Audio problem: When Eddie and Lou are taking Bart for a ride in a police car, Eddie says "I tell you, they only come out at night" the audio does not match what he is saying. (00:06:10)
Homer's Triple Bypass - S4-E11
Continuity mistake: When Krusty takes off his shirt, the superfluous third nipple and cattle skull birthmark shown in "Kamp Krusty" are not on his chest. (00:14:50)
The Ziff Who Came to Dinner - S15-E14
Factual error: When the SEC guys break up the poker game they arm their guns in the same manner you would with a shotgun (there is even the shotgun cocking sound effect) yet they are M-16 style weapons and would not need to be armed this way.
Like Father, Like Klown - S3-E6
Continuity mistake: When Krusty has a cigarette during the cartoon just before he is reunited with his father, he keeps his cigarette pack in his shirt pocket, 'close to his heart'. When Krusty and his father hug, the cigarette and the packet are gone.
Continuity mistake: When Homer is kicked out of Moe's the window behind Homer suddenly opens so Moe can throw the record out.
Factual error: If Homer's stomach was really caught on the towel rail the way they show it to be, he wouldn't be able to see the scales at all without having to look round it.
Who Shot Mr. Burns? (1) - S6-E25
Continuity mistake: In this episode, Mr Burns and Smithers are on the balcony of Burns' office. Mr Burns says "It will be like taking candy from a baby. Hey, that sounds like a laugh. Let's try it right now." We see Smithers, whose expression doesn't change as the two go back into Burn's office. In 'Who Shot Mr Burns Part 2', Smithers remembers this event, except now he looks quite shocked when Burns mentions about taking the candy.
Milhouse Doesn't Live Here Anymore - S15-E12
Continuity mistake: Right at the end Lisa is holding a doll head and some cards. There is a very brief shot of Bart, and then when it's back on Lisa she doesn't have anything in her hands when they hug. There wasn't time for them to be put down and you cannot see them anywhere either.
Hunka Hunka Burns in Love - S13-E4
Continuity mistake: In one of the shots after Mr Burns has broken the fortune cookie, the remains of it vanish from the table. There is something else on the table, nearer Smithers, but it and the cookie are very different shapes, so it's very difficult to confuse the two.
Springfield Connection - S6-E23
Continuity mistake: When Marge walks into the hair salon, she is carrying a white bag. But in the next shot when she talks to Mrs Krabappel, the bag is gone.
Much Apu About Nothing - S7-E23
Continuity mistake: When Nelson pulls back Uta, he drops a book. In the next shot, the book has gone.
Much Apu About Nothing - S7-E23
Continuity mistake: When Apu and Lisa go to the 'Immigration Centre', they pass Dr Hibbert on the right, and he has yellow skin. In the next shot, his skin returns to its proper brown colour.
Continuity mistake: Homer goes to ring the phone number '1-909-WIN-BIGG' but we see he only presses four dials, and that's all we hear too.
The Twisted World of Marge Simpson - S8-E11
Continuity mistake: At the fair, Helen Lovejoy picks up a pita and bites a corner off it. There are a few angle changes and when it goes back to her and you can see her pita looks untouched. It's unlikely she just picked up another one or finished a whole one in that time.
Continuity mistake: The Simpsons' bathroom has changed position. In 'Today I am Klown' it was opposite the top of the stairs, at the back of the house; in this episode it is at The Front of the house. (00:00:35)
I, (Annoyed Grunt)-bot - S15-E9
Other mistake: When the bullies are riding around Bart and Milhouse there are problems with their water bottles. In some frames they vanish, in some others the bullies' far legs are drawn in front of the bottle. (00:01:40)
Continuity mistake: When Krusty ends up at his old neighbourhood, the street changes between shots. E.g. the "LLBeanie" sign appears above the door of the 2nd shop and the suit of the man reading the newspaper changes from grey to blue. (00:05:25)
Lisa's Pony - S3-E8
Continuity mistake: When Homer is phoned by Lisa, the color of the phone is blue. Immediately after, it turns to green. (00:19:00)
Continuity mistake: When Homer enters Moe's tavern, Moe is playing solitaire. The cards on the table are face down/face up between shots. (00:04:45)
Tree House of Horror II: A Simpsons Halloween - S3-E7
Continuity mistake: When Bart and Lisa argue over which wish to make, the hand on the table changes position, and the thumb switches sides. (00:03:30)
Answer: I'm not sure if this is the one you're thinking of, but an episode of "Ray Bradbury Theatre," called "A Sound of Thunder," dealt with a similar matter: a group of hunters travel back in time to hunt dinosaurs, only to find things have changed when they get back because someone stepped on a butterfly.
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