Rope Around Earth

Suppose you wrap a rope tightly all the way around the earth. You then cut the rope and tie in a six feet long segment. How high off of the ground will you now be able to lift the rope, keeping it the same height all the way around?

Promiscuous Villagers

In a certain village, there are 50 couples. As it turns out, everyone in the village is having an affair. This despite a particularly gruesome custom which requires a wife, upon discovering that her husband is having an affair, to kill him the following morning. Even more oddly, the women in the town talk quite freely about their activities! In fact they are all perfectly aware that any woman who is having an affair will tell EVERY other woman in the village except, of course, for the wife of the man with whom they are having the affair. Yet life there goes on quite peacefully since no woman can know for sure that her husband is actually having an affair. One day, a well revered wise man visits the village and announces that someone is having an affair. What happens after this?

The Lame Fly Problem

A rectangular room measures 30 feet in length and 12 feet in height, and the ends are 12 feet in width. A fly, with a broken wing, rests at a point one foot down from the ceiling at the middle of one end. A smudge of food is located one foot up from the floor at the middle of the other end. The fly has just enough energy to WALK 40 feet. Show that there is a path along which the fly can walk that will enable it to get to the food.

Patriarch's Will

Here is a scenario which occurred many millenia ago: The patriarch of a wealthy family was on his deathbed and wanted to divide his gold among his eight sons who were all very very greedy. Wishing to favor the oldest son (as tradition would have it) but also to reward the more cunning of his progeny, he made the following decree:
The oldest son is to propose a plan for dividing up the gold. The sons are all to vote on this plan, and if it receives at least half of the votes (four or more) then that will be the way the gold is divided. If this plan does not receive half of the votes, the oldest son gets nothing, the next oldest proposes a plan, and there is another vote, now among the remaining seven. Again at least half of the vote (still four or more) is required, and failure removes this son from the process. This is to continue until some son's plan receives at least half of the votes of the remaining heirs.
Assuming that these sons will do anything to get the most gold possible for themselves, how much (if any) will the oldest son be able to inherit?

Clever Bartender

A man walks into a bar and after a few drinks has the following conversation with the batender: 
      Man: Try to guess the ages of my three daughters. 
Bartender: Well, I won't guess but if you give me clues, I'll try to
           figure it out.
      Man: Ok.  The sum of their ages is eighteen.
Bartender: That's not enough.  I need more information.
      Man: Well, the age of one of them is the address above the door
           outside.
Bartender: (after returning from looking at the door) That helps, but
           I still don't know all of their ages.
      Man: Alright then, my youngest girl's favorite flavor of ice 
           cream is strawberry.
At this point the bartender promptly told the man the ages of all three of his daughters. What were they?

Market Women

Two market women were selling apples, one at 3 apples for a penny and the other at 2 apples for a penny. One day when both were called away they left their stock in charge of a friend. To simplify her reckoning the friend amalgamted the stocks - there were 30 apples of each quality - and sold them all at 5 for two pennies. Thus the friend took in 24 pennies. When it came to dividing the proceeds between the two owners, trouble arose. The one who had turned over 30 apples of 3 for a penny quality demanded her due 10 pennies. The other not unreasonably asked for 15 pennies. The sum actually realized was a penny short. Where did it (the missing penny) go?

Prisoners Hats

Three men being held in a torture dungeon (or are they just physics grad students in the basement of RLM?) are brought before their captor who has decided to give them a chance to go free. He explains the rules of a game that might win them emancipation. The captor has collected five hats - three red and two white. He will blindfold the prisoners, place a hat on each one, then remove the blindfold from one man at a time. That prisoner can then see the other two hats and try to determine the color of his own. A correct answer, of course, is a ticket out of bondage. The first prisoner, upon seeing the other hats, says he doesn't know what color his hat is. When the second man's blindfold is removed, he also does not know. However, before the third prisoner's blindfold is even touched, he correctly announces the color of his hat. He is allowed to leave while the other two are forced to stay as slave labor building a new fusion reactor. What color was the last prisoner's hat, and how did he know?

Hell

You are in Hell. The devil challenges you to a game. He will think of a natural number, and you can have one guess at it each day. (Once he has thought of his number, he will never change it.) If you guess it, he'll let you out. Will you get out? (Remember - you have eternity...) Explain.

What if he instead told you that he was going to pick an integer - now will you get out? What about if he chose a rational number? A real number?

Camels and Bananas

A camel has to carry 3,000 bananas across a 1,000 mile dessert. There are two conditions: 1. The camel can only carry 1,000 bananas at a time. 2. The camel eats one banana for each mile he travels. What is the maximum number of bananas that will reach the other side of the desert? What if you have three camels?

Light Switches

A working light bulb is in a closed room with no windows. Outside the room, is a panel of three switches, one of which controls the light inside (up is on, down is off.) You may do anything you like to the three switches and then enter the room to inspect the light. After this, without any further experimentation, you must indicate which switch controls the light. What do you do?

Every Parent's Dilemma

A cake has dimensions 15'' x 15'' x 3''. It is frosted on the sides and top. How can it be divided into 5 pieces so that each piece has the same amount of cake and the same amount of frosting? What if the dimensions are changed? What about 6 pieces? 7 pieces? n pieces?

Bears

A bear walks one mile due south, then one mile due east, and then one mile due north. He ends up at exactly the same spot where he started. What color is the bear? Give a rigorous proof.

Slow Train?

A train wheel has two rims as shown below and is designed to run along a bi-level track. The outer (smaller) rim has radius 10 inches while the inner (larger) rim has radius 12 inches.
  1. How far does the outer rim roll over the course of one rotation of the wheel?
  2. How far does the inner rim roll over the course of one rotation of the wheel?
  3. How far does the train travel over the course of one rotation of the wheel?

Sad Story

A boy and his father are driving along a windy road and have a terrible accident in which the father is killed. The boy is rushed to a hospital, and the doctor enters the room only to say, ``I can't operate on this boy. He's my son.'' How can this be?

Deserted Island Medicine

Four people are stranded on a deserted island. Two of them are badly injured, and (luckily) the other two are doctors. However (not so luckily), each of the four people carries a virus which could be deadly to the other three. Each of these viruses can be transmitted by simply touching something where the contagious person had previously touched it. There are two surgical gloves. Is it possible for each of the two doctors to safely examine both injured people?

McNuggets

What is the largest number of McNuggets that can not be ordered? (McNuggets can be ordered in packages of 6, 9, or 15.)