hard to do a literal trivia game because those usually include questions about geography and celebrities. Essentially, what would be common knowledge for locals, but unfortunately these questions are impossible for the PCs to answer. 

Instead I think it would be fun to include questions about medieval science and technology (describe the process of brewing mead?). Or if the players are into making cool character builds and know lots of 5e rules maybe considering trivia about that (at what level does a fighter get a third attack?).

The problem I have (personally) with puzzle-type situations, is it challenges the players, not the characters. (I know there are people who love them and get super-engaged with them, and that’s cool, too, but it’s just not for me.) And in this case it runs the risk of you vs. players instead of characters vs. NPCs, which is an extra complication. So if I were to run something like this, it would be a bunch of abstracted skill checks to keep it about the characters. Make some history checks, arcana checks, etc. maybe run it like a skill challenge to get to the next round and see how far they can go. 
Then as they distinguish themselves, between rounds you can have some key NPCs take notice and maybe introduce themselves for a little role play interlude, some side bets, personal challenges. They can certainly be plot relevant NPCs, or it could be they make a new friend/enemy/contact in town if they are impressive enough.

Quote from theologyofbagels >>

First, I run an intrigue campaign, so the trivia night is a prime opportunity to make some NPC connections, overhear some rumors, and learn a little about the criminal underworld associated with their current quest.

If that's the goal... make the trivia night a cover for the underworld syndicate to pass coded messages -- i.e. have a round of multiple choice questions where the answers form a cypher (that the party can then crack), that sort of thing

Basically have it work on two levels for the PCs -- playing the game straight, and trying to figure out what's really going on around them

Quote from AntonSirius >>

Quote from theologyofbagels >>

First, I run an intrigue campaign, so the trivia night is a prime opportunity to make some NPC connections, overhear some rumors, and learn a little about the criminal underworld associated with their current quest.

 If that's the goal... make the trivia night a cover for the underworld syndicate to pass coded messages -- i.e. have a round of multiple choice questions where the answers form a cypher (that the party can then crack), that sort of thing

Basically have it work on two levels for the PCs -- playing the game straight, and trying to figure out what's really going on around them

 This is perfection. First, because this tavern is in fact a popular spot for syndicate activity, and second because my players freaking love challenges like this. They keep immaculate notes and have conspiracy boards, so a coded cypher for them to solve would be right up their alley. I was trying to justify why practitioners of the Quiet Arts would congregate for a pub trivia game every week, and this is exactly the answer. Thank you!

Great ideas and advice from everyone on this thread so far. I really appreciate it!

My campaign started a month ago and is set in a city whose residents have general amnesia, so the party doesn't have very much in-game knowledge. I do NOT want to ask a bunch of lore-dump questions and feed the answers to players with the best history checks

I get this, but if the setup of the campaign is a city with general amnesia, consider that your players may actually think the trivia contest is crucial to the story.  So maybe not only a lore dump, but a clue may be in order.  In addition to any minigames.

Consider using this to drop the lore you need to drop on the players.

For example, if it's near this Wizard college, consider what important things they may need to know about the college in the future - when it was founded, by whom, who is in charge, how you can enter it, what festivals or celebrations they hold - and smatter these in with the other trivia. The purpose of their entering this quiz is unlikely to be to win it, and I would add common knowledge in too, so they feel like they did something, but also they realise that they don't know about the world. For example (pulled out of my >hat< as I type):

1: What is the common name for the Noctua-Ursa? (someone might know Ursa from Ursa Minor to mean "Bear". It's an Owlbear in latin.)

2: What breath does a red dragon breathe? (Fire)

3: In what season do the Wizards celebrate the harvest by engaging in a magical giant-vegetable battle? (Autumn, which should be guessable by the harvest, but this is an amusing lore drop for them to look out for in future!)

4: How many mages are there in the high council? (12, or whatever suits. No clues here, this is for them to learn!)

5: What drink is brewed using only honey and water? (Mead)

6: How many miles is it from here to >next town<? (30? whatever is true!)

7: Why is the forest to the North considered dangerous? (It's filled with magical experiments from the mages college called Rogues, which even children know of)

8: How many schools are there in >town<? (Answer is 1, with 3 crossed out on the answer sheet, as a minor plothook that the schools have been closed)

9: How many eyes does a Beholder have? (Answer is 11)

10: How many heads does a Hydra have? (trick question!)

this way when they get the answers called out, they will learn a bit about the place, and when answering, they will learn their ignorance - like a pub quiz normally works!