Puzzles

Handshakes:

Five couples, including the host and his wife, attend a party. Throughout the evening, as introductions are made, various people shake hands (except couples do not shake each others' hands). At the end of the evening, the host discovers that each of the other nine people at the party shook hands a different number of times. How many hands did the host shake, and how many hands his wife shake?

3 Cards:

There are 3 cards: one is all red, one is all blue, and the third is blue on one side and red on the other. The cards are shuffled. You pick one at random and it is blue on the side facing you. What are the odds that it is also blue on the other side?

Five Hats:

Three wise men are lined up in single file, each wearing a white hat. They know there were 5 hats total, three white, and two black, but they can only see the hats on the men in front of them. They don't speak unless they figure out what color hat they have on. Who figures out first?

100 Switches:

You have 100 toggle switches, each connected to a different light bulb. At first, all light bulbs are off. The first person comes along, pressing every single toggle; thus turning al the blubs on. The second person comes alone, and presses every second toggle (i.e. 2, 4, 6, ..., 98, 100);thus, turning every other bulb off. The third person presses every third toggle (i.e., 3, 6, 9, ..., 96, 99); every 4th person presses every fourth toggle (i.e. 4, 8, ..., 92, 96, 100); and so on. Question: After 100 people, which bulbs will be on?

Gasoline Crisis:

Imagine that the fuel stations located on a long circular route together contain just enough fuel to make one trip around. Show that if you start at the right gas station with an empty tank, you can always make it all the way around the route.

Coins in a Row:

On a table is a row of 50 coins, of various denominations. Alice picks a coin from one of the ends, and puts it in her pocket. Then Bob chooses a coin from one of the (remaining) ends, and the alternation continues until Bob pockets the last coin. Can you show that Alice can always play so as to guarantee at least as much money as Bob?

Three bulbs:

You move into a new house, which has an incandescent light bulb in the basement. You learn that one of the three switches in the kitchen controls that light, but you do not know which one. The only way to see if the bulb is on requires going down to the basement. How can determine which switch controls the basement lamp by going to the basement exactly once?