Monday Musing: Cocktail Party Conversation Permit

Frg0061dIt is a frequently observed phenomenon that the less educated and intelligent people are, the more they tend to have decisive and strong opinions on the most complex political, philosophical, economic, and other pressing issues. You know the kind of person I am talking about, the one who is eager to quickly diagnose and solve a world problem or two with a single profound proclamation at every cocktail party. Like the two urbane and seemingly well-educated and well-dressed slightly older gentlemen I once overheard at a dinner party in Karachi (and there are plenty here in America, or anywhere for that matter) saying with great conviction (and with extremely thoughtful expressions on their faces, and in ponderous cadences, as if they were straining under the burden of a massive feat of cognitive strength and skill):

1st Guy: “Pakistan’s only problem has always been that our leaders lack sincerity.”
2nd Guy: “No, no, no. Our only problem has always been that our leaders lack committment.”

The first guy then actually carefully considered this pearl of wisdom from political philosopher and all-round theorist #2 and finally, having reevaluated his own sophisticated worldview in the light of this new gem, dumped it unceremoniously, humbly but gravely declaring defeat: “Yes… I see… you are right… it is a matter of committment.” In the throes of the cringing frustration one feels when faced with this sort of cretinism, I have sometimes felt that people should have to be licensed to spew profundities at cocktail parties, otherwise they should only be allowed to speak about either the weather or quantum theory. And the license would be received after demonstrating the ability to think about really, really, simple problems by passing a test. The idea, of course, being that if you can’t think lucidly, logically, creatively and successfully about very simple problems where all the information required to solve them is present in their statement, and which have very clear and demonstrable solutions, what the &$@# makes you think you should be engaging hard and incredibly complicated and intricate issues?

Okay, okay, for the last nine days or so I was out of town and very busy and that is my excuse for not writing a substantive column today. (Perhaps some of you noticed that I wasn’t posting all of last week?) Instead, now that I have given you some motivation to try and think about simple problems, I present a challenge to you: solve some logical and mathematical puzzles that my friend Alex Freuman sent me. Alex teaches high school physics and math at La Guardia High School here in Manhattan. (It was the model for the high school in the movie Fame.) I had seen some of the puzzles before but not others, and it took me a while to solve some of those. The first person to email me (click “About Us” at the top left of this page for my email) a full list of correct solutions, wins the privilege of writing one of our Monday columns for November 20th. Okay, so it’s not a huge prize, but hey, if you’ve got something to say, here’s your chance. And, of course, you will have earned the cocktail party conversational permit as far as I am concerned.

Screenhunter_5_7Don’t look up the solutions, and please don’t post solutions in the comments. Try to do all of them yourself. Believe me, even if you have to think for some days about a problem before you get it, there is a huge satisfaction and mental reward in doing so yourself. And you will feel more confident of yourself too. I shall, of course, trust you not to cheat. Here they are:

