Hint: The limit as the number of iterations goes to infinity may give a different answer. Try to think about this one without using math at first.
There is a rubber band attached to a wall. The rubber band is one meter long, levitating horizontally away from the wall. When I say "go", it will start stretching so the end moves at 10 meters per second. It stretches infinitely. The stretch is uniform e. There is an ant starting where the rubber band connects to the wall. It walks at. Will the ant ever reach the end of the rubber band? If so, how long will it take?
What if the rubber band doubles in length every second? A woman and her husband attended a party with four other couples. As is normal at parties, handshaking took place. Of course, no one shook their own hand or the hand of the person they came with. And not everyone shook everyone else's hand. But when the woman asked the other 9 people present how many different people's hands they had shaken they all gave a different answer.
Question this is NOT a trick! You have two very resilient dinosaur eggs. They will absorb a certain amount of force with no negative consequences, but at some point they will crack. If they don't crack, no damage is incurred. You're on a story building.
You have 20 trials you're allowed at most 20 individual egg drops and 2 eggs. Is it possible to devise a testing strategy that guarantees to tell you at exactly what floor the eggs will break? If after your first trial dropping an egg off of the balcony on one floor of the building , if the egg does not break, then you have 19 trials remaining and two eggs.
If the egg breaks, then you still have 19 trials remaining, but only one egg left. How few trials do you need? Let this number be k. Using k trials, can you solve a story building? How about ? You work with Jane. You know that she has two children.
One day you meet one at a boys' camp, and it is a boy. What is the probability that she has two boys? Where on the Earth can you walk a mile south, a mile west, and a mile north, and end up exactly where you started?
Hint: There are infinite places: find them all! What is a shape defined by a mathematical equation that has infinite surface area, but finite volume? There are two empty bags. You have 50 black and 50 white pebbles. You must put all of the pebbles into the two bags. A coin will then be flipped. If it is heads, you discard bag A. If it is tails, you discard bag B. From the remaining bag, you reach in and randomly select a pebble. If it is white, you win - black you lose. How should you put the pebbles in the bag?
Does it make a difference? This problem is hard. If this is your first look at this page, I recommend skipping it. There are a countably infinite number of mathematicians in a room.
Each has a black or white hat placed on his or her head. Color selection is random. Each mathematician can see the color of everyone else's hat, but not his or her own. At the same moment, with out communicating with each other, all must guess the color of his or her own hat.
What strategy could the mathematicians use to guarantee that only a finite number guess incorrectly? The mathematicians may talk before the hats are placed in order to agree on a strategy. You may assume the axiom of choice. Toggle navigation. Publications Logic Puzzles Teaching. Logic Puzzles This is a collection of logic puzzles.
Four Card Code You select five cards from a standard 52 card deck no jokers and place them on a table. The Othello Problem You are in a completely dark room. The 7 Hats Problem You are in a room with 7 or any other number greather than 3 friends. Hanging a Picture You have a framed painting the kind with a string coming out of the top left and attached to the top right that you want to hang on the wall using two nails, such that if you remove any one nail the painting will fall, but with both nails in the wall it will not fall.
Scales On Steroids The original version: There are nine stones. The Mountain Climber You have a rope that is feet long. The Perfectly Logical Pirates There are pirates. Black and White Hats There are mathematicians including you! Ant in a Room An ant is in one corner of a room shaped like a cube. The Four Quarters Problem You have a square cafeteria tray with four quarters on it - one in each corner.
The Two Quarters Problem Do not physically do this - it will ruin it. The Duck and the Fox There is a duck in the middle of a perfectly circular pond radius 1.
Water and Wine You have two identical glasses, filled to the same level. Car Parts If a car is traveling at 60 miles per hour, what part is stationary? Coin Flip Contest Bob flips fair coins. Rocky Waters You're in a boat with a big rock, in a swimming pool. Gamers of the Round Table You are given a round table, and a large number of coins. Fox and the Foxholes There is a straight line of N foxholes, and one fox. The Monks with Red Eyes There is an island with red and blue eyed monks.
