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You have twelve coins, one of which is known is known to be counterfeit and therefore heavier or lighter than all the others. You also have a balance-pan scale. In three weighings determine which coin is counterfeit and whether it is heavier or lighter. You have ten big vats of 10-gram coins. Except that one of the vats contains counterfeit coins, which weigh only 9 grams. You have an accurate (numerical) scale. In one wieghing, determine which vat contains the counterfeit coins. What is the smallest number of coins this requires? Same as above, except that any number of vats (including none) may contain counterfeit 9-gram coins. In one weighing, find the counterfeit vats, using as few coins as possible.	Coin weighing Light bulbs Knights & knaves Miscellaneous River crossing Census takers
In the lobby of a hotel there are three toggle switches. One is attached to a light bulb on the third floor; the other two are attached to nothing at all. You may work the three switches as often as you please, but you may visit the third floor only once. How can you tell which switch is attached to the light bulb? You have a tower with 1000 wires running from the top to the bottom. You also have a battery and a lightbulb. The wires are coaxial, i.e., a battery - wires - lightbulb connection will light the bulb. (You can tie wires together to form one long wire.) Figure out which wires at the top of the tower correspond to which wires at the bottom, in as few traversals of the tower as possible (i.e., as little stair climbing as possible). Same as above, except that now you have wires that are not coaxial, but just wires. For the light bulb to go on, you need two connections: battery - wires - bulb - wires - battery. What is the new answer for least number of tower traversals?	Coin weighing Light bulbs Knights & knaves Miscellaneous River crossing Census takers
There are three gods: True, False, and Random. They answer accordingly. but not in English! They say "da" and "ya" instead of "yes" and "no", and you dont know which is which. With three questions determine who is who. There are three gods: And, Or, and Xor. They all answer the truth, but of all the questions have been asked thus far they apply their namesake operation. Identify using three questions. Same names as above, all answer the truth. But you have to pick your questions beforehand, and they answer the operation applied to all questions. Identify in three. Same as above, except that they apply their respective operations to all of the questions not asked to them. There are three gods: Past, Present, and Future. They'll answer the truth, but: Present answers the question you are asking; Past answers the last question you asked; Future answers the next question you will ask. If you ask the first question to Past or the third question to Future, they give a random answer. Because of possible time conflicts, you must determine your questions ahead of time, rather than based on previous answers. You are, however, allowed to choose who you ask your three questions to dynamically. No time related questions are allowed (i.e., "if the answer to my q2 was no, then... otherwise ..."). They answer with das and yas like before. With three questions determine who is who.	Coin weighing Light bulbs Knights & knaves Miscellaneous River crossing Census takers
Suppose you start at the Earth's equator and travel continuously northwest until you reach the North Pole. What does your path look like? How long is your path? A man leaves his camp and walks one mile south. He then spots a bear 1 mile due east of him, and shoots the bear with his rifle. He walks the 1 mile eat to the bear, then carries the bear 1 mile north, arriving at his camp. What color is the bear? Though the previous problem has a unique correct answer, there is not a unique possible location for the camp. What is the locus of all points on a spherical earth satisfying this geometry - i.e., walking 1 mile south, one mile east, then one mile north brings you to your starting place? Consider the (perfectly spherical) globe. You want a plane to circumnavigate it (around a great circle). You have one airplane base and as many airplanes as it takes, but each plane can only hold enough fuel to get it halfway around the world. Airplanes can refuel instantaneously in midair. How can you get one plane around the world (without any of the planes crashing - they all must have enough fuel to get back to the base), using as few airplanes as possible? You have an angry couple: wife in a boat in the middle of a circular lake, and husband on the shore. The man can run on land four times as fast as the woman. Can the woman escape the lake (she can run faster than the man on land) or does she get caught by the man? Replace "four" by other values in the previous problem. Who wins? What is the transition between wife winning and husband winning? Two cars start 50 miles apart, heading towards each other. One travels at 30 mph, the other at 20 mph. At the same time that the cars start driving, a bug that can fly at 60 mph begins flying back and forth between the two cars. How far has the bug flown by the time the two cars pass each other? You have four ants at the four corners of a square, each facing the one in front of him in a cyclic order. They simultaneously begin walking towards the and in front of them (i.e., A walks towards B, B towards C, C towards D, and D towards A), always heading straight at the next ant. How far have they travelled by the time