Tasks from the slopes of Fuji
On Habré regularly mention the book by William Poundstown "How to move Mount Fuji." I read it here once and decided for the general pleasure of collecting all the tasks mentioned in it into a pile.
Please note - here are only those tasks that have the exact solution. Questions to reason, such as "how many in the world of piano tuners," I missed.
Does the sun always rise in the east?
You have six matches. Make four equilateral triangles of them.
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Let's play Russian roulette. You are tied to a chair and cannot stand up. Here is a gun. Here is his drum - there are six cartridge sockets in it, and they are all empty.
Look: I have two bullets. Did you notice that I inserted them into the adjacent drum sockets? Now I put the drum in place and rotate it. I bring the gun to your head and pull the trigger. Click! You are still alive. What a score!
Now, before we start discussing your resume, I’m going to click the hook again. Which do you prefer: so that I turn the drum again or just pull the trigger?
The mother sent her son to the river and asked him to bring home exactly 7 liters of water. She gave him two pitchers with a capacity of 3 and 5 liters.
Explain to me how a boy, using only these two jugs, can measure exactly 7 liters of water (cannot be measured by eye).
Another variant of the problem: how to measure 4 liters of water?
How many times during the day do the hands of the clocks overlap?
You have b boxes and n one dollar banknotes. Distribute the money to the boxes, which are then sealed, so that, without opening the boxes, you can pay any whole amount of dollars, starting from zero dollars and ending with n. What are the limits for b and n?
You certainly know that fish can swim in water. Now solve the problem.
Suppose we have an incomplete bucket of water. We put this bucket on the scales and find out that the weight of the bucket of water is exactly 45 pounds. Then we drop fish weighing exactly 5 pounds into a bucket.
How much will the bucket weigh with the fish now?
On the table are four cards. Each one has a letter on one side and a number on the other. Naturally, you see only one of the parties:
[A] [F] [2] [7]
Problem Condition: Determine which card (s) you need to flip to check whether the rule “If there is a vowel on one side of the card, then there is an even number on the other side of this card”.
Before you are two doors. One leads to the interviewer's office, and the other - to the street. Near the door is a consultant. He may be from our company, and maybe from a competitor company. Consultants from our company always tell the truth, consultants from competing firms always lie.
You are allowed to ask the consultant just one question in order to get into the interview room.
A person needs to cross the river wolf, goat and cabbage. The wolf should not be left with the goat, and the goat - with cabbage. In a boat a person can take only one passenger - either a wolf, or a goat, or cabbage.
How to transport all?
How can a passenger jet be weighed if it cannot be placed on a scale?
Why are manhole covers round, not square?
Why does the right and left direction change places in the mirror instead of the top and the bottom?
If you float in a boat and throw a suitcase out of it, will the water level rise or fall?
How many such places are on the globe, where, if you walk one mile to the south, then one to the east, then another one to the north, will you return to the same place where you left?
How can you cut a rectangular cake into two equal pieces after one rectangular piece has already been cut out of it? This piece can be of any size and orientation. Only one straight cut is allowed.
You have eight billiard balls. One of them is "defective" - it is heavier than the others. How is it possible to determine the defective ball in two weights on weights without weights?
You have five jars of pills. In one of the jars all the pills are “spoiled”. This can only be determined by weight. All "normal" pills weigh 10 grams, and "spoiled" - 9 grams. You have scales, and you can only do one weighing.
How can I determine which of the jars are “spoiled” pills?
In the three corners of an equilateral triangle is located on the ant. Each of the ants begins to move to another randomly selected angle in a straight line. What is the probability that none of the ants will collide with another ant?
Four dogs are in the corners of a large square. Each of the dogs begins to pursue another dog, located from it in the course of an hour hand. All dogs run at the same speed, and they constantly change the direction of their movement so as to follow strictly in a straight line the dog they are chasing.
How much time will pass until the dogs catch each other? Where will this happen?
