All complex problems are solved by sudden epiphany!
In every Locked Room Mystery, the detective can't solve the crime just by examining the relevant evidence. They always need some external inspiration (apparently) completely unrelated to everything, something along the lines of:
Sharona: Oh well, looks like you don't solve the crime. Whatever. Let's go get milkshakes. Monk: Milkshakes? The killer is lactose intolerant! So is the victim's ex-wife! Of course!
Bob: No tea please. Dave:That's how that letter found its way twenty miles to a dead man in Sudbury!
This will lead directly to The Summation, unless there's an Evidence Scavenger Hunt in between. Often the character having the epiphany will tell the person whose offhand remark inspired it that they're "a genius" or the like; the remarker will then variously nod in a befuddled manner, ask "I'm what?", or simply demand an explanation.
Named for perhaps the most famous non-detective related example, Archimedes' exclamation of "Eureka!" after jumping into a bath and realizing that held the key to the problem he was trying to solve. (See below for details.)
The Eureka Moment shows up a lot on diagnosis-mystery medical shows, such as House, in which he does it in nearly every episode, and in the first episode of Grey's Anatomy, which isn't even a medical detective show! It may also show up on telenovelas or soap operas with some ongoing secret that defines a character or even the main plot. The truth is often almost revealed several times, by easily overhead conversations or weak evidences, and each time the status quo is kept with some contrived explanation. When the Eureka Moment takes place, The Reveal is not the result of a confession or an evidence that is too solid to be ignored, but instead the result of the hero putting all the small hints toguether and figuring out the truth by himself. In this case, expect a wave of several flashbacks of those hints before the "Eureka!".
Not to be confused with a Bat Deduction. While both can initially appear almost identical, a Eureka Moment leads to a coherent chain of reasoning that the detective can explain to the bystanders later; whereas a Bat Deduction, if it gets explained at all, makes even less sense after the explanation. However, although logically sound, the Eureka Moment may be enough to convince the one who came to it, but not to convince others (specially a court of law). In this case, the character may began a quest to obtain a Smoking Gun to prove things beyond any reasonable doubt.
Often used as a Deus ex Machina, albeit one that is acceptable more often than annoying. If the detective actually takes the idea literally rather than uses it as an inspiration, that is I Was Just Joking.
Can even happen in your sleep, with Dreaming The Truth.
Has no relation to a small town full of scientists. Or mecha with surfboards.
Compare Shaggy Search Technique and You Were Trying Too Hard. See Placebo Eureka Moment for when there wasn't any external inspiration, but they act like there was, and Love Epiphany, when the insight gained is that one party loves the other.
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Detective Conan does this a lot. One time, he solved a murder he had been stuck on just by Ran mentioning that she had bought new pajamas for them. (To be more specific, she mentioned that said pjs were a matched pair, leading Conan to realize that the murderer was a set of identical twins.)
Conan is a brilliant detective hiding in the body of a small child, so if he figures something out before anyone else, it's not uncommon for him to subtly lead those around him to their own Eureka Moment, rather than raising suspicions about himself by voicing his deductions directly.
There is also a certain part wherein Heiji and Conan figures out the dying message, but start, at Heiji's prompting, to subtly lead Kazuha to deciphering the dying message to make her feel better, as she was about to cry.
As well as Conan deliberately inducing Eureka moments to Kogoro due to Kogoro's It's Personal attitude after the murder of one of his old judo teammates.
In an episode of Cardcaptor Sakura, wherein two of her magical allies were cursed with Shapeshifter Mode Lock and could not effectively masquerade as non-magical, she had no idea how to break the spell until she was given a dead crab as a gift. She was inspired by the crab's shell which protects it "just like a shield!" and deduced that she could use her Shield card to protect her allies from the curse's outside interference long enough for them to transform.
This was also how she first defeated the Watery card.
In Utawarerumono (the visual novel), Hakuoro studies a strand of Mutikapa's fur in frustration, unable to explain why the creature fled the previous night when it had him and Eruruw in its grasp. Unable, that is, until Aruruw drenches him and the fur in tea and this trope ensues.
Arguably the entire point of Yakitate!! Japan, but applied to making bread. For example, one of these moments leads to the creation of Kazuma's "sushi-style" melon bread.
In Magic Kaito, another work by author Gosho Aoyama (author of the aforementioned Detective Conan), Kaito figures out how an old, poor magician got a parliamentary pen after a kidnapper claiming to be the Kaitou Kid took the Prime Minister after Aoko tells him the key to beating the boss of their video game is to defeat the king controlling him first.
Pokémon: Ash Ketchum has one of these in pretty much every Gym battle to turn it around (granted, his epiphanies don't alwaysmake the mostsense). If you see him panicking because his opponent is pulling some kind of unbeatable schtick, then snap his head up with a thousand-yard stare and say "wait, that's it!", chances are someone's about to get their ass beat.
Sometimes his companions do it too. May and Dawn get a few of these during their contest battles.
Invoked by Cameron by snapping his headband back if he's in a pinch during a Pokemon battle. It doesn't always work though.
Togusa gets two of these moments, early in the first season of Ghost in the Shell: Stand Alone Complex, and in both of them, he's in a bathroom, looking at the mirror. In the second, the mirror is crucial: He realizes that, from the batch of pictures he's examining, all of them show a mirror or other reflective surface, but the camera isn't visible in any of them.
Occurs in PatlaborThe Movie. Our heroes are trying to find out what's causing some Humongous Mecha with a new OS to go out of control. While taking a break while the investigation seems to be leading nowhere, Noah sees a dog barking at something they can't see & mentions they can hear things humans can't, cluing Shinohara in to the fact that ultrasonic waves are the culprit.
Shinohara actually gets two of these in the film. The second comes when they learn that the evil genius who created the program had a plan to make all the robots in Tokyo to go crazy at once & are trying to figure out what could cause enough ultrasonic waves to make it happen. When a whistling tea kettle causes a nearby figurine in a glass case to shake they realize it's sympathetic vibrations in skyscrapers caused by the wind. Fortunately, it'd take a mind-bogglingly huge building & incredibly powerful winds for a city-wide disaster to occur. Unfortunately a giant factory complex has just finished construction in Tokyo Bay & a record-breaking typhoon is due to hit in three days...
Naru has these regularly in Ghost Hunt, usually due to an offhand comment from Mai.
Happens twice in Fullmetal Alchemist, within the same scene. When trying to decipher a code from Scar's brother's notes, the group takes a break to reassemble Al who is in 'pieces', like a puzzle. May rips the bindings of the notes and with Scar and Marcoh's help reassemble it to see the nationwide transmutation circle (which Ed and Al figured out beforehand). When trying to think there was a silver lining in this, Yoki sneezes and causes the papers to shift. They're annoyed that the papers have been 'flipped over', and Al comes to his realization that the papers needed to be turned over to see the other half of the hidden message: a second transmutation circle which can be used to reverse the effects of the first.
