# Riddles

Riddles are a core part of Paradigm's culture. Here's a few of our favorites.

1.

Imagine an island where 100 perfectly logical prisoners are trapped. Each has green eyes, but they don't know it. They can't see their own reflection, and they can't communicate, though they see each other daily. The only way to escape is to approach the guards at night, but they can only leave if they're certain they have green eyes; otherwise, they're thrown into a volcano. One day, they are told, 'At least one of you has green eyes.' On the 100th morning after this, all the prisoners have escaped.Answer

On the 100th day, all prisoners realize they have green eyes and leave. Each day, they observe that no one has left, eliminating the possibility of fewer green-eyed people. By day 99, each person knows there are at least 99 green-eyed people. Seeing 99 others with green eyes, they deduce their own eyes must be green, leading to everyone leaving on day 100.2.

There is a lightbulb inside a closet. The door is closed, and you cannot see if the light is on or off through the door. However, you know the light is off to start. Outside of the closet, there are three light switches. One of the door light switches controls the lightbulb in the closet. You can flip the switches however you want, but once you open the door, you can no longer touch the switches. How do you figure out without a doubt which of the three light switches controls the light?Answer

Turn on switch 1 for a few minutes, then turn it off. Turn on switch 2. Open the door. If the bulb is on, switch 2 controls it. If it's off but warm, switch 1 controls it. If it's off and cool, switch 3 controls it.3.

You are given a set of scales and 12 marbles. The scales are of the old balance variety. That is, a small dish hangs from each end of a rod that is balanced in the middle. The device enables you to conclude either that the contents of the dishes weigh the same or that the dish that falls lower has heavier contents than the other. The 12 marbles appear to be identical. In fact, 11 of them are identical, and one is of a different weight. Your task is to identify the unusual marble and discard it. You are allowed to use the scales three times if you wish, but no more. Note that the unusual marble may be heavier or lighter than the others. You are asked to both identify it and determine whether it is heavy or light.Answer

Divide the marbles into three groups of four. Weigh two groups against each other. If they balance, the odd marble is in the third group. If not, it's in the heavier or lighter group. Take the group with the odd marble, divide it into three groups (three marbles, one single), and repeat the process. The final weighing will identify the odd marble and whether it's heavy or light.4.

I have three envelopes, into one of them I put a $20 note. I lay the envelopes out on a table in front of me and allow you to pick one envelope. You hold but do not open this envelope. I then take one of the envelopes from the table, demonstrate to you that it was empty, screw it up and throw it away. The question is would you rather stick with the envelope you have selected or exchange it for the one on the table. Why? What would be the expected value to you of the exchange?Answer

You should switch. The probability of winning increases from 1/3 to 2/3 by switching. The expected value of switching is $13.33, while staying has an expected value of $6.67.5.

A farmer is on his way back from the market, with him he has a fox, a chicken and some grain. When he reaches a river crossing he must use a small boat only big enough for him and one other item. Unfortunately if the fox is left alone with the chicken it will eat it, as will the chicken eat the grain. Explain how the farmer can cross the river.Answer

1) Take chicken across. 2) Return alone. 3) Take fox across. 4) Bring chicken back. 5) Take grain across. 6) Return alone. 7) Take chicken across.6.

100 black and white socks are in a drawer. How many socks must you pull out before you are guaranteed to have a pair? Generalize to socks of N different colors.Answer

For 2 colors, you need to pull out 3 socks to guarantee a pair. For N colors, you need to pull out N+1 socks to guarantee a pair.7.

A lady has two children. One is a boy. What are the chances of the other child also being a boy? How does this change if you are told the oldest child is a boy?Answer

If one child is a boy (without specifying which), the chance of the other being a boy is 1/3. If the oldest child is a boy, the chance of the other being a boy is 1/2.8.