From Los Angeles to New York, the train leaves at a constant speed of 15 miles per hour. Simultaneously, from New York to Los Angeles on the same way a counter train leaves at a speed of 20 miles per hour. At the same moment, a bird flies out of Los Angeles from the station and flies strictly over the railway track towards New York at a speed of 25 miles per hour.
As soon as the bird reaches the train that departed from New York, it immediately turns around and flies in the opposite direction at the same speed until it meets the train that left Los Angeles, after which it turns around again and flies in the opposite direction. So she flies back and forth between the two trains until they collide.
What distance will the bird fly?
You have 26 constants, denoted by letters from A to Z. Let A be 1. The value of the next constant will be determined by the sequence number of this letter in the English alphabet raised to the power corresponding to the value of the previous constant.
This means that the value of B (second letter) = 2 to the power of A = 2 to the power of 1 = 2. C = 3 to the power of B = 3 squared = 9, etc.
Find the exact numerical value of the expression: (XA) x (XB) x (X-C) ... (XY) x (XZ).
Develop a number system with a base of minus 2. Write the numbers from 1 to 10 in this system.
You have two vessels and 100 marbles, fifty of which are red, and the second half is blue. In a random order, choose one of two vessels, from which one then randomly selects and pulls out one ball.
How to distribute the balls in the vessels so that the probability of getting the red ball was maximum? (All one hundred balls should be put in the vessels.) What will be the probability of a random selection of a red ball, if you use your scheme?
One of your employees insists on being paid in gold. You have a gold bar, the cost of which corresponds to the seven-day salary of this employee. He is already marked up in seven equal pieces.
If you were allowed to make only two cuts of the ingot, and the worker needs to pay at the end of each day, how can this problem be solved?
You have a jar in which pills of three colors: red, green and blue. With your eyes closed, you need to get two jelly beans out of the jar so that they are of the same color.
How many dragees do you need to get to be sure that there are two of the same color among them?
You have three fruit baskets. In one of them - only apples, in the other - only oranges, finally, in the third - both apples and oranges. You do not see what kind of fruit inside the baskets. Each basket has a well-marked label, but the information on it is incorrect. With your eyes closed, you are allowed to remove one fruit from one basket and then examine it.
How can you determine that in each of the baskets?
In a village where fifty married couples live, each of the husbands cheated on his wife. Each of the women in this village, as soon as one of the men has betrayed his wife, immediately learns about it (everyone knows how quickly gossip is spread in small towns), unless this is her own husband (everyone will be the last to know about their troubles) .
The laws of this town require that a woman who has received evidence of her husband’s infidelity, kill him on the same day. None of the women can disobey. One day the queen, famous for her infallibility, comes to the town. She announces to residents that at least one of the men in the town has committed adultery. What will happen?
An evil demon caught many gnomes (their exact number is unknown). After that, during a “briefing on hiring” to his company, the demon attached a red or green gemstone to each of the dwarves on his forehead. The demon tells each of his new slave-dwarf that now he has a precious stone on his forehead that cannot be removed. Neither the demon nor the other gnome will say what color this stone is (gnomes are strictly forbidden to talk).
The stones of one of the two colors signify dwarves who sympathize with spies sent to the demon's company, and stones of a different color are attached to the forehead of the unfortunate prisoners who do not sympathize with spies. The demon does not want to tell this gnome what stone he has on his forehead, and he will never tell him about it at all. At this "instructing" ends. Every morning the gnomes are built. This is done so that the demon can count them and make sure that none of the dwarfs has escaped.
One day the gnomes tired of the demon, and he decided to get rid of them. He announces to the dwarves that he will let them all go free, if they are able to correctly determine what color is attached to each of them on the forehead stone. He gives them one clue: there is at least one dwarf with a green stone and one with a red stone.