Evil characters also can have this, as Iznogoud proves:
Iznogoud: Do you have an idea how I could get rid of the Caliph, Wa'at Alaaf?
Wa'at Alaaf: Sir, I'd rather feed my tongue to a cat!
Iznogoud(thinking): Tongue to a cat... the cat is a predator... the tiger is also a predator... the tiger is a man-eater... the Caliph is a man... so the tiger is a Caliph-eater!
Iznogoud(loud): That's a great idea of you, organizing a tiger hunt!
Wa'at Alaaf: A tiger hunt? Me? What?
In almost every issue of The Maze Agency, some seemingly random comment or event will start the gears spinning in Gabe's head and cause him to suddenly see the solution to the current mystery.
Ultimate Spiderman: Kong (the Dumb Muscle school bully, nothing less) remembered several things: the incident of Peter Parker with the weird spider in Oscorp, that Pete suddenly became a great basketball player, that he broke Flash's hand, that Spider-Man debuted at the wrestling club shortly after the gang visited it, that Spider-Man showed up to save the school when the Green Goblin began to attack it... and Kong suddenly realized it: Peter Parker is Spider-Man!
In one issue of Amazing Spider-Man, Reed Richards and Iron Man have literally minutes to defuse a bomb left by Doctor Octopus that would destroy New York City, while the rest of the Fantastic Four and The Avengers look on. Johnny Storm suggests that Spider-Man could come up with a "fallback plan", prompting Spider-Man to point at a clock reading nearly 2 am, saying "There's no time for me to come up with a fallback...fall back...THAT'S IT!" He then syncs the bomb's electronic timer with its main OS system's internal clock. Turns out it was the first Sunday in November, meaning the end of Daylight Savings Time, when people set their clocks back an hour or "Fall back". By syncing the bomb's timer, Spider-Man was able to extend the countdown by an hour, giving the FF and the Avengers enough time to take the bombs out to sea to explode harmlessly.
A commercial for Shell gasoline shows a scientist trying to figure out how to explore for petroleum reserves underwater without building multiple derricks and drilling dozens of holes. While at a restaurant with his son, his son is using a bendy-straw to suck up the last bits of milkshake at the very bottom of the glass. He comes up with the idea for a "bendy-drill".
In Earth And Sky, Twilight first starts working on how to help non-pegasus ponies fly after her conversations with Pinkie and Pumpkin, but doesn't settle on the idea of using artificial wings until she sees Sweetie Belle wearing a pair as part of her Princess Celestia costume.
Ralph realizes who Vanellope really is when he sees the Sugar Rush video game and discovers her picture on the console.
Vanellope gets her own when she sees Ralph hammering away at a seemingly indestructible Jawbreaker. The second she mentions it's indestructible, Ralph splits it in half, and Vanellope's eyes light up.
In Toy Story 2, Buzz is trying to figure out who stole Woody with only a Vanity License Plate to go by; LZTYBRN. The other toys are dismissive of this, and Mr. Potato Head just says, "Let's leave Buzz to his toys." This makes Buzz realize that the kidnapper is the Al from Al's Toy Barn, especially when they make Etch-a-sketch do a composite of the suspect in a chicken suit (which he found out when he picked up a feather that came out of Al's trunk Buzz was chasing his car earlier)..
Princess Atta: I mean, even Hopper's afraid of birds— Flick: Haha, yeah, I — say that again?
In Pinocchio, just when Pinocchio and Gepetto think they'll never escape Monstro the whale, Gepetto says that they can start a fire to cook some fish. This gives Pinocchio the idea to start a fire that will make Monstro sneeze.
Film - Live Action
In Blazing Saddles, Hedley Lamarr's evil plans all come as a result of these.
Ryan, mulling over how Capt. Ramius could manage to get the rest of his crew off the sub as part of his plan to defect.
"We don't have to figure out how to get the crew off the sub. He's already done that, he would have had to. All we gotta do is figure out what he's gonna do. So how's he gonna get the crew off the sub? They have to want to get off. How do you get a crew to want to get off a submarine? How do you get a crew to want to get off a nuclear sub..."
Midshipman Blakeny: "It's a rare phasmid, sir... It's an insect that disguises itself as a stick in order to confuse its predators."
Captain Jack: (later) A nautical phasmid, Doctor... I intend to take a greater interest in the bounty of nature from now on. I had no idea that a study of nature could advance the art of naval warfare! Now to pull this predator in close and spring our trap.
The entire plot of National Treasure hinged on the protagonist's ability to solve puzzles and uncover secrets in this manner. Granted, the hint was invariably directly related to the solution ("Pass and Stow" was written on an ad for the Liberty Bell, the correct time was drawn on a famous picture of the building it was stored in, etc), far more so than typically for this trope.
I, Robot features several effective examples of this trope including... "...the right man for the job...?" "...follow the bread crumbs..." "...I think my father wanted me to kill you..." etc,etc.
In Die Hard With A Vengeance, running into a band of under-aged looters alerts John McClane (Bruce Willis) that the villain's apparent plan is likely a distraction.
An unexpected source: in Godzilla Vs Biollante, a scientist works out the flaw of a recent attempt to neutralize Godzilla via Applied Phlebotinum by seeing dry ice hauled about for emergency refrigeration in the wake of the unthwarted monster attack.
In Independence Day, a scientist's father tells him to dress more warmly so he doesn't catch a cold, which gives him the idea to disrupt the aliens' force fields by uploading a virus into the mothership's computers in a reference to The War Of The Worlds.
There's a more serious case later in the movie: Nash realizes he hallucinates when he realizes in all the years he's seen them, the little girl never ages.
After two weeks of poring over a cartouche in the original Stargate movie, Daniel Jackson finally figures out that the symbols aren't hieroglyphs, but star constellations when he sees a picture of Orion on a guard's newspaper and recognizes the shape as one of the symbols.
Played with in Pi: Faith in Chaos. Mathematician Max consults with his mentor Sol about a numerical sequence he's trying to solve. Sol tells him the story about Archimedes (see below). Which leads to this exchange:
SOL: So what's the lesson? MAX: That an answer will come. SOL: No, it's the wife! You have to relax!
The revelation that the aliens reproduce quicker with fire is brought about by Orlando Jones' character dropping a lit cigarette into a petri dish, seconds after explaining how he hasn't smoked in years.
The alien's weakness to selenium is discovered by Julianne Moore taking off her jacket, revealing her periodic table of elements t-shirt.
Note that Evolution was, in fact, a pastiche of these kinds of movies.
In Real Genius, Knight's frustration at his sabotaged laser failing leads him to a freezer, where he realizes "It must be frozen..." and invents an entirely new and better laser using a frozen core.
