Train Problem

The official math problem of TV Land, meant to resemble grade school mathematics problems. "If train A leaves Tropetown at 11:00 towards Idiomopolis that is 32 miles away, moving 50 miles an hour, while train B leaves from Idiomopolis towards Tropetown at 60 miles an hour at 11:30..."

The question is either when/if the two trains will reach a certain location, or numbers of passengers at time/place Z. (Where at each station, the movement of people on and off the train is described with bits of math). Occasionally subverted by making the question irrelevant to the setup details. The third type is where/when the two trains will meet, in which case it is almost certainly Denver. Whatever the intended answer, it's quite likely someone will start asking for entirely irrelevant details.

One variant that pops up occasionally is two trains leave from City A and City B at different times at different speeds. Which train is closer to City A when they meet? If the trains have met, they're in the same place, so neither is "closer" to City A.

Another variant is to have the question answered in a humorous or joking way, revealing the problem was not mathematical but merely the set-up to a humorous trick-question. Although not train-related, a common example is something like: "One man leaves Tropetown going 80 mph. Another leaves Idiompolis going 75 mph. Where do they meet? IN JAIL! Because the speed limit is 55!"

Often times, vital pieces of information are left out, such as the distance between the cities, or that Train B makes a 20 minute stop in Clicheville. To show a character is smart they'll come up with these vital missing facts as part of their answer. The most common answer is normally something along the lines of "Who is the idiot that designed this railroad?"

Examples of Train Problem include:

Comic Books

Very common in Zipi y Zape, where they get those as asignements half the time and one entire long story is set around solving one. They do this when they aren't asked simple multiplication. Which they have trouble with anyway.

Film

In Wayne's World 2, in a vague dream sequence: "Ask me a question", "Ok, if a train leaves a station...", "No, a question about your life"
Mentioned during Roman Moronie's trial in the movie Johnny Dangerously.

Maronie: That's a farging trick question!

Used in Deskset with Spencer Tracy and Katherine Hepburn; once seriously, a second time as a joke when they were all a little tipsy during the Christmas Party.
The problem is poked fun at with a non-train variant in Sky High as something of a Funny Background Event. Will is having a quiet, introspective moment while his friends are in the background doing homework.

Ethan: Your hero flies north at 300 miles per hour for 15 minutes. His archenemy is tunneling south at 200 miles per hour for 10 minutes. Assuming your hero has X-ray vision, how long before he realizes he's going the wrong way?

Literature

A variation of this problem, using riverboats instead of trains, is faced by Lawrence Pritchard Waterhouse when he joins the navy in Neal Stephenson's Cryptonomicon. He answers with a multi-page dissertation on the problem, attempting to take into account such variables as the speed of the current at different points of the river and the effect it has on the absolute velocity of the vessels. This isn't Obfuscating Stupidity of any kind; the actual Train Problem is so beneath his genius that it doesn't occur to him that it could challenge anyone, so he honestly assumed that he was supposed to consider the effects of the current on the vessel and the windspeed on its cross-section. He sends a copy of his answer to a mathematics quarterly and gets it published. Meanwhile, the navy has put him in the ship's band because their standardized test has shown he's too dumb to do anything else.
The Dodecahedron presents a similar puzzle to Milo, Tock and the Humbug in The Phantom Tollbooth, but one that contains cars, roads, and insufficient information. Tock solves it anyway.
One of the Shadowrun short stories in Wolf & Raven starts out with Wolf about to be run down by a car, and feeling like he's trapped in "one of those math problems".
A (nonfiction) book parodies of one of these was used to demonstrate how the two hemispheres of your brain work: (paraphrased)

"Train 1 leaves (City A) towards (City B) at 10:45 and travels at 55 miles per hour, while Train 2 leaves (City B) towards (City A) at 11:15 and travels at 45 miles per hour. How much is the lunch special on the second train?"

Live-Action TV

This happened at one point on Dharma and Greg.
Sabrina the Teenage Witch also used it, with the added twist that she had to then prevent them from colliding....with her house. (She fails)
One of the characters on the The Wire tries to use one of these questions (substituting cars) for his 8th grade math class, but cannot finish the question because the students demand to know what types of cars they are and what neighborhoods they left from.
- In an earlier season, a juvenile drug runner can't solve a Train Problem, but when it's recast in terms of drug purchases and resupplies, he answers it perfectly. When asked to explain this, he replies, "Count be wrong, they fuck you up."
On Saved by the Bell, one of these problems turned up on the SAT.
Star Trek: Enterprise "Shuttlepod One" Trip starts rattling off a Train Problem while trying to determine what's wrong with the pod, only to exclaim that he never could work those kind of questions out. Our Chief Engineer, ladies and gentlemen.
- Sci Fi Debris was greatly annoyed at this event. The chief engineer of Earth's most advanced starship can't do one-dimensional algebra.
In the episode of NCIS where Abigail Breslin guest stars, a hostage is being held somewhere near railroad tracks and the team's hacked web cam tells them when a train passed her. When they start doing the math to figure out where the train was at that time, Tony admits he'll have to apologize to his high school math teacher, whom he told he would never have to use this problem in Real Life.
"A Virgin train is traveling at 120 miles per hour between London and Manchester, at what time will it be canceled?"

