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Puzzles 
Any group of 6 or more contains either 3 mutual friends or 3 mutual strangers. Prove it.
See Answer
Take a person X. Of the five other people, there must either be at least three acquaintances of X or at least three strangers of X. Assume wlog that X has three strangers A,B,C. Unless A,B,C is the required triad of acquaintances, they must include a pair of strangers, wlog A,B. Then X, A, B is the required triad of strangers, QED.

Two fifty foot ropes are suspended from a forty foot ceiling, about twenty feet apart. Armed with only a knife, how much of the rope can you steal?
See Answer
Almost all of it. Tie the ropes together. Climb up one of them. Tie a loop in it as close as possible to the ceiling. Cut it below the loop. Run the rope through the loop and tie it to your waist. Climb the other rope (this may involve some swinging action). Pull the rope going through the loop tight and cut the other rope as close as possible to the ceiling. You will swing down on the rope through the loop. Lower yourself to the ground by letting out rope. Pull the rope through the loop. You will have nearly all the rope.

Two people are talking long distance on the phone; one is in an East Coast state of the US, the other is in a WestCoast state of the US. The first asks the other "What time is it?", hears the answer, and says, "That's funny. It's the same time here!"
See Answer
One is in Eastern Oregon (in Mountain time), the other in Western Florida (in Central time), and it's daylightsavings changeover day at 1:30 AM.

You have ten boxes; each contains nine balls. The balls in one box weigh 0.9 kg; the rest weigh 1.0 kg. You have one weighing on an accurate scale to find the box containing the light balls. How do you do it?
See Answer
Number the boxes 09. Take 0 balls from box 0, 1 ball from box 1, 2 balls from box 2, etc. Now weigh all those balls and follow this table:
If odd box is Weight is
0 45 kg
1 44.9 kg
2 44.8 kg
3 44.7 kg
4 44.6 kg
5 44.5 kg
6 44.4 kg
7 44.3 kg
8 44.2 kg
9 44.1 kg

Suppose that it is equally likely for a pregnancy to deliver a baby boy as it is to deliver a baby girl. Suppose that for a large society of people, every family continues to have children until they have a boy, then they stop having children. After 1,000 generations of families, what is the ratio of males to females?
See Answer
The ratio will be 5050 in both cases. We are not killing off any fetuses or babies, and half of all conceptions will be male, half female. When a family decides to stop does not affect this fact.

The "house" rolls two 20sided dice and the "player" rolls one 20sided die. If the player rolls a number on his die between the two numbers the house rolled, then the player wins. Otherwise, the house wins (including ties). What are the probabilities of the player winning?
See Answer
It is easily seen that if any two of the three dice agree that the house wins. The probability that this does not happen is 19*18/(20*20). If the three numbers are different, the probability of winning is 1/3. So the chance of winning is 19*18/(20*20*3) = 3*19/200 = 57/200.

Given two random points x and y on the interval 0..1, what is the average size of the smallest of the three resulting intervals?
See Answer
You could make a graph of the size of the smallest interval versus x and y. We know that the height of the graph is 0 along all the edges of the unit square where it is defined and on the line x=y. It achieves its maximum of 1/3 at (1/3,2/3) and (2/3,1/3). In between these positions the surface forms a series of planes. Thus the volume under it consists of 2 pyramids each with an altitude of 1/3 and an (isosceles triangular) base of area 1/2, yielding a total volume of 1/9.

A man walks into a bar and asks the barman for a glass of water. The barman pulls out a gun and points it at the man. The man says 'Thank you' and walks out.
This puzzle has claims to be the best of the genre. It is simple in its statement, absolutely baffling and yet with a completely satisfying solution. Most people struggle very hard to solve this one yet they like the answer when they hear it or have the satisfaction of figuring it out.
See Answer
A man walks into a bar and asks the barman for a glass of water. The barman pulls out a gun and points it at the man. The man says 'Thank you' and walks out.
This puzzle has claims to be the best of the genre. It is simple in its statement, absolutely baffling and yet with a completely satisfying solution. Most people struggle very hard to solve this one yet they like the answer when they hear it or have the satisfaction of figuring it out.

A man is lying dead in a field. Next to him there is an unopened package. There is no other creature in the field. How did he die?
See Answer
The man had jumped from a plane but his parachute had failed to open. It is the unopened package.
This is sometimes given with the following rather elegant clue  as he approached the centre of the field he knew he was going to die. This is another of the top classics which is right up there with 'The Man in the Bar'. If the solver is thinking along the wrong lines (i.e. in the two dimensions of the ground) then the lateral jump to the third dimension can be tough to make.

Five pieces of coal, a carrot and a scarf are lying on the lawn. Nobody put them on the lawn but there is a perfectly logical reason why they should be there. What is it?
See Answer
They were used by children who made a snowman. The snow has now melted. Another change of state puzzle. After this you should be on the lookout for them!

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