In order to gain freedom, the dwarves during the morning building must (they still cannot talk) give the demon the right signal: all the dwarves with the red stone in their forehead must fail one step, and all those who have the green stone must remain in build If they do not make any mistakes at the same time, all the dwarves will be able to go home and work again at their favorite coal mines. If they make a mistake, everyone will be executed on the spot.
The time that is given to the gnomes to determine the color of the stones is not limited. They all have flawless logic and really want to go home. What do they need to do?
Four tourists need to cross the river at night on a suspension bridge. The bridge is already very dilapidated, there are holes in the flooring, and it can withstand no more than two people at a time (if there are more than two people on the bridge, the bridge will collapse). Tourists need to light the road with a flashlight - otherwise they may fall through a hole in the bridge decking and die, but they have only one flashlight.
These four people move at different speeds. Adam can cross the bridge in one minute, Larry - in two minutes, Edge needs five minutes, the slowest of all Bono - it will take him ten minutes to cross.
bridge.
Exactly seventeen minutes later, the bridge will collapse. How can all four have time to cross it?
There are three switches in the corridor. One of them turns on the light in a room at the far end of the corridor. The door to this room is closed, and you cannot see whether the light is on or not. You need to understand which of the three switches controls the lighting in that room.
How can you reliably determine this by only entering the room once?
You play the game with only one other player. The game begins on an empty rectangular table, and you have an unlimited supply of coins in denominations of twenty-five cents. Each player in turn puts one coin on any place on the table. The only rule is that you must put your coin in such a way that it does not touch any other coin that is already on the table.
You and your opponent take turns in putting the coins in until almost the entire table is full of them. The player who does not have the opportunity to make a move by the rules loses. You go first. What strategy will you choose to play?
Five pirates on the island must divide among themselves hundreds of gold coins. They divide their prey like this: the older pirate suggests how to divide the prey, and then everyone votes, agreeing with his proposal or not. If at least half of the pirates vote in favor, they will divide the coins as suggested by the older pirate, but if not, they kill the older pirate and start over. The eldest pirate (of those who survived) proposes a new plan, they vote for him according to the same rules, and then either divide the loot, or kill the older pirate.
The process continues until a plan is adopted.
Suppose you are a senior pirate. How do you propose to split the prey? (All other pirates are greedy, they think very logically, and they all want to live.)
In one of the schools there is such a ritual, held on the last day of classes: students go to the hall and stand near their lockers, which store clothes. On the first whistle, each student opens his locker, on the second whistle, students close even lockers (that is, lockers numbered 2, 4, 6, etc.). On the third whistle, the students change the position of the doors of every third cabinet, that is, if it was open, it is closed, and if it is closed, it is opened. This happens with lockers number 3, 6, 9, etc. On the fourth whistle, the state of the door of every fourth locker changes, the fifth whistle of every fifth locker, etc.
Suppose for simplicity that this is a small school and there are only 100 lockers. For the hundredth whistle, the student who stands next to the cabinet under the hundredth number (and only this student) changes the position of the door of this cabinet. How many cabinets are open after this?
You have two pieces of Bickford fuse. Each of them burns for exactly one hour, but the pieces may not be identical and do not necessarily burn at a constant speed: there are fragments that burn quickly, and there are those that burn slowly.
How do you know that forty-five minutes have passed, using only these pieces of fuse and lighter?
You are in a boat in the exact center of a perfectly round lake. On the shore of the lake goblin. The goblin is plotting something evil against you, but he does not know how to swim and he does not have a boat either. If you manage to moor to the shore, and the goblin will not be able to wait for you there and immediately grab it, you will always be able to run away from it on the ground and break free.
Here is the condition of the problem: a goblin can run at a speed four times higher than the speed of your boat. He has impeccable eyesight, he never sleeps and thinks very logically. He will do his best to catch you. How could you get away from the goblin?
Please note - here are only those tasks that have the exact solution. Questions to reason, such as "how many in the world of piano tuners," I missed.
Task 1
Does the sun always rise in the east?