An executioner lines up 100 prisoners single file and puts a red or a blue hat on each prisoner's head. Every prisoner can see the hats of the people in front of him in the line - but not his own hat, nor those of anyone behind him. The executioner starts at the end (back) and asks the last prisoner the colour of his hat. He must answer 'red' or 'blue.' If he answers correctly, he is allowed to live. If he gives the wrong answer, he is killed instantly and silently. (While everyone hears the answer, no one knows whether an answer was right.) On the night before the line-up, the prisoners confer on strategy to help them. What should they do?Answer

The last prisoner counts the number of blue hats he sees. If it's odd, he says 'blue'. If it's even, he says 'red'. Each subsequent prisoner can then deduce their own hat color based on the previous answers and the hats they see in front of them.9.

A traveler meets a pair of twins at a fork in the road: one path is correct and one incorrect. One of the twins always tells the truth (knight), and the other one always lies (knave). Which single question can the traveler ask one of the twins to determine the correct path?Answer

Ask either twin: 'If I were to ask your twin which path is correct, what would they say?' Then take the opposite path.10.

A mother is 21 years older than her child. In exactly 6 years from now, the mother will be exactly 5 times as old as the child.Answer

The child is currently 5 years old and the mother is 26 years old.11.

Suppose you have a hotel which has one floor with infinite number of rooms in a row and all of them are occupied. A new customer wants to check in, how will you accommodate her? What if infinite number of people want to check in, how will you accommodate them? Suppose infinite number of buses arrive at the hotel, each having infinite number of people, how will you accommodate them?Answer

1) Move everyone to the next room, freeing room 1. 2) Move everyone to twice their room number, freeing all odd-numbered rooms. 3) Assign each bus a prime number, move current guests to powers of 2, new guests to powers of their bus's prime.12.

A group of 5 people want to keep their secret document in a safe. They want to make sure that in future, only a majority (>=3) can open the safe. So they want to put some locks on the safe, each of the locks have to be opened to access the safe. Each lock can have multiple keys; but each key only opens one lock. How many locks are required at the minimum? How many keys will each member carry?Answer

10 locks are required. Each person carries 6 keys. This ensures any 3 people can open all locks, but no 2 people can.13.

After the revolution, each of the 66 citizens of a certain city, including the king, has a salary of 1. King cannot vote, but has the power to suggest changes - namely, redistribution of salaries. Each person's salary must be a whole number of dollars, and the salaries must sum to 66. He suggests a new salary plan for every person including himself in front of the city. Citizens are greedy, and vote yes if their salary is raised, no if decreased, and don't vote otherwise. The suggested plan will be implemented if the number of 'yes' votes are more than 'no' votes. The king is both, selfish and clever. He proposes a series of such plans. What is the maximum salary he can obtain for himself?Answer

The king can obtain a maximum salary of 32. He can propose a plan where he gets 32, 33 citizens get 1, and 32 citizens get 0. This will pass with 33 yes votes and 32 no votes.14.

There is a building of 100 floors. If an egg drops from the Nth floor or above it will break. If it's dropped from any floor below, it will not break. You're given 2 eggs. How do you find N in the minimum number of drops?Answer

Use a decreasing step method. Start at floor 14, then 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, 100. If the first egg breaks, use the second egg to check each floor below. This method requires at most 14 drops.15.

You are in a game against devil, on a perfectly round table and with an infinite pile of pennies. He says, 'OK, we'll take turns putting one penny down, no overlapping allowed, and the pennies must rest flat on the table surface. The first guy who can't put a penny down loses.' You can go first. How will you guarantee victory?Answer

Place the first penny exactly in the center of the table. Then mirror every move the devil makes on the opposite side of the center. This guarantees you'll always have a move as long as the devil does.16.

An employee works for an employer for 7 days. The employer has a gold rod of 7 units. How does the employer pay to the employee, so that the number of employee's rod units increases by one at the end of each day? The employer can make at most 2 cuts in the rod.Answer

Cut the rod into pieces of 1, 2, and 4 units. Give 1 unit on day 1, 2 units on day 2, and 4 units on day 4. On other days, take back smaller pieces to give the larger piece.17.

Three athletes (and only three athletes) participate in a series of track and field events. Points are awarded for 1st, 2nd, and 3rd place in each event (the same points for each event, i.e. 1st always gets 'x' points, 2nd always gets 'y' points, 3rd always gets 'z' points), with x > y > z > 0, and all point values being integers. The athletes are named Adam, Bob, and Charlie: Adam finished first overall with 22 points, Bob won the Javelin event and finished with 9 points overall, Charlie also finished with 9 points overall. Who finished second in the 100-meter dash (and why)?Answer

Adam finished second. The only way to get these scores is if x=5, y=3, z=1. Adam must have won 4 events and come 2nd in one to get 22 points. Since Bob won Javelin, the event Adam came 2nd in must be the 100-meter dash.18.