The Three-Way Duel There are three people who want to have a duel. The Fuses You have two lengths of fuse. The Hungry Cow You have a cow in a circular pen radius 1. The Weird Warden There is a prison with inmates.
The Really Really Big Urn You have a very big urn, and pebbles numbered with the natural numbers 1, 2, Working on his commission, Jonny asks Spencer to pass him the pair of plyers. Spencer asks where he came up with the idea, and when Jonny tells her that he was inspired by Rosewood, Spencer questions him, to which Jonny mentions that the whole town is strung together with whispers.
Out, Damned Spot. At Hollis College, Jonny listens as Spencer questions whether he usually paints in the dark. Spencer notes that she used to come to Hollis a lot when she was a kid and that her parents would take her and Melissa to a summer concert series back when she thought she was going to be a famous violinist.
Rolling his eyes, Jonny turns his headlight on and tells Spencer that he usually manages. As Spencer asks whose going to see it, Jonny tells her the higher ups will, before telling Spencer to catch. Spencer quickly catches the can of spray paint Jonny has thrown, before telling him that throwing things at people on ladders is not cool. Jonny and Spencer have a moment, before backs away and hands Jonny the head torch.
Climbing the ladder, Jonny takes another spray can from his back pocket and after looking down to Spencer, he begins to spray the roof. After the two share a look, Jonny says goodnight, before walking out the French doors. Pretty Isn't the Point. Walking down the street, Jonny tells Spencer that if that was awkward, she could have just introduced him to Toby as the guy who lives in the barn. Noticing something in a window, Jonny stops, but Spencer, unaware, keeps on walking.
In the art gallery, Jonny and Spencer stand before two pieces of their vandalism. When the gallery owner comes up behind them, saying that the pieces are compelling, Jonny and Spencer turn. The gallery owner explains that someone from Hollis saved them from hauled off in a garbage truck, so technically they are his until someone buys them. As Spencer questions why, and if something broke, Jonny says that he just needs her toolbox. Saying that he can take the door of its hinges, Jonny listens as Spencer tells him that before he plans on stealing back his own work, he should let her talk with her mom because he must have some legal recourse.
Spencer asks which one, and when Jonny tells her it was the corner rabbit, Spencer exclaims that she painted that one. Again asking Spencer where her toolbox is, Jonny watches as Spencer goes to grab it. Pulling up outside of the art gallery, Jonny gets out of his van and runs up the steps to the gallery. As he struggles with picking the lock, Spencer comes up behind him and asks if she can help.
After Ali's funeral, the Liars all receive the same text from "A": I'm still here, bitches, and I know everything. The Jenna Thing. To Kill a Mocking Girl. Can You Hear Me Now? The Liars try to avoid "A"'s messages by blocking unwanted emails and calls on their cell phones. When Spencer is escorting Wren to his motel room, "A" calls in a song dedication to the radio station, to which Hanna is listening.
Reality Bites Me. Later, "A" sends Hanna the photo booth pictures of Emily and Maya, exposing to her that Emily is gay. Ezra sees a text from "A" on Aria's phone that hints at their relationship and, assuming she has told her friends about it, breaks up with her. There's No Place Like Homecoming. At the end of the episode, the black hooded figure is seen vandalizing a Rosewood town sign. The figure spray paints over the last digit in the population census the 8 in and changes it to It is possible that "A" is referring to Emily, but this is unknown.
The Homecoming Hangover. Emily sees a smashed-up motorcycle in front of Toby's house, and when she tells her friends about it, "A" texts her, "Thanks for getting Toby out of my way. Spencer receives a text from "A" threatening her to do Ali's memorial "right.
The Perfect Storm. At the end of the episode, the black gloved figure anonymously sends a file to the Rosewood Police Department. The file contains footage of Alison the night she disappeared.
Before filling in Anonymous in the top left corner of the envelope, "A" scrawls the letter 'A', then a few seconds later, fills in the remaining letters. This leaves no doubt that "A" and the black gloved figure are one and the same. Keep Your Friends Close. In the same episode, at the climax of the episode, Hanna seems to figure out who "A" is while spying in the forest. But before she can reveal the identity of "A," she is run down by a car while running to the other liars and loses consciousness.