In A Few Good Men, Tom Cruise briefly halts a brainstorming session with the rest of the defense team to look for his lucky baseball bat, which Demi Moore has innocently placed in the closet. Staring into the closet prompts a Eureka Moment that reveals an important fact about the case—the murder victim's clothes were hanging in his closet; if he had really been due to transfer to another post the next morning—as his CO has claimed—his things would have been packed, and his closet empty. Prompting the line "he really does think better with that bat."
In War Games, during the tense final scene, when Falkin tries to access Joshua with his password and finds it's been taken out, David asks what they're going to do. Falkin says "I don't know. Do you?", then for some reason, Jennifer says "I told you not to play games with that thing". So, David thinks for a moment, then says "It's games. GAMES!", and proceeds to play games with Joshua.
In Blood Work, the detective realizes the meaning of the Code Killer's message ("903 472 568") looking at a check he wrote for his neighbor, Jasper "Buddy" No one, after someone else has pointed out that the message doesn't contain a 1.
The title characters have been racking their brains trying to figure out where the "Galaxy" is. Their only hint is that they were told it's "on Orion's Belt" by a dying alien. When Agent J sees Frank barking at a cat on the street, he realizes the alien meant bell, not belt—the cat's name is Orion, and the "Galaxy" is a trinket the size of a marble that is on the cat's collar where a bell would be.
And it happens again later on, when the MIB are trying to figure out where the Bug might get a ship, and Jay spots the mural of the World's Fair, where they'd stashed a pair of flying saucers decades before.
Double Subverted in the sequel, where Will Smith's character starts to decode a ridiculously complicated chain of arrows in a pizza shop which leads him nowhere. He missed it at the first arrow. However, the free-association links he found do turn out to be quite close to the actual plot.
Based on the Romance of the Three Kingdoms saga, the Chinese movie Red Cliff has Boisterous Bruiser Zhang Fei, not known for his tactical genius, but still unwittingly blurting out vital insights more than once. Towards the end of the movie, his comment on the enemy battle fleet ("those ships could give us more firewood than we'll use in our lifetime") leads to Zhou Yu's plan to set the entire fleet on fire.
Who Framed Roger Rabbit: As Eddie Valiant leaves a movie theater, a newsreel tells the news of how Maroon Cartoons was sold to Cloverleaf Industries, the same company that bought the Red Car trolley line and bid for ownership of Toontown. He runs back in and shouts, "That's it! That's the connection!"
Dogma has one of these near the end; Jay tells Bethany (while trying to get it on, because the world's about to end) about a boardwalk he once took a girl to on a date. Bethany, after learning that the boardwalk has Skeeball, realizes that "John Doe Jersey," of whom her minister was sermonizing about earlier, is God, trapped in a comatose mortal body.
Hey, Jay was a prophet (or is that profit?), after all.
L: change the WorLd has one character hiding clues in math problems. L figures out the solution when accidentally given a clue.
In The Line Of Fire: Clint Eastwood's character figures out the meaning of a word with seven letters after a chance remark by a character played by Joshua Malina.
Polly: Have you looked at this? There's markings on it, like a ruler. And there's a moon and a star.
Joe: "And all I ask is a tall ship, and a star to steer her by." A star. He wasn't talking about a place, he was talking about a star! Rana is a star! Ancient sailors used to navigate by using the night sky. They could determine their position by the moon and the stars. The Vikings were known to create maps for certain stars, latitude tables that required a key to decipher them. The key was called Jacob's staff. This has to be the key!
In Hot Fuzz, Nicholas Angel's suspicions are stumped by the question of how one person could be in multiple places at once. When he and Danny Butterman go visit the convenience shop, the conversation that takes place between Danny and the clerk causes Nicholas to suspect multiple killers:
What's more, the shop clerk who triggered the Eureka Moment is one of the killers herself. Oops.
Major League: The Indians' manager is about to send hopelessly wild pitcher Ricky Vaughn back to the minor leagues. During their conversation, the manager off-handedly mentions another pitcher who went down to the minors and had a successful career. Vaughn squints in the pitcher's direction, and the manager realizes all his problems are related to poor vision.
In K-9 Dooley is growing frustrated with a case he's working. His girlfriend Tracy told him that he needed to relax, and the answer would come to him. Not much later, things are getting romantic between the two, when lo and behold, Dooley has an epiphany and must rush out the door with his four-legged sidekick.
William has one in The Name of the Rose book considering the secret of the library. Adso remembers how Salvatore said "tertius equi", which is Canis Latinicus for "The third of horse" (when he meant "the third horse"). William concludes: "the first and the seventh of the four" really means "the first and the seventh of the word four", and "four" is "quattuor" in Latin, so you have to push the letters Q and O!
They had a minor one earlier, when Adson dreamed a story similar to the "Coena Cypriani", a kind of The Bible parody. Which helps William to remember that there was a book in the library consisting of four texts, one of them a comment for the Coena Cypriani, another one the book they're looking for.
Happens to Tony Stark in The Avengers, when he realizes that Loki is using Stark Tower to launch the alien invasion.
In Ghostbusters II, after the particle throwers have no effect on the slime encasing the art museum Egon figures out it's powered by all the negative emotion in the city. He starts to talk about needing a massive infusion of positive energy to shatter the shell.
Egon: We need something that everyone in this town can get behind. [looks down] We need...a symbol! Ray: Something that appeals to the best in each and every one of us. Egon: Something good. Winston: Something decent. Peter: Something pure. [close up of the Statute of Liberty image on Ecto-1A's license plate]
In The Man Who Knew Too Much, Ben McKenna follows the lead of spy Louis Bernard's dying words, "Ambrose Chappell", to search for the kidnappers of Ben's son. Ben visits a taxidermy shop owned by a man named Ambrose Chappell, only to learn that Mr. Chappell has no association with the criminals. As Ben's wife, Jo, waits anxiously for him to return to their hotel room, one of her friends, Val, asks her the name of the person Ben's searching for. Val mistakenly calls him "Church", so after Jo corrects him with the word, "Chappell", she pauses, then exclaims, "It's not a man, it's a place! It's Ambrose Chapel!" Jo's friends proceed to help her find the address of this chapel in the phone directory, before she leaves them to search there.
Iron Man 2 : While trying to perfect the arc reactor, Tony suddenly finds inspiration in his father's miniature model of the first Stark Expo. Of course, his dad did that quite on purpose during the video he left behind for Tony.
Howard Stark:This is the key to the future, Tony. Camera cut to the City of the Future Camera cut to the Unisphere Camera cut to a prototype arc reactor
It seems that they did some research on the missing element. The element in question (118) does exist as "Ununoctium" though it is radioactive and only lasts for a very short time before decaying. It can only be synthesized with the help of a particle accelerator.
I am jumping up and down because at the very moment Dick Clark said the word "Go," it was like an invisible hand reached out and snatched away my veil. And for almost a minute, I understood everything. When that veil isn't hanging down right in front of a person's face, a minute is long enough to realize a lot of things.