Flatland is a plane extending infinitely in all directions. It has an infinite number of airfields no two of which are exactly the same distance apart. At a point in time a plane will take off from each airport and land at it's nearest neighbouring airport. What is the maximum number of aeroplanes that may land at any single airfield?Answer

The maximum number of planes that can land at a single airfield is 5. This occurs when the airfield is at the center of a regular pentagon formed by its 5 nearest neighbors.19.

I have three envelopes, into one of them I put a $20 note. I lay the envelopes out on a table in front of me and allow you to pick one envelope. You hold but do not open this envelope. I then take one of the envelopes from the table, demonstrate to you that it was empty, screw it up and throw it away. The question is would you rather stick with the envelope you have selected or exchange it for the one on the table. Why? What would be the expected value to you of the exchange?Answer

You should switch. The probability of winning increases from 1/3 to 2/3 by switching. The expected value of switching is $13.33, while staying has an expected value of $6.67.20.

You have a string-like fuse that burns in exactly one minute. The fuse is inhomogeneous, and it may burn slowly at first, then quickly, then slowly, and so on. You have a box of matches, and no watch. How do you measure exactly 30 seconds? If you had 2 fuses could you measure 45 seconds?Answer

Light both ends of the fuse. When they meet, 30 seconds have passed. For 45 seconds with 2 fuses: Light both ends of one fuse and one end of the other. When the first fuse is gone, light the other end of the second fuse. When it burns out, 45 seconds have passed.21.

You are standing at the center of a circular field of radius R. The field has a low wire fence around it. Attached to the wire fence (and restricted to running around the perimeter) is a large, sharp-fanged, hungry dog. You can run at speed v, while the dog can run four times as fast. What is your running strategy to escape the field?Answer

Run straight to the edge of the field. When you reach it, run along the perimeter. The dog will catch up, but you'll have covered more than 1/4 of the circumference, allowing you to escape.22.

There are 25 horses among which you need to find out the fastest 3 horses. You can conduct a race among at most 5 to find out their relative speed. At no point, you can find out the actual speed of the horse in a race. Find out the minimum no. of races which are required to get the top 3 horses.Answer

7 races are needed. First, run 5 races of 5 horses each. Then race the winners of these 5 races. Finally, race the 2nd and 3rd from the winner's original race with the 2nd from the final race.23.

A duck is sitting at the center of a circular lake. A fox is waiting at the shore, not able to swim, wishing to eat the duck. The Fox can move around the whole lake at a speed four times the speed at which the duck can swim. The duck can fly, but only once it reaches the shore of the lake, it can't fly from the water directly. Can the duck always reach the shore without being eaten by the fox? Note: This is an old duck, and cannot take a flight while swimming. The duck cannot submerge in the water.Answer

Yes, the duck can always escape. It should swim directly towards the shore. By the time it reaches the shore, the fox will have run less than 1/4 of the lake's circumference, allowing the duck to fly away safely.24.

How large of a group is required such that there is a greater than 50% chance that one of the people in the group has a birthday matching any other person?Answer

23 people. This is known as the Birthday Paradox. With 23 people, the probability of a shared birthday is about 50.7%.25.

What is the most money in coins you can have in your pocket and not be able to break a dollar?Answer

$0.99 in coins: 3 quarters, 2 dimes, and 4 pennies.26.

You are dealt 2 cards from a deck. What is the probability that both of your cards are aces?Answer

The probability is 1/221 or about 0.45%. There are 4 aces out of 52 cards, so the probability of drawing an ace, then another ace, is (4/52) * (3/51) = 1/221.27.

