Riddles and puzzles

Below are some interesting logic puzzles. (I don't include the solutions.)

Lost in an amusement park

You are at a large amusement park trying to find a friend who's wandered off on his/her own. Your friend, who is not looking for you, is constantly and randomly wandering around (i.e. randomly picking a direction to walk), and could be anywhere in the park by now. Would your odds of finding your friend be greater if you stood still or also wandered around in the same matter, and why?

Some additional assumptions: Foot traffic concentration is uniformly distributed throughout the park. Field of vision is always unobstructed. Both you and the friend move at the same walking speed. You do NOT have unlimited time.

The puzzle of 100 hats

One hundred persons will be lined up single file, facing north. Each person will be assigned either a red hat or a blue hat. No one can see the color of his or her own hat. However, each person is able to see the color of the hat worn by every person in front of him or her. That is, for example, the last person in line can see the color of the hat on 99 persons in front of him or her; and the first person, who is at the front of the line, cannot see the color of any hat.

Beginning with the last person in line, and then moving to the 99th person, the 98th, etc., each will be asked to name the color of his or her own hat. If the color is correctly named, the person lives; if incorrectly named, the person is shot dead on the spot. Everyone in line is able to hear every response as well as hear the gunshot; also, everyone in line is able to remember all that needs to be remembered and is able to compute all that needs to be computed.

Before being lined up, the 100 persons are allowed to discuss strategy, with an eye toward developing a plan that will allow as many of them as possible to name the correct color of his or her own hat (and thus survive). They know all of the preceding information in this problem. Once lined up, each person is allowed only to say “Red” or “Blue” when his or her turn arrives, beginning with the last person in line.

Your assignment: Develop a plan that allows as many people as possible to live. (Do not waste time attempting to evade the stated bounds of the problem — there’s no trick to the answer.)

Three gods

On the summit of this remote volcano, you realize a few things through divine intervention.

First, you know that one of the three gods always tells the truth, another always lies, and the third will respond to questions randomly. Therefore, let us call the gods True, False, and Random.

The gods speak a different language. They understand all languages perfectly well, but only answer questions with either ja or da, the words for yes and no. You do not know which god is which, and you do not know which word means yes and which word means no.

Finally, you have an existential problem on your hands. You may ask three yes-or-no questions, each one directed to only one god, and only that god will answer with either ja or da. If you can determine the identities of the three gods, they will send you on your way with their blessing, and you can be assured of a prosperous and fulfilled life. If you fail to determine the identities of the gods, however, they will be less generous in their treatment. The volcano pit smokes and glows red beside you.

With your three questions, how do you figure out which god is True, which is False, and which is Random?

Unfair coin

Imagine you have an unfair coin, one that does not land on each side 50 percent of the time. How could you use this coin to simulate the 50/50 odds of a fair coin flip?

Blue eyes

A group of people with assorted eye colors live on an island. They are all perfect logicians--if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.

The Guru is allowed to speak once (let's say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:

"I can see someone who has blue eyes."

Who leaves the island, and on what night?

Dystopic birth policy

In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country?

Coaster game

I propose a game: You and I take turns placing identical circular coasters on a circular table. The coasters cannot overlap. Whoever places the last coaster wins the game.

You can pick who goes first.

There is a simple strategy which ensures you will always win the game regardless of the table size and coaster size. What is it?

(The table can hold at least one coaster. We have plenty of coasters.)

Coins in the dark

There are 26 coins lying on a table in a totally dark room. Ten are heads and 16 and tails. In the dark you cannot feel or see if a coin is heads up or tails up, but you may move them or turn any of them over. Separate the coins into two groups so that each group has the same number of coins heads up as the other group.

Poisoned wine

An evil king has 1000 bottles of wine. A neighboring queen plots to kill the bad king, and sends a servant to poison the wine. The king's guards catch the servant after he has only poisoned one bottle. The guards don't know which bottle was poisoned, but they do know that the poison is so potent that even if it was diluted 1,000,000 times, it would still be fatal. Furthermore, the effects of the poison take one month to surface. The king decides he will get some of his prisoners in his vast dungeons to drink the wine. Rather than using 1000 prisoners each assigned to a particular bottle, this king knows that he needs to murder no more than 10 prisoners to figure out what bottle is poisoned, and will still be able to drink the rest of the wine in 5 weeks time. How does he pull this off?

Greedy pirates

A pirate ship captures a treasure of 1000 golden coins. The treasure has to be split among the 5 pirates: 1, 2, 3, 4, and 5 in order of rank. The pirates have the following important characteristics: infinitely smart, bloodthirsty, greedy. Starting with pirate 5 they can make a proposal how to split up the treasure. This proposal can either be accepted or the pirate is thrown overboard. A proposal is accepted if and only if a majority of the pirates agrees on it. What proposal should pirate 5 make?

Love in Kleptopia

Jan and Maria have fallen in love (via the internet) and Jan wishes to mail her a ring. Unfortunately, they live in the country of Kleptopia where anything sent through the mail will be stolen unless it is enclosed in a padlocked box. Jan and Maria each have plenty of padlocks, but none to which the other has a key. How can Jan get the ring safely into Maria’s hands?

Ants on a stick

You pick up a meter stick with 100 ants on it. Each ant walks 1 cm/s toward an end of the stick, and it reverses direction any time it encounters another ant. What's the longest amount of time you'd have to wait for all ants to fall off of the end of the stick?

Free throws

You shoot free throws at 70%. In order to escape a death sentence you received in Basketballandia for traveling you have to demonstrate that your free throw rate is at least 66%. You are offered the choice of taking 3 shots or 6 shots to demonstrate this. Which do you choose and why?