Music

Cirno's Perfect Math Class (and the accompanying Fan Vids) does the "getting on/getting off" version with a bus in Gensokyo. The answer is zero, because there aren't any buses in Gensokyo. Then a train runs over her.
Makes an appearance in the final moments of "Fernando"'s confused ramblings at the end of "You Look Mahvelous!" by Billy Crystal:

If a train is leaving Chicago at 11 AM in the morning, with five thousand pounds of bluefish, and another train is leaving Phoenix at exactly the same time, and Bert Convy is still hosting Tattletales...

Newspaper Comics

In one Far Side strip, titled "'Math Phobic's Nightmare'", has a man at the gates of the Fluffy Cloud Heaven, with an angel asking him this question.
Calvin and Hobbes: Calvin's method for answering a question of this sort at school is to turn it into a detective mystery in his head. It doesn't quite work, as the answer he comes up with is "a billion".
FoxTrot did one where Jason listed all the "assumptions" he had to make to give them the answer they wanted (such as the train remaining on the track, not going "as the mole digs", all clocks being accurate, ignoring relativistic effects, etc.)

Radio

Adventures in Odyssey: Connie reads some kids in Whit's End a riddle about an electric train that involves the speed and direction of the train and of the wind. The question is, in which direction is the train's smoke blowing? The answer, of course, is that electric trains don't have any smoke. Connie explodes when the kids don't get it.
Often pondered by the navigation officer, Sub-Lieutenant Phillips, on The Navy Lark (among many other things, including references to Noddy's Big Book of Boats). The answer is usually, "Left hand down a bit."

Recorded and Stand Up Comedy

Endlessly parodied by satirists when given any opportunity to put the boot into the people running the train service.

Web Animation

One Homestar Runner sick filler had Homestar subbing in for Strong Bad's "word problem" and using one of these.

Web Comics

Played with in this strip of A Modest Destiny, with a problem involving "Horse A", "Horse B", "Stop C", "Rider Z", "Cow Q", "Town LMNOP", "fuzzy dice J" and "coffee C".
One of the more recent strips in the Jackie's Fridge universe has Miss Masters teaching at 'Coleman Elementary', and she starts to use this problem on her kindergartners, stops, and rephrases it to "If J.Lo and Ben Afleck leave their respective movie shoots, and head toward each other, going x MPH..." etc. The class can solve that one.
Irregular Webcomic 1611

Western Animation

The second episode of The Simpsons, "Bart the Genius," had it as one of the questions on the intelligence test. Complete with Dream Sequence.
Done in one episode of Dog City with two protagonists tied up on the bridge that both trains would soon cross.
One of the questions used in Darkwing Duck to overload the brain of a super-genius who used a special ray gun to increase his IQ when he had a clear head. DW and Launchpad bombarded him with questions while they fired the ray at him, causing his head to a-splode.
In an episode of Clone High dealing with mandatory testing, a Magical Truckdriver gives Gandhi a Train Problem - actually a Trucker and Randy Housewife Problem, asking him to calculate where the two could meet for a "guaranteed" encounter.
This was the question that caused Doug to fail a math test.
When Oscar Proud decided to help his daughter with her math homework in The Proud Family, this question comes up. He concludes that both trains will meet at the crash site.
An episode of The Grim Adventures of Billy & Mandy had a temporary intelligent cat (It Makes Sense in Context...kind of) trying to tutor Billy with a problem like this, but Billy's more worried about what color the train is and who's on it ("Can it be clowns?"), rather than anything relevant to the problem.
The Powerpuff Girls once had to face one of these problems in order to figure out where a train collision might occur. As they struggle to find an answer, Blossom delivers a solution: "We're superheroes! Let's just find the two trains and stop them!"
- Him also gives Blossom this riddle in a Your Worst Nightmare scenario he's conjured for her, as her worst nightmare is failing an upcoming test. The question following the set-up is, "Which train will get to Cuba first?"

Blossom: Neither -- trains don't go to Cuba!

In one episode of Big Guy and Rusty the Boy Robot, Rusty ends up fighting an enemy in cyberspace- the Quark building's mainframe, to be precise. In order to buy him some time until Big Guy can be uploaded and help out, Dr. Slate freezes the computers (And the action) by inputting one of these problems into the system. She reveals that that she got kicked out of a computer club for doing that.

Video Games

Larry Butz of Phoenix Wright: Ace Attorney mentions that he's terrible at these problems when he's trying to change the subject to avoid a question.