Agatha Christie loved doing this in the Hercule Poirot novels. Very frequently, he fits the pieces together after inspiration from a chance remark by one of the other characters; to Poirot's credit, he never fails to explain precisely how it helped him break the case.
Every Cat Who... novel by Lillian Jackson Braun features Koko the Siamese doing something odd which eventually leads to Qwilleran having a Eureka Moment. How plausible Koko's behavior is, either taken at face value or with the strong hint he's trying to give clues, varies considerably.
Lampshaded in The Burglar Who Traded Ted Williams: While Bernie Rhodenbarr is talking with a friend, she mentions the Cat Who... series, when his cat suddenly launches into strange behavior and Bernie solves the crime. Subverted, in that he'd already solved the crime, but didn't care about revealing the result. When the cat acted, as if on cue, he decided to play along.
In A Song of Ice and Fire, Sansa supplies her father with one when she says that Joffrey (who she has a mad crush on, the poor girl) is nothing like his father, Robert. This finally makes Eddard realize that none of Cersei's children take after Robert — though all of Robert's bastards do — and the kids are most likely the product of incest between Cersei and her brother, Jaime.
Tyrion gets one two books later, when the readers have all pretty much forgotten about the never-quite-solved mystery of who sent the assassin with the dagger to try to kill Bran from the first book, when he hears Joffrey bragging about how he's very familiar with expensive things like Valyrian steel. Because the chapter is told from Sansa's point of view, the readers only find out what's on his mind when he gets her alone for long enough to ask "Hey, do you know if Joffrey had any reason not to like your brother?
In American Gods, this is how Shadow figures out where a small-town serial killer has been hiding his victims' bodies, acting on a comment from a god who knew what was going on, and was trying to clue him onto it.
This is nearly Elijah Baley's entire modus operandi in Isaac Asimov's Robot novels. The third and final book in the series even barely gets away with justifying it.
In The Caves Of Steel, Lije Baley figures out the murder mystery when his partner casually brings up Enderby's glasses. All pieces of the puzzle then come together: Lije realizes that Enderby (who was earlier established to be psychologically incapable of killing a human) accidentally murdered a person, instead of destroying their robotic lookalike, because he couldn't see clearly when his glasses shattered.
Asimov also wrote an essay about how this sort of thought process works called "the Eureka effect". It's even made it into a few high school literature books.
An extremely complicated version of this trope is pretty much the premise of Chasing Vermeer.
Timothy Zahn likes this trope. Or, more accurately, this trope fits his style. Many of Zahn's original works are First-Person Smartass, and the settings are similar to Hitchcock-style suspense/intrigue mysteries, so there's guaranteed to be one when the protagonist finally puts the pieces together. (Though the reader, unless he figures it out too—which is sometimes possible and sometimes not—won't know until The Summation.) Some examples:
The Icarus Hunt has at least two: One when a comment makes the protagonist realize another person's murder was connected with something completely different than he'd been assuming; another when a different comment triggers a flashback. The Summation indicates he may have had another couple more along the way that the audience wasn't even privy to.
The Quadrail Series tends to have one per book, minimum; most notably, the moment in Night Train to Rigel when Compton figures out how the FTL trains work.
This even shows up in his Star Wars Expanded Universe work, though those get more promptly revealed to the reader: in Survivor's Quest, a random comment by Luke triggers a flashback for Mara, in which she recognizes a major inconsistency in some of their companions' back story.
And finally, one for Leia in The Last Command. Talon Karrde mentions how he was a Chimaera prisoner, and for Leia, "suddenly, all the pieces had fallen into place...and the picture they formed was one of potential disaster."
Aside from the few times they recognize the answer instantly, the characters in The Da Vinci Code seem to rely solely on this trope to solve all the various puzzles and sub-puzzles. For the last puzzle before the book's climax, we don't actually get to see the main character work out the answer.
Nero Wolfe does this quite a bit. He's got all the pieces, but can't make them fit together, Archie (or one of the suspects) says something that causes him to look at one particular thing in a different way, and everything falls into place. He will often admit to Archie when it was something that he said or did; if it was someone else, he saves the information for the Parlor Scene. Both Archie and the reader know one has just occurred when Nero starts doing his "lip exercises".
Subverted, while lampshading the subversion, in Mil Millington's "Love And Other Near-Death Experiences", in which the central character has a startling realization that he recognizes would qualify as a Eureka Moment - except that nobody did or said anything to trigger it, it just happened without prompting.
In Harry Potterand the Half-Blood Prince, a sarcastic comment by Harry ("Fifty-seventh time lucky?") about his inability to get a crucial memory of Voldemort from Slughorn inspires Ron to suggest Harry use his luck potion, which proves effective. A similar scene occurs in the movie, only (in keeping with the screenwriter's general inclinations) reversed: Ron's comment inspires Harry.
In Deathly Hallows, we're treated to Harry's entire train of thought as a broken talking stone gargoyle triggers his memories and he realizes exactly where the last Horcrux is hidden.
In Memory, Sorrow, and Thorn, near the climax of the third novel, Binabik the troll has a classic Eureka Moment during his party's escape from the Norns beneath Asu'a, as a result of an offhand comment from the dwarrows who are aiding them. Unconventionally, this leads not to the solution to the plot but the horrifying realization that the heroes have been doing the Big Bad's bidding the entire time.
In the grand tradition of mystery novels, happens frequently in Ngaio Marsh's Inspector Alleyn series. An interesting departure is that the Eureka moment for Alleyn usually occurs near the middle of the book, with the reader catching up as more evidence is gathered.
Played straight at the end of the book, when remarks of Detritus and Fred Colon make Vimes realize the Big Bad had been rubbing the hints of his plot in Vimes' face, assuming Vimes wouldn't get them.
Lampshaded in that "eureka" is a Ephebian word meaning "Bring me a towel".
Subverted in Reaper Man; Inventor Ned Simnel is trying to think up a way for his combination harvester to move without a horse when a blast of steam goes off. Ned just gets annoyed on how he's always getting interrupted when trying to think. According to The Discworld Companion, this has happened over a 150 times.
Subverted in Moving Pictures. CMOT Dibbler has been selling advertising space in some pictures, but is told to stop. Then he hears about a mistake someone made while editing a film, leaving in a frame which was just a picture of gold. Nobody noticed, but everyone was thinking about gold when they left the theater. It looks like he's about to figure out a way to sell advertising after all... but he puts in five minutes of film depicting what he's trying to advertise, on the reasoning that if one frame can do that, five minutes worth must work wonders.
In Dorothy L. Sayers's Whose Body?, Lord Peter Wimsey's moment of revelation is explicitly compared to staring at jumbled letters until they formed a word of their own inclination.
In Remote Man, Ned comes up with his plan to bring down Laana's smuggling operation when he's browsing in a dime store. He picks up a pin cushion that he discovers has a music box inside playing a familiar tune: The Entertainer. This brings back memories of watching The Sting with his father, which in turn sparks his plan, which is to invent a person, a rich businessman with his own fabricated website, to be a client of Laana.