Just then, the girls receive a text from "A" claiming responsibility for the hit and run. In the hospital, after her accident, Hanna tells the girls that Noel Kahn wrote the message on the back of Ezra's car and he is presumed to be "A". However, this is proved untrue when "A" plants the answers to three midterm exams in Noel's locker, because he was going to tell the principal that Aria and Ezra were dating.
Salt Meets Wound. Hanna gets back to her house and Mona decides to throw a "Welcome Back Party". After the party, the money Ashley had stolen from the bank is missing and "A" can now blackmail Hanna. If she does what they say, Hanna will get her Mother's money back. In the same episode, "A" decides to break up Spencer and Alex by confirming Alex's opportunity to go to Sweden, even though Alex did not want to go. Spencer, being the only one that knew about this, was blamed.
Alex has yet to be seen in the series since. We then see "A" slipping the hundred dollar bills from Hanna's kitchen into a creepy clown piggy bank. Know Your Frenemies. Later, "A" decides to give them the entire video of Alison's disappearance. That is when the liars suspect Ian as the murderer of Alison. Careful What U Wish 4. At the end of the episode, it is revealed that "A" had been among the guests at the school's dance marathon.
Clearly, "A" had gone to the dance in order to dictate to Hanna her dance instructions for the evening, and deliver Hanna's paycheck to her coat pockets. It becomes clear that "A" had commissioned the bead lady to fashion a fake Alison bracelet and a fake Jenna bracelet, placing the order under Spencer Hastings' name.
At the end of the same episode, the eccentric old woman prepares a cup of tea for her house guest. The black gloved figure takes it from her as the lady comments on the beautiful black gloved figure's eyes.
She also tells the guest not to worry Je Suis une Amie. At the end of the episode, the black gloved figure is listening to a record playing a French tutorial for English speakers, while arranging some clue-like objects.
There is a wrench, a pipe, a knife, a rope resembling a noose, and a marble box which rests on a candlestick, which seems fiddled with somehow.
The record skips, repeating "Je Suis" I am three times. The New Normal. Apparently, "A" has a sense of humor. At the end of the episode, the black gloved figure approaches Mrs. Potter's grave, which reads The Badass Seed. In the last scene after the trophy fallout , it is presumably showing a room in "A"s house. There are four cages, each containing a white rat, animal feed, and a wheel.
Each cage has the name of each liar taped to the top of it. The cage labeled 'Spencer' is missing a rat, however. Earlier, "A" had planted a phony trophy with rats blood in the props room compartment in the school's drama room for the liars to find. A Person of Interest. The Neumann's bag, which Jenna had taken from Ian, is filled with ice on the bed of the motel room that "A" must have rented.
A recording of Jenna's flute music is playing on the stereo, whereas Spencer actually thought Jenna was practicing inside. At the end of the episode, the black gloved figure is sitting comfortably in a room somewhere, beside a bowl of popcorn, watching the footage of the girl's reaction to "A"'s text on Spencer's front lawn, which let them know that Spencer is now a person of interest in Alison's murder investigation.
Someone to Watch Over Me. At the end of the episode, we see "A" preparing a gift for Hanna; a glass heart with Hanna's name on it, smashed by "A" with a hammer and some crazy glue with a custom made note: Dear Hanna, try putting the pieces back together -A. Monsters in the End.
At the end of the episode, the black gloved figure is seen swiping the key from underneath Ezra's doormat. For Whom the Bell Tolls. Spencer is saved from being murdered by Ian by a hooded figure that is believed to be "A. Although, she didn't kill Ian and saw him walk away. After Ian disappears from the church, "A" sends the girls a text assuring them that their nightmare is not over. Sullivan's office for the liars to see during their session. At the end of the episode, "A" is seen in Emily's home erasing her computer hard drive which contains all the videos.
When the liars sit at separate lunch tables, "A" sends them all the same text: Look at you, all alone in a crowd. I win! Xoxo - A. My Name Is Trouble. After Emily tears up the fake Danby letter and tosses it in her trash can, an unharmed replica of the letter gets sent to the Field's residence.