Task 2
You have six matches. Make four equilateral triangles of them.
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Task 3
Let's play Russian roulette. You are tied to a chair and cannot stand up. Here is a gun. Here is his drum - there are six cartridge sockets in it, and they are all empty.
Look: I have two bullets. Did you notice that I inserted them into the adjacent drum sockets? Now I put the drum in place and rotate it. I bring the gun to your head and pull the trigger. Click! You are still alive. What a score!
Now, before we start discussing your resume, I’m going to click the hook again. Which do you prefer: so that I turn the drum again or just pull the trigger?
Task 4
The mother sent her son to the river and asked him to bring home exactly 7 liters of water. She gave him two pitchers with a capacity of 3 and 5 liters.
Explain to me how a boy, using only these two jugs, can measure exactly 7 liters of water (cannot be measured by eye).
Another variant of the problem: how to measure 4 liters of water?
Task 5
How many times during the day do the hands of the clocks overlap?
Task 6
You have b boxes and n one dollar banknotes. Distribute the money to the boxes, which are then sealed, so that, without opening the boxes, you can pay any whole amount of dollars, starting from zero dollars and ending with n. What are the limits for b and n?
Task 7
You certainly know that fish can swim in water. Now solve the problem.
Suppose we have an incomplete bucket of water. We put this bucket on the scales and find out that the weight of the bucket of water is exactly 45 pounds. Then we drop fish weighing exactly 5 pounds into a bucket.
How much will the bucket weigh with the fish now?
Task 8
On the table are four cards. Each one has a letter on one side and a number on the other. Naturally, you see only one of the parties:
[A] [F] [2] [7]
Problem Condition: Determine which card (s) you need to flip to check whether the rule “If there is a vowel on one side of the card, then there is an even number on the other side of this card”.
Task 9
Before you are two doors. One leads to the interviewer's office, and the other - to the street. Near the door is a consultant. He may be from our company, and maybe from a competitor company. Consultants from our company always tell the truth, consultants from competing firms always lie.
You are allowed to ask the consultant just one question in order to get into the interview room.
Task 10
A person needs to cross the river wolf, goat and cabbage. The wolf should not be left with the goat, and the goat - with cabbage. In a boat a person can take only one passenger - either a wolf, or a goat, or cabbage.
How to transport all?
Task 11
How can a passenger jet be weighed if it cannot be placed on a scale?
Task 12
Why are manhole covers round, not square?
Task 13
Why does the right and left direction change places in the mirror instead of the top and the bottom?
Task 14
If you float in a boat and throw a suitcase out of it, will the water level rise or fall?
Task 15
How many such places are on the globe, where, if you walk one mile to the south, then one to the east, then another one to the north, will you return to the same place where you left?
Task 16
How can you cut a rectangular cake into two equal pieces after one rectangular piece has already been cut out of it? This piece can be of any size and orientation. Only one straight cut is allowed.
Task 17
You have eight billiard balls. One of them is "defective" - it is heavier than the others. How is it possible to determine the defective ball in two weights on weights without weights?
Task 18
You have five jars of pills. In one of the jars all the pills are “spoiled”. This can only be determined by weight. All "normal" pills weigh 10 grams, and "spoiled" - 9 grams. You have scales, and you can only do one weighing.
How can I determine which of the jars are “spoiled” pills?
Task 19
In the three corners of an equilateral triangle is located on the ant. Each of the ants begins to move to another randomly selected angle in a straight line. What is the probability that none of the ants will collide with another ant?
Task 20
Four dogs are in the corners of a large square. Each of the dogs begins to pursue another dog, located from it in the course of an hour hand. All dogs run at the same speed, and they constantly change the direction of their movement so as to follow strictly in a straight line the dog they are chasing.
How much time will pass until the dogs catch each other? Where will this happen?