  1. You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it doesn’t have to burn at a uniform rate. In other words, half the rope may burn in the first five minutes, and then the other half would take 55 minutes. The rate at which the two ropes burn is not necessarily the same, so the second rope will also take an hour to burn from one end to the other, but may do it at some varying rate, which is not necessarily the same as the one for the first rope. Now you are asked to measure a period of 45 minutes. How will you do it?
  2. You have 50 quarters on the table in front of you. You are blindfolded and cannot discern whether a coin is heads up or tails up by feeling it. You are told that x coins are heads up, where 0 < x < 50. You are asked to separate the coins into two piles in such a way that the number of heads up coins in both piles is the same at the end. You may flip any coin over as many times as you like. How will you do it?
  3. A farmer is returning from town with a dog, a chicken and some corn. He arrives at a river that he must cross, but all that is available to him is a small raft large enough to hold him and one of his three possessions. He may not leave the dog alone with the chicken, for the dog will eat it. Furthermore, he may not leave the chicken alone with the corn, for the chicken will eat it. How can he bring everything across the river safely?
  4. You have four chains. Each chain has three links in it. Although it is difficult to cut the links, you wish to make a loop with all 12 links. What is the fewest number of cuts you must make to accomplish this task?
  5. Walking down the street one day, I met a woman strolling with her daughter. “What a lovely child,” I remarked. “In fact, I have two children,” she replied. What is the probability that both of her children are girls? Be warned: this question is not as trivial as it may look.
  6. Before you lie three closed boxes. They are labeled “Blue Jellybeans”, “Red Jellybeans” and “Blue & Red Jellybeans.” In fact, all the boxes are filled with jellybeans. One with just blue, one with just red and one with both blue and red. However, all the boxes are incorrectly labeled. You may reach into one box and pull out only one jellybean. Which box should you select from to correctly label the boxes?
  7. A glass of water with a single ice cube sits on a table. When the ice has completely melted, will the level of the water have increased, decreased or remain unchanged?
  8. You are given eight coins and told that one of them is counterfeit. The counterfeit one is slightly heavier than the other seven. Otherwise, the coins look identical. Using a simple balance scale, can you determine which coin is counterfeit using the scale only twice?
  9. There are two gallon containers. One is filled with water and the other is filled with wine. Three ounces of the wine are poured into the water container. Then, three ounces from the water container are poured into the wine. Now that each container has a gallon of liquid, which is greater: the amount of water in the wine container or the amount of wine in the water container?
  10. Late one evening, four hikers find themselves at a rope bridge spanning a wide river. The bridge is not very secure and can hold only two people at a time. Since it is quite dark, a flashlight is needed to cross the bridge and only one hiker had brought his. One of the hikers can cross the bridge in one minute, another in two minutes, another in five minutes and the fourth in ten minutes. When two people cross, they can only walk as fast as the slower of the two hikers. How can they all cross the bridge in 17 minutes? No, they cannot throw the flashlight across the river.
  11. Other than the North Pole, where on this planet is it possible to walk one mile due south, one mile due east and one mile due north and end up exactly where you began?
  12. I was visiting a friend one evening and remembered that he had three daughters. I asked him how old they were. “The product of their ages is 72,” he answered. Quizzically, I asked, “Is there anything else you can tell me?” “Yes,” he replied, “the sum of their ages is equal to the number of my house.” I stepped outside to see what the house number was. Upon returning inside, I said to my host, “I’m sorry, but I still can’t figure out their ages.” He responded apologetically, ‘I’m sorry. I forgot to mention that my oldest daughter likes strawberry shortcake.” With this information, I was able to determine all of their ages. How old is each daughter? I assure you that there is enough information to solve the puzzle.
  13. The surface of a distant planet is covered with water except for one small island on the planet’s equator. On this island is an airport with a fleet of identical planes. One pilot has a mission to fly around the planet along its equator and return to the island. The problem is that each plane only has enough fuel to fly a plane half way around the planet. Fortunately, each plane can be refueled by any other plane midair. Assuming that refuelings can happen instantaneously and all the planes fly at the same speed, what is the fewest number of planes needed for this mission?
  14. You find yourself in a room with three light switches. In a room upstairs stands a single lamp with a single light bulb on a table. One of the switches controls that lamp, whereas the other two switches do nothing at all. It is your task to determine which of the three switches controls the light upstairs. The catch: once you go upstairs to the room with the lamp, you may not return to the room with the switches. There is no way to see if the lamp is lit without entering the room upstairs. How do you do it?
  15. There are two gallon containers. One is filled with water and the other is filled with wine. Three ounces of the wine are poured into the water container. Then, three ounces from the water container are poured into the wine. Now that each container has a gallon of liquid, which is greater: the amount of water in the wine container or the amount of wine in the water container?

In case no one gets all the answers right, the highest score wins. In the case of a tie, whoever gets me the next correct answer first wins. And keep in mind that by no means am I suggesting that everyone should get the solutions of all the problems. Some of them are hard, and if you can’t figure them out, don’t worry about it. But keep trying! Thanks for sending the problems, Alex, and sorry but you are disqualified.

Ready, set, go!

UPDATE: We have a winner!

UPDATE 2: Answers here.

My other Monday Musings can be seen here.