The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Homer and Flanders drop their presents, the presents move around and change colours between shots. (00:10:20)
Factual error: When Rod and Todd Flanders are roasting marshmallows in the car, Todd's marshmallow was the only one above the lighter, even though both of them have turned black. (00:17:40)
Continuity mistake: John is wearing Homer's old "Pin-pals" bowling shirt when the Simpsons first meet him, but only after we see the back of the shirt does the name "Homer" appear over John's breast-pocket; clearly there was nothing there during their entire previous conversation. And even if this is a deliberate "mistake" for comic purposes, in 7th season episode"Team Homer", the very same bowling shirt is ripped to shreds when Homer is being mauled by Mr.Burns' dogs at the very end.
Tree House of Horror II: A Simpsons Halloween - S3-E7
Audio problem: In the opening when Marge is warning us about how bad this years episode is, her mouth barely matches what she says.
Homer vs. Lisa and the 8th Commandment - S2-E13
Continuity mistake: When Homer and Marge are discussing about unhooking cable, the time on the clock on Homer's bedside table changes between shots.
Homer vs. Lisa and the 8th Commandment - S2-E13
Continuity mistake: When all the kids are watching TV, we see Wendell, Martin and Milhouse. Absurdly they appear to be right below the ceiling in this shot - when in every other shot they are sitting on the sofa or standing.
Burns Verkaufen der Kraftwerk - S3-E11
Audio problem: When Horst is checking the power plant, he sees some radioactive liquid pouring out of the chimney, he says "That shouldn't be" but his mouth doesn't move when he says "be". When he spots the raccoons he speaks but his mouth does not move at all. And at the end when Mr Burns is mocking the German men, Horst says "Burns, stop it." but his mouth doesn't move.
The Last Temptation of Homer - S5-E9
Continuity mistake: When Homer and Barney are at Moe's, Barney drinks his beer and then his cup his empty, and Homer's is full. Then later, Homer's cup is empty and Barney's cup has magically refilled itself.
I'm Goin' to Praiseland - S12-E19
Continuity mistake: When Flanders' girlfriend pulls up at the end, from one angle she's wearing a pink shirt with what looks like a white vest over top. But from another angle her shirt is all white.
Tree House of Horror VI - S7-E6
Continuity mistake: In the 'Nightmare on Evergreen Terrace' segment, in Bart's dream you see a shadow of what looks like someone with four knife-fingers. Then when we see who it is, it is Willie holding a rake. But the rake has five tines, not four, so the shadow is wrong. (00:07:00)
Continuity mistake: When Marge arrives after the song, Flanders and Lovejoy are talking to her, and behind them you can see a crowd of people. The crowd then disappears in one shot, only to reappear a shot later.
Homer Badman - S6-E9
Continuity mistake: Homers car gets blown over by a helicopter, and gets dented, and all the windows smash. But in the next overhead shot the car is parked on the side of the road, perfectly intact.
Audio problem: Just after Homer says "About 3 Weeks" in reply to Bart's question about how long Grandpa lived with them, the family all start laughing. However, the second time Homer laughs, his mouth doesn't move.
Days of Wine and D'Oh'ses - S11-E18
Continuity mistake: When Moe tells Homer to sit on Barney's seat in front of the beer tap, Homer is sitting on a stool next to it. Homer suddenly switches seats between shots and the other seat disappears.
Continuity mistake: In the episode which recalls how Homer and Marge got married, Homer writes a note to Marge on a piece of paper entitled 'From The Pen Of Patty' he SCRIBBLES out Patty's name and writes his own, but when Marge reads the note, 'Patty' only has a single straight line through it. The word "HOMER" also changes.
The Old Man and the C Student - S10-E20
Continuity mistake: Springy the Olympic spring that Homer makes has 9 coils. Then we see the large version that has many coils, at the Olympics meeting. Homer's original spring is in his hand and now it has many coils also. When it cuts back to the stage, the large spring has few coils, like the very first time we saw Springy.
Springfield Connection - S6-E23
Continuity mistake: When the Simpsons are eating and the family are praising Marge, Homer's plate is first seen to have a small amount of yellow food and a bit of pink food (meat?).In a later shot when Homer steals and uses Marge's pepper spray he has a large pile of yellow food.
Tree House of Horror II: A Simpsons Halloween - S3-E7
Continuity mistake: While Bart is in school, Sherri and Terri are making him comfortable. One of the twins vanishes for one shot, and their skin color keeps changing from white to yellow. (00:11:05 - 00:12:15)
Tree House of Horror II: A Simpsons Halloween - S3-E7
Audio problem: Right after Flanders makes his first wish, when Moe chases the alien, Moe is talking, but his mouth doesn't move. (00:08:00)
Continuity mistake: At the Penitentiary gates, Nelson's position changes between shots. The whole group are near the gates, then far away, then Nelson is separate from the group when he throws the key.
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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