Earlier, Ned overhears a conversation mentioning "an anchorman in Kingston". When he later learns that Kingston is the capital of Jamaica, he realises that what he heard was "a nanka-man", nanka being a local term for the Jamaican Boa.
When Chichikov gets the idea for his scam in Dead Souls.
In Heroes Die, Toa-Sytell has one when he reads the Monastic records on Caine and realises that he's an Aktir.
Live Action TV
The Dutch Detective show Baantjer featured, in each and every episode, a Eureka moment when the protagonists were in their usual bar, when the barman makes an offhand remark. The Euraka moment always looks and sounds the same too. Eventually Lampshaded, when the barman asks if he doesn't deserve be put on the police payroll.
Jonathan Creek actually used this so much that at one stage another character actually anticipated that a totally random object would trigger a Eureka Moment in Jonathan, and so caused a distraction.
Hong Kong prime time dramas seems to love this trope. Case in point, the period drama Song Shijie, where the eponymous character is a court lawyer who seems to be surrounded by people and incidents that randomly give him pivotal clues.
In Angel this is lampshaded when Angel announces that they're waiting for Wesley's Eureka Moment - at which point Wesley promptly shouts "Eureka!"
Angel: "Listen, listen, listen... What are you listening for?" Fred: "The click. When it all comes together and makes sense. There's like a click in your brain and then you understand things again."
In the pilot episode of Bones, Brennan is sitting down to have a drink and discussing a book written by the episode's original patsy, when they come across the patron saint of fish, and Brennan realizes who the real baddie was, based on the fact that he kept tropical fish tanks that used a certain type of diatomaceous earth, and runs off to take him down by herself.
The episode "The Maiden in the Mushrooms" has a C-plot involving Hodgins using the last of Finn's late grandmother's special hot sauce and trying to recreate the recipe using the lab equipment to analyze its contents. He succeeds and Finn starts digging into the catfish that's just like his grandmother used to make, and Hodgins tells him to slow down or he'll choke on a bone. Finn says that that's not a worry with a baby catfish because their bones are too malleable. Cue realization that the victim was young enough that hanging wouldn't necessarily have snapped her hyoid bone.
This trope was regularly parodied in the 1960s Batman TV show. "That's it, Robin! The man in the grey suit was whistling 'The Star-Spangled Banner' BACKWARDS! The Joker's lair must be in the old fireworks factory! To the Batmobile!"
Castle is beginning to show an aptitude for Eureka Moments, usually inspired by his daughter. In a reversal, he gave one to his daughter in one episode.
Castle and Beckett will often have these moments simultaneously (or near simultaneously) in order to demonstrate how they click (in more ways than one). For one example, Beckett was writing something on The Big Board when she suddenly had a brainwave... and Castle at the exact same time ran in from the elevator, having obviously had the exact same brainwave.
Happens with astonishing regularity in The Closer. Brenda has them all the time... in fact, she had one when she was at her father's hospital bed, and another when she was trying on her wedding gown.
In a CSI NY episode, Hawkes watches a Jennifer Lopez video during his lunch break. While admiring her, um, assets, he remembers they are insured, helping him figure out the case - it's an insurance scam.
Doctor Who has it happen so many times, it's practically a drinking game at show marathons.
The Talons of Weng-Chiang makes fun of the word itself: the Doctor claims that it translates from Greek as "This bath is too hot."
Sheriff Carter solves problems with literal Eureka Moments, followed by saving the town with an It's Up to You sequence. (Especially ironic since he's the supposedly only normal person in a town full of geniuses.) Sometimes, however, the Eureka Moment moments come from the geniuses after the sheriff suggests a Muggle solution.
Inverted in Everybody Loves Raymond, when Ray lies that he has a great Christmas present ready for Debra, which he got the idea for on a trip they took to Manhattan a while back when she shouted "Oh my god!". On his brother's advice he gets her a First Edition copy of To Kill A Mockingbird, then has to blunder his way through an explanation of what she shouted "Oh my god" at in Manhattan that caused him to think of the book.
In Farscape: Not so much a mystery as a scientific puzzle, but while John tries to figure out how to save Earth from the Scarrans at the end of the regular series, Aeryn tells him not to beat himself up over it. Because of the way she phrased it, he gets an idea for a solution, kisses her, tells her to "Never change!", and runs out. Aeryn gets a half-pleased, half-bemused look on her face.
Hogan's Heroes: Colonel Hogan gets most of his ideas this way. A typical scenario involves pacing while the others discuss a problem. He then pauses, smiles, and says "Wait a minute," before outlining the plan.
House does this every episode, most of the time inspired by Dr. Wilson. It's actually stated on the show that House will seek out Wilson to talk about unrelated subjects because it helps to jog his mind and inspire Eureka Moments. The show isn't above lampshading it in other ways, either:
Wilson: "Who the hell chats about their blood type? You had to have tested me."
House (to himself): "There's no reason to ask anyone their blood type..."
House pauses, and as usual starts staring into space as if he's working something out in his head.
Wilson: "You're about to run out of here now, aren't you?"
House gets up and leaves immediately.
Or, in another episode:
Wilson: ...and you aren't even listening to me anymore, because I just gave you the solution to your case, right? And now you are going to leave without saying anything else.
House (as he leaves Wilson's office): No.
House once actually showed up outside Wilson's apartment after they'd had a falling out in the hope that if he could get Wilson talking long enough, he'd eventually say something that would give him an epiphany. It didn't work.
Parodied in one episode of Scrubs using paint. It was a parody of House.
Scrubs uses this regularly as well. JD often sees his friends solve their problem of the week and realizes this can be applied to his own situation. Beautifully lampshaded here:
JD: Oh, I'm just doing this thing where I use a slice of wisdom from someone else's life to solve a problem in my own life.
Jordan: Seems coincidental.
JD: And yet I do it almost every week.
Done frequently in Inspector Morse with a inconsequential remark by Sergeant Lewis providing the inspiration for Morse to solve the mystery.
A staple on Midsomer Murders, where DCI Barnaby regularly receives the inspiration for solving the murder case from unrelated conversations, often in chats with his wife about whatever happens to be their personal side story (preparations for a county fair, etc.) in the episode.
Monk: In one episode, the date on a ketchup bottle let him work out a whole case he hadn't even known existed! In earlier seasons, this frequently led to some humorous non-sequiturs, such as "If he's 37 years old, then I know how the judge was killed."
Happens in nearly every episode of NUMB3RS, with camera effects indicating the mathematical wheels turning in Charlie's head. Usually followed up with the strangely PBS-like explanation of the relevant analogy.
In Prison Break, Mahone puzzles over a clue to where Michael is going that ends with the word "woods." After seeing the badge of a cop whose last name is Rivers, he realizes that the clue does not refer to a place but a person.
