Brain Teasers & Puzzles

A park contains paths that intersect at various places. The intersections all have the properties that they are 3way intersections and that, with one exception, they are indistinguishable from each other. The one exception is an intersection where there is a restaurant. The restaurant is reachable from everywhere in the park. Your task is to find your way to the restaurant.
The park has strict littering regulations, so you are not allowed to modify the paths or intersections (for example, you are not allowed to leave a note an intersection saying you have been there). However, you are allowed to do some bookkeeping on a pad of paper that you bring with you at all times (in the computerscience parlance, you are allowed some state). How can you find the restaurant?
You may assume that once you enter an intersection, you can continue to the left, continue to the right, or return to where you just came from.
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The harmonic series–that is, 1/1 + 1/2 + 1/3 + 1/4 + …–diverges. That is, the sum is not finite. This is in difference to, for example, a geometric series–like ½^{0} + ½^{1} + ½^{2} + ½^{3} + …–which converges, that is, has a finite sum.
Consider the harmonic series, but drop all terms whose denominator represented in decimal contains a 9. For example, you’d drop terms like 1/9, 1/19, 1/90, 1/992, 1/529110. Does the resulting series converge or diverge?
Hint: More generally, you may consider representing the denominator in the base of your choice and dropping terms that contain a certain digit of your choice.
Consider again the harmonic series, but drop a term only if the denominator represented in decimal contains two consecutive 9’s. For example, you’d drop 1/99, 1/992, 1/299, but not 1/9 or 1/909. Does this series converge or diverge?Submit Solution 816.9K views
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You have 240 barrels of wine, one of which has been poisoned. After drinking the poisoned wine, one dies within 24 hours. You have 5 slaves whom you are willing to sacrifice in order to determine which barrel contains the poisoned wine. How do you achieve this in 48 hours?
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A ledge is 1 meter long. On it are N lemmings. Each lemming travels along the ledge at a constant speed of 1 meter/minute. A lemming continues in the same direction until it either falls off the ledge or it collides with another lemming. If two lemmings collide, they both immediately change their directions. Initially, the lemmings have arbitrary starting positions and starting directions. What is the longest time that may elapse before all lemmings have fallen off the ledge?
Hint: Suppose the ledge is not horizontal, but is leaning. A lemming now travels up the ledge at a speed of 1 meter/minute and travels down the ledge at a speed of 2 meters/minute. What is the longest time before all lemmings have fallen off?
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Two witches each makes a nightly visit to an allnight coffee shop. Each arrives at a random time between 0:00 and 1:00. Each one of them stays for exactly 15 minutes. On any one given night, what is the probability that the witches will meet at the coffee shop?
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There are five holes arranged in a line. A hermit hides in one of them. Each night, the hermit moves to a different hole, either the neighboring hole on the left or the neighboring hole on the right. Once a day, you get to inspect one hole of your choice. How do you make sure you eventually find the hermit?
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The house next door has some new neighbors. They have two children, but you don’t know what mix of boys and girls they are. One day, your wife tells you “At least one of the children is a girl”. What is the probability that both are girls?
Your wife then tells you “The way I found out that at least one of the children is a girl is that I saw one of the children playing outside, and it was a girl”. Now, what is the probability that both are girls?
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A particular basketball shootout game consists of a number of duels. In each duel, one player is the challenger. The challenger chooses another player to challenge, and then gets one chance to shoot the hoop. If the player makes the shot, the playing being challenged is out. If the player does not make the shot, or if the player chooses to skip his turn, then the game continues with the next duel. A player wins when only that player remains.
One day, this game is played by three players: A, B, and C. Their skill levels vary considerably: player A makes every shot, player B has a 50% chance of making a shot, and player C has a 30% chance of making a shot. Because of the difference in skill levels, they decide to let C begin, then B, then A, and so on (skipping any player who is out of the game) until there is a winner. If everyone plays to win, what strategy should each player follow?
[For this followup question, it will be helpful to have a paper and pen–not because the calculations are hard, but because it helps in remembering the numbers.]
If A, B, and C follow their winning strategies (as determined above), which player has the highest chance of winning the game?
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You’re on a government ship, looking for a pirate ship. You know that the pirate ship travels at a constant speed, and you know what that speed is. Your ship can travel twice as fast as the pirate ship. Moreover, you know that the pirate ship travels along a straight line, but you don’t know what that line is. It’s very foggy, so foggy that you see nothing. But then! All of a sudden, and for just an instant, the fog clears enough to let you determine the exact position of the pirate ship. Then, the fog closes in again and you remain (forever) in the thick fog. Although you were able to determine the position of the pirate ship during that fogfree moment, you were not able to determine its direction. How will you navigate your government ship so that you will capture the pirate ship?
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A (presumed smart) insurance agent knocks on a door and a (presumed smart) woman opens. He introduces himself and asks if she has any children. She answers: 3. When he then asks their ages (which for this problem we abstract to integers), she hesitates. Then she decides to give him some information about their ages, saying “the product of their ages is 36”. He asks for more information and she gives in, saying “the sum of their ages is equal to our neighbors’ house number”. The man jumps over the fence, inspects the house number, and the returns. “You need to give me another hint”, he begs. “Alright”, she says, “my oldest child plays the piano”. What are the ages of the children?
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