Emily scrambles to her trash can to find the torn up letter, but instead finds a letter from "A. Later on in the episode, after Spencer pawns Melissa's engagement ring for what she hopes is only 48 hours, "A" finds the cash to buy it from the Pawn Shop owner. After Hanna's visit to Dr. Although the message isn't signed by "A", it is pretty obvious based on Ezra's degree and the text message, that "A" ravaged her office after Hanna's visit.
At the end of the episode, "A" stages the suicide of Ian Thomas. When the girls run to Melissa's aid, "A" puts Ian's phone in Spencer's bag and leaves unnoticed. This is the same phone "A" was using when texting Melissa, pretending to be Ian. The Devil You Know. It is revealed that "A" wrote Ian's suicide note. As Spencer is about to, "A" calls Ian's phone which Melissa finds and becomes convinced that Spencer was pretending to be Ian.
In the same episode, "A" reveals to the girls via video that Ian may not have killed Alison. It reveals the full video of her and Ian on the night of her disappearance. It shows Ian getting up and walking away from a giggly, happy Alison, smiling at the camera. They go in search of the projector, but "A" runs away, projector in hand. At the end of the episode, "A" tampers with the tapes so that in the middle of the fashion show, the screen picture changes to make Alison look like the devil, with fire around and peeling, black eyes.
The screen flashes, "the bitch is dead," and the music changes from sentimental to hellish rock. The girls flee from the runway as well as Jessica DiLaurentis. Spencer turns it off, leaving the audience ill stricken. At the end of the episode, "A" is seen using a syringe to siphon liquid Human Growth Hormone HGH , and injecting it into Emily's tube of pain relief cream. While Emily is with Samara and her friends, "A" sends her a message, ordering her to get Zoey's phone number.
Emily obviously had not wanted to, but since "A" had the results that showed the presence of the hormones in her body, she had no choice. Touched by an 'A'-ngel. Emily opens an individual-sized box of Alpha-Bits cereal at school to find it only contains "A's," and "A" has left her a message in a plastic capsule used for cereal box toys. Later, "A" crashes Emily's massage, giving her the massage instead of the professional, without Emily realizing until she sees the letter "A" etched into the condensation of her water glass.
At the end of the day, "A" takes Emily's psychological therapy session when Emily doesn't show up. It is revealed that "A" has been a consistent patient of Dr.
While Dr. Sullivan is perusing the files in her office, a black hooded figure is seen standing outside her window, watching the woman inside. Sullivan scans "A's" file, which shows that "A" has an obsession with revenge and refers to her enemies as "nosey bitches. Sullivan calls the girls, wishing to tell them who "A" is. They show up to her office, but she is missing. Something "A" takes credit for in a text message. At the end of the episode, the black hooded figure is seen removing a sound device, which had been planted under the bobble head Freud in Dr.
Sullivan's office. Over My Dead Body. Sullivan in a booth at an unknown diner. Sullivan tells her captor that she's done everything they've asked.
Full of money? Sullivan looks disgusted and leaves. Next, a waitress approaches the table where "A" is sitting, fills her coffee cup, and inquires how her pretty-eyed patron is doing. In this prequel episode, "A" sends Alison two texts. One when she is in the costume store, and one at the end. From the second text, it becomes clear that "A" had posed in the doll-mask and burlap cowl while in The Haunted House and possibly also in the costume store.
Alison had gotten away though when she kneed "A" and ran for it. Through Many Dangers, Toils, and Snares.
The liars trick "A" by pretending to fight with each other over "evidence" of "A's" identity, which in reality doesn't exist.
They hope this will reveal "A's" identity to them by forcing "A" to meet them in person. This plan fails, but they come away with "A's" cell phone. At the beginning of the episode, someone calls "A's" phone from a blocked number. When Emily is volunteering at a crisis center to fill her community service, she reads a transcript that sounds much like the situation that happened with Hanna, Aria, Spencer and Emily.