Shawn Spencer in Psych frequently has major Eureka Moments very early in the episode. Usually he's figured out the particularly far-fetched explanation very early (such as figuring out that a victim had been bitten by a T-Rex...don't ask), but the whole episode then revolves around him and Gus trying to find the evidence to prove the outlandish theory correct. Reverse Detective Work, maybe?
This trope regularly occurred on Remington Steele, with Steele being reminded of an old movie which somehow resembles the case he is working on. Sometimes used as a Red Herring when Steele's movie references end up providing a plausible but entirely incorrect solution.
In one episode of Seven Days, a scientist discovers a cure for cancer by seeing some water on her morning jog. The discovery is apparently averted when Frank goes back and interrupts her right before she sees the water- but she discovers it anyway after receiving a similar Eureka Moment from some coffee Frank gives her.
Stargate SG-1. If Jack O'Neill is going to solve the episode's problem without shooting his gun a bunch, he'll say something random and Carter'll build a solution out of it.
An episode of The West Wing has Josh struggle all episode with trying to determine a solution for an unfavorable rider to a long-fought for bill that will result in a piece of land being used for strip-mining. After wracking his brains all episode without success, a chance comment from Donna about antiquated computer systems reminds him of a long conversation with the President earlier in the episode about national parks, and he comes up with the idea of using the Antiquities Act to have the President declare the region a national park.
Josh and Donna had a few of these over the series. Another one concerned judicial nominees after the death of a Supreme Court Justice, and the declining health of the Chief Justice. Although all the characters were unhappy that they couldn't get the nominees they wanted confirmed, it wasn't until Donna told Josh the story of how her parents got their cats (they went to an animal shelter, and they couldn't decide between two, so they got both) that he has the idea of letting the Republicans name whoever they want for the open seat, if they will accept the Democratic nominee for Chief Justice.
Spoofed in Garth Marenghi's Darkplace ("The Apes of Wrath") when people start devolving into monkeys due to contaminated water. The hero suddenly puts all the pieces together (the fact that his friends turned into monkeys after drinking a cup of water, the only two people who haven't turned into monkeys aren't drinking the water, and that the water's a sickly green color) and concludes... that he's thirsty.
Battlestar Galactica ("Scattered"). Gaeta comes up with the idea to network Galactica's computers by lining up soap bars in the latrine.
On Good Eats, Alton was agonizing on what to do for an avocado-themed episode (while his assistants are suggesting varieties of guacamole) when one rolls into a sink full of water. He shouts "Eureka!" when he sees that the avocado floats.
Ultraman Mebius has such a moment in episode 34, when watching Ryuu light a campfire by twirling a stick on a piece of wood allows him to develop the drill kick technique he uses to defeat the KaijuOf The Week
In the Criminal Minds episode "Compulsion", Gideon has a friendly conversation with a student who broke up with his girlfriend for another guy. The student says that his ex told him he would suffer the wrath of God, leading to the conclusion that the episode's unsub is religiously motivated.
And in a later episode, Reid figures out how to get back a lost childhood memory while having a chat with a hooker in front of an automatic poker machine.
Girl: So, did it work? Did she quit smoking? Because I've tried everything. The patch, pills...
Reid: You should try hypnosis. They've had great results with... hmm. [Gets up and walks away.]
Girl: Hey, wait, you won, like, two thousand dollars on this machine!
Reid: (distracted) Keep it.
Morgan: You do realize that you just gave two thousand dollars to a hooker, right? What were you talking about, anyway?
Reid: How to quit smoking.
In NCIS, McGee got one while taking a polygraph test when the examiner mentioned that the regulars on her morning train were worried when they didn't see her.
The heroine of Ghost Whisperer seems to have moments like this now and then when trying to figure out the identity of the Ghost of the Week.
In the Murdoch Mysteries episode "The Kissing Bandit", Julia Ogden's sister is trying to explain to Murdoch that, even though Julia has settled for marrying someone else, that doesn't mean she doesn't love Murdoch, if he would only act on his own feelings, rather than resign himself to the situation. She then uses the reaction of a woman kissed by the Bandit as an example of being swept away by one's feelings.
Murdoch: I wasn't aware I had resigned myself to anything, Miss Ogden. But you have given me some insight into this case.
Ruby Ogden: I have? I didn't intend to.
Classic example, from M*A*S*H; Hawkeye and BJ challenge Winchester to join them in a poker game with Col. Potter, Major Houlihan, Father Mulcahey and Radar, only to have Winchester way ahead halfway through. While Winchester is taking a break, the others complain about his annoying whistling, with Radar commenting that he keeps whistling even louder when he's bluffing. A few seconds later...
Hawkeye: A-ha! He whistles louder...
All: ...When he's bluffing!
In the Parks and Recreation episode "Eagleton", Leslie is struggling to overcome her rival Lindsay in the neighboring town, who put up a fence in the middle of a shared public park. Ann offers to beat Lindsay with a baseball bat, which gives Leslie an idea. Cut to a few days later, when Leslie has turned Pawnee's side of the park into a baseball field using the fence as a convenient boundary.
In Homeland, Carrie has one when watching some musicians play at a bar and realising that Brody doing the same might be a signal.
In the Power Rangers In Space episode "Five of a Kind", T.J.'s watching video footage of the Rangers' last fight against the Psycho Rangers (who the Rangers have struggled badly against up to this point, as each Psycho copies their individual fighting style) when the color disappears from the video. The monochrome footage gives him the idea to disguise the entire team in the same color uniform, so the Psychos won't know who is who.
Frequently used in Columbo where the detective would see something that would inspire him and would only be revealed at the end, once that critical detail had been used to lure the killer into revealing their crime unwittingly.
This is also a Once an Episode occurrence on the 1975-6 Ellery Queen TV series. In the final act Ellery will be in some mundane activity, usually with his police Inspector father, say "Of course!" and break the fourth wall to ask the audience if they've solved the mystery.
The midseason finale of the final season of Breaking Bad ends with Hank realising his brother-in-law Walt was Heisenberg all along.
All the characters of Graduados, except the seniors and the teenagers, attended high school together in the eighties, and they are now in their forties. Unknown to them, Patricia Longo is Jimena Benitez, the fat girl of the class that everybody insulted and mocked. She got thin, she changed her name... and wants revenge. Several hints of her true identity were released here and there. And one day... a comment from Patricia left Loli pondering. She had a wave of flashbacks in the night, and got her Eureka Moment: Patricia is Jimena!
Padre Coraje has a Costumed Non Super Hero, Coraje, who poses as the priest of the village. Clara Guerrico remembered several details about both of them, and realized his secret identity.