Task 21
From Los Angeles to New York, the train leaves at a constant speed of 15 miles per hour. Simultaneously, from New York to Los Angeles on the same way a counter train leaves at a speed of 20 miles per hour. At the same moment, a bird flies out of Los Angeles from the station and flies strictly over the railway track towards New York at a speed of 25 miles per hour.
As soon as the bird reaches the train that departed from New York, it immediately turns around and flies in the opposite direction at the same speed until it meets the train that left Los Angeles, after which it turns around again and flies in the opposite direction. So she flies back and forth between the two trains until they collide.
What distance will the bird fly?
Task 22
You have 26 constants, denoted by letters from A to Z. Let A be 1. The value of the next constant will be determined by the sequence number of this letter in the English alphabet raised to the power corresponding to the value of the previous constant.
This means that the value of B (second letter) = 2 to the power of A = 2 to the power of 1 = 2. C = 3 to the power of B = 3 squared = 9, etc.
Find the exact numerical value of the expression: (XA) x (XB) x (X-C) ... (XY) x (XZ).
Task 23
Develop a number system with a base of minus 2. Write the numbers from 1 to 10 in this system.
Task 24
You have two vessels and 100 marbles, fifty of which are red, and the second half is blue. In a random order, choose one of two vessels, from which one then randomly selects and pulls out one ball.
How to distribute the balls in the vessels so that the probability of getting the red ball was maximum? (All one hundred balls should be put in the vessels.) What will be the probability of a random selection of a red ball, if you use your scheme?
Task 25
One of your employees insists on being paid in gold. You have a gold bar, the cost of which corresponds to the seven-day salary of this employee. He is already marked up in seven equal pieces.
If you were allowed to make only two cuts of the ingot, and the worker needs to pay at the end of each day, how can this problem be solved?
Task 26
You have a jar in which pills of three colors: red, green and blue. With your eyes closed, you need to get two jelly beans out of the jar so that they are of the same color.
How many dragees do you need to get to be sure that there are two of the same color among them?
Task 27
You have three fruit baskets. In one of them - only apples, in the other - only oranges, finally, in the third - both apples and oranges. You do not see what kind of fruit inside the baskets. Each basket has a well-marked label, but the information on it is incorrect. With your eyes closed, you are allowed to remove one fruit from one basket and then examine it.
How can you determine that in each of the baskets?
Task 28
In a village where fifty married couples live, each of the husbands cheated on his wife. Each of the women in this village, as soon as one of the men has betrayed his wife, immediately learns about it (everyone knows how quickly gossip is spread in small towns), unless this is her own husband (everyone will be the last to know about their troubles) .
The laws of this town require that a woman who has received evidence of her husband’s infidelity, kill him on the same day. None of the women can disobey. One day the queen, famous for her infallibility, comes to the town. She announces to residents that at least one of the men in the town has committed adultery. What will happen?
Task 29
An evil demon caught many gnomes (their exact number is unknown). After that, during a “briefing on hiring” to his company, the demon attached a red or green gemstone to each of the dwarves on his forehead. The demon tells each of his new slave-dwarf that now he has a precious stone on his forehead that cannot be removed. Neither the demon nor the other gnome will say what color this stone is (gnomes are strictly forbidden to talk).
The stones of one of the two colors signify dwarves who sympathize with spies sent to the demon's company, and stones of a different color are attached to the forehead of the unfortunate prisoners who do not sympathize with spies. The demon does not want to tell this gnome what stone he has on his forehead, and he will never tell him about it at all. At this "instructing" ends. Every morning the gnomes are built. This is done so that the demon can count them and make sure that none of the dwarfs has escaped.
One day the gnomes tired of the demon, and he decided to get rid of them. He announces to the dwarves that he will let them all go free, if they are able to correctly determine what color is attached to each of them on the forehead stone. He gives them one clue: there is at least one dwarf with a green stone and one with a red stone.