When the person calls back again, Spencer and Emily's boss tells them to listen in and take notes. They realize that the caller is Lucas and he must be a helper for "A. Let the Water Hold Me Down. Hanna and Mona get into an argument, resulting in Hanna running into the bathroom crying, she quiets down when she hears someone else enter, after a while, she notices the sink is overflowing.
She sees a small boat floating in it and a piece of paper that says, "Life is but a dream, Hanna. And I'm your worst nightmare - A". At the end of the episode, "A" is seen climbing some scaffolding and loosening a bolt in the wall of an unknown location, which is later revealed to be the barn in Spencer's backyard.
The Blond Leading the Blind. Toby has a broken arm and Spencer ends the relationship. The final picture "A" burns up is not seen. A Kiss Before Lying. At the end of the episode, "A" is seen unlocking the drawer of Mr. Hastings' desk and taking a gun out. At the end of the episode, "A" watches Emily through her window as she tries to call Maya. Breaking the Code. At the end of the episode, a gloved figure is seen shooting at a target, with Mr. Hastings' stolen gun. Father Knows Best.
At the end of the episode, "A" is seen putting two quarters into the newspaper holder for the day's copy of the Rosewood Observer. She takes a paper, featuring a huge picture of Maya on the cover, with the headline "Missing Girl. Eye of the Beholder. It is assumed by the girls that "A" was the one who texted Jenna and brought her over to Jason's house, shortly after setting it on fire, with the intention for Jenna to die there.
Hanna pulls Jenna out after coming back for Ali's bag that Jason gave the Liars. The house blows up, but only the downstairs is damaged. The final scene shows a gloved figure rummaging through the wreckage from the fire, and planting a police badge at the scene.
If These Dolls Could Talk. At the end of the episode, Martha tells "A" she kept her end of the deal, but the giant mess she has to clean up is much more than she bargained for. Seth is seen staring intently at "A. The Liars take a room in the Lost Woods Resort motel. They stay in room 1 the same room Alison stayed in while she was alive while, "A," lurking in the shadows, weaves in and out of their room without the girls ever knowing.
She even visits while Hanna is in the shower, though "A's" intentions are unknown. She also spies on them from room 2, where "A" has presumably rented her own room. At the costume ball, Mona shares with the Liars a story of how Alison, dressed as Vivian Darkbloom, had been spying on someone in a costume shop the week before she died. This prompts Spencer to realize that Alison had been searching for "A" all along and must have rented Room 1 in the motel in order to spy on the person in Room 2 - "A!
Spencer is freaked out; the room is covered with pictures of all the Liars and Alison. There are Bratz Dolls that resemble each of the five girls perched in "A's" dollhouse. Then, when Spencer spots a gum wrapper of the same type that Mona offered her, in Alison's stolen diary, she suddenly becomes suspicious.
When she sees the bag of cashmere sweaters, the understanding that Mona is "A" dawns on her. She turns around to confront Mona, who is now wearing her signature black hoodie. Mona knocks her out with a flashlight in return. Next, Spencer finds herself riding in the front passenger's seat of Mona's car. Mona threatens her to join The "A-Team", or die at gunpoint from Mr.
Spencer gets Mona to reveal her motive. She is angry at the liars for stealing Hanna, the only friend she ever had. She claims to be over Alison and the way she had treated her. Spencer uses her phone to convey the message to the other liars, and Mona, without knowing it, is caught on video conversation for the other three liars to hear. Hanna is devastated when she learns that Mona is "A".
Spencer lets the girls know where Mona is taking her by warning Mona that if she doesn't slow down, she won't make it to Look Out Point.
The girls move fast to go rescue their friend. Hanna catches up to the other two with her car and stops just short of hitting "A" with her car in anger. The other girls jet out of the car to help, but before they can, Spencer unintentionally knocks Mona off the cliff during the struggle, falling several feet into a state of unconsciousness.
Later, the police declare "A" alive, and Dr. Sullivan returns to town to comfort the girls, revealing that "A" was threatening her son which forced her to flee. The police take her to a room with a one-way mirror. The girls look on with Dr.
0コメント