they meet at the center of the square? A deer and a lion are trapped in a circular cage. They both walk at the same speed. Can the lion catch the deer? A dog is chasing a cat. They both run at the same speed. The cat initially heads in the direction perpendicular to the dog, and always walks in that direction; the dog always walks straight towards the cat. What happens? Although Mt. Everest, at 29,028 feet above sea level, is the highest mountain in the world, it is not the farthest from the center of the earth. The earth's bulge at the equator pushes Chimborazo in Ecuador, at 20,561 feet above sea level, farther. (How much?) Now suppose you run a water pipe from Everest to Chimborazo. Which way would the water flow? You have a cake that you want to split fairly between n people, in the sense that each of them thinks they have at least 1/n of the cake. (They may judge the sizes of portions of the cake differently.) For two people, this can be accomplished by having one person cut the cake exactly in half according to his point of view, and then the second person picks the bigger piece (according to his point of view). Then each thinks they have at least half of the cake.   1. How do you divide a cake between three people?   2. How do you divide a cake between n people? Subdivide a square into 8 acute triangles. Subdivide an obtuse triangle into (finitely many) acute triangles. You have an L-shaped piece of paper (i.e., six sides, all at right angles). Using a straightedge and compass, draw a line cutting its area in half. Repeat, using only a straightedge. Repeat, using only a straightedge and ensuring that you divide the paper into only two (connected) pieces. You have an infinite checkerboard, with a piece on every black square below the x-axis. Pieces jump diagonally as in checkers, and you remove the piece being jumped over. How high above the x-axis can you get a piece? Repeat, with a piece on every square below the x-axis and pieces jumping horizontally or vertically. You have an integer written out in decimal. When you chop off hte last digit and put it on the front, this doubles the integer. What is the smallest possible alue for the integer? Same as above, except that it halves the integer.	Coin weighing Light bulbs Knights & knaves Miscellaneous River crossing Census takers
You hve a cup of coffee and a cup of milk. You take a teaspoon of milk and put it in the coffee. You take a teaspoon of the milk-coffee mixture and put it back in the milk. Now, is there more milk in the coffee or more coffee in the milk? You have an old-fashioned milk bottle (think: 2-liter bottle with a flat bottom), with a neck narrower than the body. It contains milk, with a layer of cream on top, in the neck. Now you shake up the bottle so the milk and cream are mixed together. Does the pressure on the bottom of the bottle increase, decrease, or say the same? What happens to a match in zero gravity? You have a helium balloon in a car. The car turns right. What happens to the balloon? A farmer has a fox, chicken, and sack of grain that he wants to take across a river. He can only take one of the three with him in the rowboat, but if the fox is left alone with the chicken it will eat it, and if the chicken is left alone with the grain it will eat it. How does he get across without losing any of his possessions? Four people want to cross a rickety bridge at night, but they only have one flashlight. Any time people cross the bridge, they must carry the flashlight with them; but the bridge only holds two people. The people walk at different rates: it takes them 1, 2, 5, and 10 minutes, respectively, to cross the bridge; if two people cross the bridge together they must travel at the slower of their rates. How can they cross taking the least amount of time? Three innocent bystanders and three cowardly axe murderers are trying to cross a river, but they only have a boat that can hold two people. If at any time the axe murderers outnumber the innocents (on some side of the river), they will slaughter them. The axe murderers are, however, friendly, so that they will follow the bystanders' instructions (with regards to the boat). How can everyone get across the river safely?	Coin weighing Light bulbs Knights & knaves Miscellaneous River crossing Census takers
A census taker approaches a house and asks the occupants how many children they have. They reply, "We have three children. The product of their ages is 36 and the sum of their ages is my house number." The census taker looks at their house number, comes back, and says, "I still can't figure it out." They respond, "My oldest daughter as red hair." The census taker says "Thank you" and walks away. How old are the children? Replace 36 by 72 in the above problem. C(ensus taker) and (Mrs.)S, two old friends, see a lady with her two children. S knows the lady and her children well.   C: How old is the lady?   S: The product of their ages is 2450 and the sum is your age.   C: Oh, I cannot figure out.   S: Of course, you can't. I am older than the lady.   C: OK. What the the ages of all five people involved? I pick two numbers between 3 and 98, inclusive, and tell their sum to S(teve) and the product to P(aula).   S: I know you don't know what the two numbers are.   P: Well, in that case, I know what the two numbers are.   S: OK, now I know what the two numbers are. What are the two numbers?	Coin weighing Light bulbs Knights & knaves Miscellaneous River crossing Census takers
puzzles.html ~ Andrei Gnepp ~ gnepp@fas.harvard.edu $Date: 1998-11-09 21:29:21-05 $