FYS: Puzzles and Paradoxes--Problem Set 6
Complete exactly three of the following five problems.
1. Answer the following riddles. Briefly justify how your answer fits all the data.
(a) What fills the sky, tears up new roots, shakes all foundations, but cannot be seen by eyes or touched by hands?
(b) What goes uphill and downhill and yet never moves?
(c) I am sitting above a horse which was not born, but whose mother I hold in my hand.
(d) What is it that's blind with an eye in its head, though the race of mankind its use cannot spare; spends its life in clothing the dead, though always itself is naked and bare?
(e) There be two sisters, of whom the one gives birth to the other, while she herself, after giving birth, is brought forth by the other.
Note: Full credit is possible for getting at least four out of five correct, but any incorrect answer will result in a score less than ten.
2. Suppose the head of a pin has been divided into an infinite number of regions. Suppose for each region a single angel is dancing in that region. Now suppose there are an infinite number of pins, each with an infinite number of regions each occupied by a single dancing angel.
Is it possible for all the angels on all the pins to move to the first pin and still have exactly one angel dancing in each region on the first pin? Justify your answer.
3. Two men enter the Last Chance Saloon. They both order and receive identical drinks. For some unknown reason they get into a serious argument. Frustrated, one gulps down his drink and storms out of the saloon never to return even though he lives another 40 years. Fuming, the other idly sips his drink and then, twenty minutes later, keels over dead. What happened?
Write a best explanation solution to this situation. Justify that your solution is indeed a best explanation solution.
4. Week 1: You receive, on Monday, a letter on official stationary from someone who claims to be a stockbrocker who has discovered a surefire method for predicting the weekly outcome of the stockmarket and various stocks. The letter makes the following prediction—the stock market will be up by the end of the week.
Week 2: You receive, on Monday, another letter from the stockbroker, reminding you of his surefire method and mentioning that last week the stockmarket did indeed go up. But in case you are not convinced he makes the following prediction—the stock market will be down by the end of the week.
Week 3: You receive, on Monday, yet another letter asking you to consider investing some money with his firm—after all the stock market did end down last week. But just to convince you further the letter makes the following prediction—the stock market will be down by the end of the week.
Week 4: You receive, on Monday, another letter pointing out that the market was down again last week and suggesting that investing with his firm is clearly a winning proposition. Still, if you are hesitant, the letter provides yet another prediction—the stock market will be up by the end of the week.
Week 5: The market was up last week and on Monday you receive a letter. Four weeks in a row—amazing it says. No other method in history has gotten week advance predictions right four times in a row! Still not convinced? What if my method gets it right five times in a row? I predict the market will be up by the end of this week.
Week 6: Sure enough the market is up. This week’s letter arrives promptly on Monday with instructions for how to open a $500 dollar minimum account with his firm, so that you too can take advantage of his surefire method and he can make even more money on the 2% commission on your trades in index funds, i.e. conglomerate funds that are designed to go up and down in lockstep with the market. You are sure to make money with his guaranteed method. And by the way, the market will be up again by the end of this week.
Week 7: The market is up—what are you waiting for?
What is the best explanation for his uncanny predictions? A. He has a method for accurately predicting whether the market will be up or down at the end of the week. B. He beat the odds (1 in 64) and managed to guess the direction of the market six weeks in a row. C. Something else. Defend and justify your answer. [If you pick C, you had better specify what this something else is.]
5.[Bonus Point Problem]: You are the last person in line out of 129 passengers waiting to get on a plane. Unfortunately for you the first guy in line is Crazy and once on the plane will randomly pick any one of the 129 seats. No one else wishes to confront Crazy Guy, so all the other passengers will act as follows: If one's own assigned seat is available, one will sit there, but if one's own assigned seat is taken, one will randomly pick from the remaining seats.
What is the probability that you will be able to sit in your own seat? Justify your answer.
Due: Friday, February 24 at the beginning of class.