There are 2 drawers. The first drawer contains only black balls. The second contains 50% black balls and 50% white balls. There are an equal number of balls in each drawer. I pick a ball at random and it is black. What is the probability that the ball came from the first drawer?Answer

2/3. Let's say each drawer has 100 balls. Drawer 1 has 100 black, Drawer 2 has 50 black and 50 white. Total black balls = 150. Probability of picking from Drawer bility of picking from Drawer 1 given a black ball = 100/150 = 2/3.28.

Four people come to a river in the night. There is a narrow bridge, and it can only hold two people at a time. They have one torch and, because it's night, the torch has to be used when crossing the bridge. Person A can cross the bridge in 1 minute, B in 2 minutes, C in 5 minutes, and D in 8 minutes. When two people cross the bridge together, they must move at the slower person's pace. The question is, can they all get across the bridge if the torch lasts only 15 minutes?Answer

Yes, they can cross in exactly 15 minutes: A&B cross (2 min), A returns (1 min), C&D cross (8 min), B returns (2 min), A&B cross (2 min). Total: 15 minutes.29.

There are three boxes, one containing only apples, the second only bananas and the third both apples and bananas. The boxes were once correctly labeled, but unfortunately the labels have been moved so that each box is now incorrectly labeled. By observing just one piece of fruit drawn from one of the boxes, can you determine the contents of each box?Answer

Yes. Pick from the box labeled 'Mixed'. It must contain either only apples or only bananas. If it contains apples, then the box labeled 'Bananas' must be mixed, and the one labeled 'Apples' must contain only bananas. If it contains bananas, the opposite is true.30.

You have 10 stacks of 10 gold coins. All of the coins in one of these stacks are counterfeit, all the other coins are not. A real coin weighs 10 grammes. A counterfeit coin weighs 11 grammes. You have a modern scale that provides an accurate readout. What is the minimum number of weighings needed to determine which stack is fake?Answer

One weighing is sufficient. Number the stacks 1 to 10. Take 1 coin from stack 1, 2 from stack 2, 3 from stack 3, and so on. Weigh them all together. The excess weight over 550g (10+20+30+...+100) will indicate which stack is counterfeit.31.

Suppose you roll a fair 6-sided die until you've seen all 6 faces. What is the probability you won't see an odd numbered face until you have seen all even numbered faces?Answer

The probability is 1/16. You need to roll 2, 4, and 6 before rolling any odd number. The probability of rolling an even number first is 1/2, then 1/3 for the second even number, and 1/4 for the third. So, (1/2) * (1/3) * (1/4) = 1/16.32.

Jim will roll a fair, six-sided die until he gets a 4. What is the expected value of the highest number he rolls through this process?Answer

The expected value is 5. There's a 1/6 chance of rolling a 4 on the first try (highest = 4), 1/6 chance of highest being 5, and 4/6 chance of rolling a 6 before a 4.33.

A tree doubled in height each year until it reached its maximum height over the course of ten years. How many years did it take for the tree to reach half its maximum height?Answer

9 years. If it doubled each year for 10 years, then after 9 years it was half its final height.34.

1024 pirates stand in a circle. They start shooting alternately in a cycle such that the 1st pirate shoots the 2nd, the 3rd shoots the 4th and so on. The pirates who got shot are eliminated from the game. They continue in circles, shooting the next standing pirate, till only one pirate is left. Which position should someone stand to survive?Answer

Position 1024 (or the last position). This is because 1024 is a power of 2 (2^10). In each round, the number of pirates is halved, and the survivor is always in the last position of the remaining group.35.

There are two jars, one contains water, and the other contains wine (equal volumes). A certain amount of water is transferred to the wine, then the same amount of the mixture is transferred back to the water beaker. Is there now more water in the wine than there is wine in the water? Assume there is no expansion, compression, spillage or any chemical reaction.Answer

The amount of water in the wine is exactly equal to the amount of wine in the water. The total volume of liquid in each jar remains constant, so whatever volume of wine is in the water jar must be replaced by an equal volume of water in the wine jar.36.

Watermelon is 99% water. I have 100 pounds of watermelon. After a week, drying in the sun, the shriveled watermelon had only dried down to being 98% water. What is the total weight of the watermelon now?Answer

50 pounds. Initially, there was 1 pound of non-water content. After drying, this 1 pound represents 2% of the total. So, 1/0.02 = 50 pounds total.