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. (00:14:30)
Lisa's Date with Density - S8-E7
Continuity mistake: When Jimbo and Co are 'coleslawing' Skinners house, he wakes up when coleslaw hits his window. When the shot moves to outside, there isn't any coleslaw on any of the windows.
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.
Continuity mistake: When Wolfcastle says that 'Jay Sherman insulted him, now he will die', for a few shots the shadows are on the floor before they magically disappear. Also, the shadow of the gun does not match the real gun.
Continuity mistake: When Homer is driving home in a bad mood (just before he gets stopped by Chief Wiggum), you can see that there is a pattern on the side of the car. In a next shot when Wiggum is walking alongside the car there is no pattern.
Continuity mistake: When Homer sees the hotdog that Duffman holds up near the end of the episode, the order of the condiments on the hotdog change in subsequent shots.
Continuity mistake: After Bart attempts to escape by using spray paint cans as rockets, his feet alternate between clean and spray-painted for a few shots.
Continuity mistake: At the very end of the episode, when Homer is being carried off by the crowd, you can see that the crowd carrying Homer, the people are small, so you'll have to look closely, but right before the credits, Marge's necklace changes from red to white, back to red and then finishes with white all in one shot. (00:21:00)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Millhouse, Bart and Lewis are talking about the fake Santa, Millhouse's eyebrows go from blue to black to blue again. (00:14:15)
Continuity mistake: When Bart and his friends are making copies of the photo of Homer, the banner across the copier says "Copies 5 cents," but when they go to deposit the money it says 10 cents.
Homer's Barbershop Quartet - S5-E1
Audio problem: When the Bee Sharps are singing 'Goodbye my Coney Island Baby' next to the Statue of liberty, you can hear Chief Wiggum's voice singing with the other members of the group, but he got chucked out of the group earlier on in the episode. (00:14:20)
Homer's Triple Bypass - S4-E11
Continuity mistake: Homer is interrupted from praying by a nurse who comes into his room and says, "Ssh." However, when she says this, the door is closed, and it is only in the next shot where the door is open immediately and she is pointing to the NO PRAYING sign. (00:14:20)
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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