A simultaneous example occurs in the Square One TV Mathnet mystery "The Case of the Mystery Weekend". While playing a game of color 8-ball, where the player must sink balls in the order red-orange-yellow-green-blue-purple-black, George Frankly realizes that guests have been disappearing in a similar order, going by the decorations of the rooms they were assigned. Later, fellow guest and sax player Miles Reed reveals he deduced the same thing, presumably inspired by "Somewhere Over the Rainbow", the song Reed had been practicing throughout the weekend.
Charles: You have that ability, peculiar to the mediocre, of making the obvious statement that brings out the genius in a genius.
Parodied in the first season finale of Telltale's Sam And Max Freelance Police series. Most fans had figured out by this point who the season villain was, so at the beginning of the episode Sam is trying to figure out who the villain is. has to drop increasingly obvious clues before Sam finally figures it out.
The most triumphant example in the first game has to be in case 4. You're backed into a corner, you have no proof that the witness isn't who he says he is, and it looks like you're going to lose. von Karma then says something as a joke which suddenly turns everything around: "Perhaps you would like to cross-examine the parrot for some comic relief?"
This trope is probably why the Edgeworth-based spin-off game makes his Catch Phrase "Eureka!" (Though quite a few fans thought this was a little Narm-y.) He actually does get one in AAI case 2, when Zinc LeBlanc falls over the railing in the plane's cargo bay.
At several points in AAI Edgeworth will hit a brick wall with his confrontations, but then the player is reminded of past innocuous scraps of conversation, then links together half a dozen or so pieces of logic, which leads to a whole new perspective on the case. Sufficiently awesome music starts up to accompany this.
Heavy Rain - The clues start coming together in the last act for the characters (and, by extension, the player) this way.
In Assassins Creed II, Ezio and an ally are trying to figure out how to get into the Doge's palace to prevent an assassination plot. The friend remarks that the walls are so high, "only a bird could get over them". Ezio hears this and mutters "A bird... yes...", then runs off to ask Leonardo da Vinci about that flying machine of his...
Leonardo later has his own Eureka Moment after they discover that said flying machine doesn't get very far on its own. In his frustration, he throws a piece of paper into the fire, where it starts to rise from the heat, inspiring the crew to set up bonfires across Venice.
Bartolomeo inspires Ezio to another Eureka Moment when he realizes that they have no chance of storming Baron de Valois's fortress to rescue Pantasilea. He starts wondering if it would just be best to just give himself up, leading Ezio to his idea of disguising the mercenaries as French soldiers and pretending that he surrendered himself to get past the gates.
In Nine Hours, Nine Persons, Nine Doors, Junpei looks at Snake's dead body, whilst searching the Shower room for Clover, and notices his left arm's bone: the real Snake's left arm was in fact prosthetic. Snake was thus alive, and Junpei slowly realizes who did it.
In one ending, at least. In another, Clover makes the connection.
In Dawn of War 2: Retribution, Kaptain Bluddflagg has one of these when he figures out where Big BadKyras is hiding in the campaign map.
Mr. Nailbrain: Maybe we should go 'ide out on dat place dey dakka'd before. Dey won't waste good dakka, right?... Bluddflagg: Hold on a squig! If dat Kyras-git is anglin' ta zog da hole sektor... Mr. Nailbrain: Right... Bluddflagg: And Kyras is in da Sektor... Mr. Nailbrain: Okay... Bluddflagg: And Kyras don't want ta get zogged... Mr. Nailbrain: Yeah... Yeah... Right... Bluddflagg: And dose big humie ships won't dakka da place dey dakka'd before... Mr. Nailbrain: Yeah?! Bluddflagg: Den Kyras... would go... ta da place... dey won't dakka! ... Which is dat place! beat Mr. Nailbrain: ...Still not following ya kap'n.
RuneScape: During the events leading up to the destruction of the old wizards' tower, mages were baffled on how to make new teleportation spells. The Red apprentice was working on a different problem; how to increase the effectiveness of combat spells by bypassing defenses. He had an idea he thought would work, but couldn't figure out why it wasn't. So he showed it to the Grey apprentice, who had the Eureka Moment; his idea - sending the spell through a second plane with Law magic - was the solution to the teleportation problem.
In "Ayla and the Birthday Brawl" in the Whateley Universe, Phase has a Eureka Moment in the middle of the final battle of the story. It nearly turns into a Brick Joke when we don't get to hear the deduction until well into the next novel, as the attack used by the Necromancer tells Phase that there is a link between the Necromancer and Hekate, another of their Big Bads.
In the Batman The Animated Series, virtually every conversation Batman has with Alfred while pondering the crime du jour leads to a Eureka Moment (and an "Alfred! You're a genius!" declaration, and a bemused "Of course, sir" response).
One memorable conversation involved the Riddler and a series of riddles. Batman and Alfred are staring at the riddles and Alfred makes the comment that the riddles all involve numbers. From that, Batman turns the numbers into Roman numerals and then simultaneously, both declare, "Of course! The Department of Motor Vehicles!" (suddenly seeing the Roman numerals DMV together could easily be a Eureka Moment for any American.)
Another has Batman deducing where Count Vertigo's hideout is when Alfred tells him that he soon won't know his right hand from his left. Yes, he was confused about which direction Vertigo escaped in because his senses were scrambled, and he had actually gone in the opposite direction Batman had originally thought.
In yet another (Killer Croc's appearance on the scene), Batman realizes the nature of his foe due to Alfred's offhand comment that he was going to keep dinner in a crock pot to keep it warm.
In Arthur, Buster Baxter solved a case involving missing quarters thanks to a comment from Alan, a.k.a. "Brain".
Used on several of the earlier Sideshow Bob episodes, with "Big shoes to fill" and "Tonight on MacGyver" being two of the phrases that trigger Bart to figure out Bob's evil schemes.
Not to mention, in the episode where Sideshow Bob is the detective for once, this is how he discovers the culprit: "A mechanic?"
In one flashback episode, several people congratulated Homer because Marge had become pregnant (with Maggie). Homer, himself unaware of the pregnancy, misinterpreted these comments (even the ridiculously direct ones) as regarding his new job. Then one character congratulates him on the new job, prompting Homer to respond, "New job? MARGE IS PREGNANT!?" That's "ridiculously direct", as in "Hey, Homer, way to get Marge pregnant!" "This is getting increasingly abstract, but yes, I do enjoy working at the bowling alley"
When the family is trying to figure out a way to keep Apu from being deported, Abe mentions he could try living in a balloon. Lisa exclaims "That's it!", leading Bart to declare that she's as dumb as him. She explains that she connects him being a grandfather with grandfather clause, with his statement being irrelevant.
Captain Flamingo runs on this trope. Every use of the eponymous character's "Bird Brain" is either this, or a full-on I Was Just Joking, depending on which would be funnier. Either way, it's played for laughs, as the ideas he gets are usually ridiculous... but always manage to work.
According to Animaniacs, Albert Einstein has conceived his famous mass?energy equivalence formula (you know, E=mc^2) while watching Wakko writing "Acme" backwards (his "a" looked like a "2").