In order to gain freedom, the dwarves during the morning building must (they still cannot talk) give the demon the right signal: all the dwarves with the red stone in their forehead must fail one step, and all those who have the green stone must remain in build If they do not make any mistakes at the same time, all the dwarves will be able to go home and work again at their favorite coal mines. If they make a mistake, everyone will be executed on the spot.
The time that is given to the gnomes to determine the color of the stones is not limited. They all have flawless logic and really want to go home. What do they need to do?
Task 30
Four tourists need to cross the river at night on a suspension bridge. The bridge is already very dilapidated, there are holes in the flooring, and it can withstand no more than two people at a time (if there are more than two people on the bridge, the bridge will collapse). Tourists need to light the road with a flashlight - otherwise they may fall through a hole in the bridge decking and die, but they have only one flashlight.
These four people move at different speeds. Adam can cross the bridge in one minute, Larry - in two minutes, Edge needs five minutes, the slowest of all Bono - it will take him ten minutes to cross.
bridge.
Exactly seventeen minutes later, the bridge will collapse. How can all four have time to cross it?
Task 31
There are three switches in the corridor. One of them turns on the light in a room at the far end of the corridor. The door to this room is closed, and you cannot see whether the light is on or not. You need to understand which of the three switches controls the lighting in that room.
How can you reliably determine this by only entering the room once?
Task 32
You play the game with only one other player. The game begins on an empty rectangular table, and you have an unlimited supply of coins in denominations of twenty-five cents. Each player in turn puts one coin on any place on the table. The only rule is that you must put your coin in such a way that it does not touch any other coin that is already on the table.
You and your opponent take turns in putting the coins in until almost the entire table is full of them. The player who does not have the opportunity to make a move by the rules loses. You go first. What strategy will you choose to play?
Task 33
Five pirates on the island must divide among themselves hundreds of gold coins. They divide their prey like this: the older pirate suggests how to divide the prey, and then everyone votes, agreeing with his proposal or not. If at least half of the pirates vote in favor, they will divide the coins as suggested by the older pirate, but if not, they kill the older pirate and start over. The eldest pirate (of those who survived) proposes a new plan, they vote for him according to the same rules, and then either divide the loot, or kill the older pirate.
The process continues until a plan is adopted.
Suppose you are a senior pirate. How do you propose to split the prey? (All other pirates are greedy, they think very logically, and they all want to live.)
Task 34
In one of the schools there is such a ritual, held on the last day of classes: students go to the hall and stand near their lockers, which store clothes. On the first whistle, each student opens his locker, on the second whistle, students close even lockers (that is, lockers numbered 2, 4, 6, etc.). On the third whistle, the students change the position of the doors of every third cabinet, that is, if it was open, it is closed, and if it is closed, it is opened. This happens with lockers number 3, 6, 9, etc. On the fourth whistle, the state of the door of every fourth locker changes, the fifth whistle of every fifth locker, etc.
Suppose for simplicity that this is a small school and there are only 100 lockers. For the hundredth whistle, the student who stands next to the cabinet under the hundredth number (and only this student) changes the position of the door of this cabinet. How many cabinets are open after this?
Problem 35
You have two pieces of Bickford fuse. Each of them burns for exactly one hour, but the pieces may not be identical and do not necessarily burn at a constant speed: there are fragments that burn quickly, and there are those that burn slowly.
How do you know that forty-five minutes have passed, using only these pieces of fuse and lighter?
Task 36
You are in a boat in the exact center of a perfectly round lake. On the shore of the lake goblin. The goblin is plotting something evil against you, but he does not know how to swim and he does not have a boat either. If you manage to moor to the shore, and the goblin will not be able to wait for you there and immediately grab it, you will always be able to run away from it on the ground and break free.
Here is the condition of the problem: a goblin can run at a speed four times higher than the speed of your boat. He has impeccable eyesight, he never sleeps and thinks very logically. He will do his best to catch you. How could you get away from the goblin?
Source: https://habr.com/ru/post/89831/
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