In a similar episode, the Warners were cleaning Beethoven's chimney (badly, of course) and humming an obnoxious tune that went a little something like "Hm hm hm hmmmmmmmm". As they're leaving, ol' Ludwig finally has the inspiration for his Fifth Symphony. The next house on the Warners' route is "that Van Gogh grouch again", and Wakko picks a sunflower to help cheer him up.
In an episode of The Raccoons, Cyril is accused of stealing a gem during a blackout. Bert is trying to solve the crime but is stumped until he sees Cedric drop his Nerd Glasses in the river. It's then that he realizes that the gem was not stolen but knocked in an aquarium, and was rendered invisible by water refraction, same as Cedric's lenses.
Jimmy Neutron had a recurring delayed reaction to this. When the heroes have exhausted all their obvious options, Jimmy starts thinking about the episode's events. One completely ordinary-seeming event would pop into his mind, he'd shout "Brain Blast!", and he'd come up with a solution based on that event.
Spoofed in an episode of Johnny Bravo where Johnny's enlisted the help of Adam West to find his missing Momma. They stop for lunch at a Chinese restaurant and West interprets his fortune cookie message ("Your heart's afire"), via Batmanlogic, to mean that Momma Bravo's being held prisoner at the golf course.
Sent up in a Running Gag on the South Park episode "Cancelled". The astronomer investigating Cartman's satellite dish implant draws several correct conclusions after increasingly absurd free association sessions based on conversations with Chef.
Mr. Garrison struggled to invent a new vehicle until Mr. Hat commented "Boy, that Enrique Iglesias can sure gyrate his hot ass around" and decided on a gyroscope.
When Kim visited her cousin Larry, he commented that Monty Fiske, the archeologist they had assisted earlier, was a master of Tai Shing Pek Kwar or Monkey Kung-Fu, prompting Kim to realize that he was the villain Monkey Fist.
Ron lapses into one when Kim mentions "Homefield" in "Ill Suited".
Farnsworth: Amy, technology isn't intrinsically good or evil. It's how it's used. Like the Death Ray. Amy: Maybe you're right. I guess science has to keep pushing forward, with the hope that... That's it! Pushing forward. Professor, I know how to restart the Earth! Farnsworth: Pssh. You can't suddenly know something by assembling a committee of words... That's it! I'll assemble your committee!
In Kung Fu Panda, Po opens the scroll containing the secret to limitless power and kung fu itself to find nothing but a blank, reflective foil, which seems to imply that obtaining limitless power is impossible. He then meets his father who says that the secret ingredient to his popular "Secret Ingredient Soup" is also nothing; it's special because people believe it is. This leads Po to having his Eureka Moment: that limitless power resides within oneself (hence the reflection), and that "there is no secret ingredient".
Finn from Adventure Time has one in which the word "RENCE" on one half of a murder weapon makes him think of Clarence, whom he had met several minutes prior.
In another episode, Finn solves a case when he's tipped off by the sentence, "This place could use a scarecrow."
Terrytoons' TV character Tom Terrific will usually get a Eureka Moment when Mighty Manfred the Wonder Dog says something off the cuff to try to weasel out of a situation. In "Go West, Young Manfred," Tom is looking for a way to get back to the settler with his cap full of gold:
Manfred: Please, Tom. Don't saddle me with your problems.
Tom: Saddle. Pony. Pony Express! That's it, Manfred! We'll go by Pony Express!
In The Powerpuff Girls episode "Mommy Fearest", after the girls are grounded by the Professor's new wife Ima Goodlady, Blossom gets a call from the Mayor saying that Sedusa is robbing him of his jewels. There, Blossom puts the pieces together about this coincidence and deduces that the two of them are one and the same and it was all part of Sedusa's scheme to keep them away while she committed crime.
Blossom: Don't worry, Mayor. We'll catch Sedusa...as soon as she gets home.
The word "Eureka" originates from ancient Greek, meaning "I have found it!" Archimedes was consulted to figure out how to tell whether a crown that the king had commissioned was made from the pure gold he had supplied to the jeweler, or whether the jeweler had substituted silver for gold to make a crown the same weight. When Archimedes took a bath, some of the bathwater overflowed the edge of the tub, prompting him to realize that an object displaces its equal volume in water. Jumping out of the water, Archimedes shouted "Eureka!" and ran to report this discovery to the king. Naked. The subject of volume displacement in physics is, fittingly enough, known as "Archimedes' Law" because of this.
Newton's understanding of gravity supposedly came after an apple fell on his head. (Which never actually happened, but nobody cares about that.)
The accepted explanation is usually that he came up with the law while thinking about apples falling from trees (or perhaps even saw an apple fall to the ground), but the story about one actually falling on his head is just fluff. Newton himself told this version later in life.
A similar tale surrounds the invention of the sewing machine: the inventor dreamed of spears with holes in them stabbing towards him, and realized upon waking that the eye of a needle to fit into a sewing machine had to be in the tip.
Supposedly, Eli Whitney came about the idea of the cotton gin by watching a cat wash itself.
According to IMDB, composer Bill Conti had a rocking theme all ready for the Training Montage, but couldn't figure out what to call it. Director Avildsen, impressed by the nameless piece, agreed that "It should be almost like Rockyis flying now."
Watson and Crick, co-discoverers of the DNA molecule's double-helix structure, got the idea after noticing a spiral staircase. It was a completely unlikely solution based on the limited evidence.
They actually got the idea from Rosalind Franklin's research. Not so much a Eureka Moment as it was intellectual theft.
The inventor of Genetic Fingerprinting, the DNA identifying technique used by forensic investigators and dramas all over the world, Professor Sir Alec Jeffreys, had such a moment (and it is frequently described as a Eureka Moment) while looking at an X-ray film image of a DNA experiment at the University of Leicester. Noticing the obvious similarities and differences between the DNA of family members, within half an hour he saw the potential.
Parodied by Discworld, by suggesting that DNA might have been faster but only licensed to carry 14 people, if he'd been looking at an elevator during the Eureka moment.
Gauss once wrote in his diary "EYPHKA! num = Δ + Δ + Δ," thinking he'd proven Fermat's famously unproven polygonal number conjecture (yes, that's right, he did it more than once) for n = 3; he later realized that the proof in his head needed significant ironing out, but for an open question, he got it quickly enough.
According to Cognitive Psychology, when it comes to problem solving those "Eureka Moments" are actually called insights which are deep, useful understandings of the nature of a problem. However, compared to the typical slow approach to a problem solution, insights often occur abruptly and almost without warning. Tasks that involve insight usually require something new and non-obvious to be done and in most cases they are difficult enough to predict that the initial solution attempt will be unsuccessful. This can lead to the so called "AHA-experience" where the solution pops up all of a sudden.
Merv Griffin came up with the name of his popular quiz show format Jeopardy! after a network exec told him that the show, piloted under the name That's the Question, needed "more jeopardies".