75 logical thinking questions

Source: Internet
Author: User

---- It is said that it is normal to perform 10 IQ statements, 30 IQ statements, 60 iq statements, and rare talents!


[1] Suppose there is a pond with an infinite amount of water in it. There are two empty kettles with 5-litre and 6-litre capacities respectively. The question is how to get 3 litre water from the pond with only the two kettles.

[2] Zhou Wen's mother is a member of Yulin cement plant. One day, Zhou Wen came to the laboratory to do his work. After that, I want to go out and play. "Wait, mom will take a test of your question." She went on to say, "You see these 6 glasses are used only for testing. The first three are filled with water, and the last three are empty. Can you move only one glass cup to separate the cup filled with water from the empty one? "Zhou Wen, who loves his brains, is a famous" clever "in the school. She just wanted to do it for a while. Please think about how "Xiao smart" works?




[3] three young men fell in love with a girl at the same time. In order to decide who can marry the girl, they decided to fight with a hand-gun. Xiao Li's hit rate is 30%, Xiao Huang is better than him, and the hit rate is 50%. The best gunman is Xiao Lin. He never misses the mistake and the hit rate is 100%. Because of this obvious fact, for the sake of fairness, they decided to follow the order: Xiao Li shot the gun first, Xiao Huang second, and Xiao Lin finally. Then the loop is like this until there is only one person left. Who has the greatest chance to survive these three individuals? What policies should they adopt?



[4] Two prisoners are in the same room. Every day, the prison will provide a pot of soup for the room so that the two prisoners can score points by themselves. At first, these two people often have disputes because they always think that the other party has more soup than their own. Later, they found a perfect solution: let another person select first. So the dispute is settled. But now there is another new prisoner in this cell. Now there are three people here. You must find a new way to maintain peace between them. What should I do?

Press: psychological problems, not logical problems


A goes down first, B goes down, C gets, A gets, B gets

[5] place n round coins of the same size on the desktop of a rectangle. Some of these coins may be incomplete on the desktop, or some may overlap with each other. When another coin is placed and its center is inside the desktop, the new coin must overlap with some of the original coins. Prove that the entire desktop can be completely covered with 4 n coins

[6] a ball and a straight ruler with a length of about 2/3 in the diameter of the ball. How do you measure the ball's radius? There are many ways to see who is more clever


The ball is placed on the ground, and the ruler is on the side of the ball. It is 90 degrees angle to the ground,

[7] five Yuan coins of the same size. What should we do if we need to contact each other?


Positions are similar to vertices of the positive Pentagon.

[8] card guessing Problems
Mr. S, Mr. P, and Mr. Q, they know that there are 16 cards in the drawer of the table: red peach A, Q, 4 black peach J, 8, 4, 2, 7, 3 K, Q, 5, 4, 6 blocks A, 5. Professor John picked a card from the 16 cards and told Mr. P about the number of cards and the color of the card to Mr. Q. Professor John asked Mr. P and Mr. Q: Can you tell from the known points or colors what the card is? So Mr. S heard the following conversation: Mr. P: I don't know this card.
Mr. Q: I know you don't know this card.
Mr. P: Now I know this card.
Mr. Q: I know.
After hearing the above conversation, Mr. s thought about it and correctly launched the card.

Excuse me: What is this card?

Box 5


[9] A professor of logic has three students, and all three of them are very smart!
One day, the professor gave them a question. The professor pasted a piece of paper on each person's head and told them that each person's paper had a positive integer, and the sum of two numbers is equal to the third one! (Each person can see the other two numbers, but cannot see their own)
The professor asked the first student: Can you guess your number? Answer: No. Ask the second question. No. The third question. No. ask the first question. No. The second question. No. The third question. I guess it is 144! The professor smiled with satisfaction. Can you guess the number of the other two people?

[10] A car crashed and escaped from a city.
The city only has two colors of cars, blue 15% green 85%
A person saw the incident at the scene.
He testified that it was a blue car.
However, according to the expert's on-site analysis, the possibility of that condition being correct was 80%.

So what is the probability of a car being a blue car?


80%, do not be confused by the first two proportions

[11] A person who has 240 kilograms of Water wants to transport to a dry area to make money. He can carry up to 60 kilograms each time, and each forward one kilometer must consume 1 kg of water (even water consumption ). Assume that the price of water is 0 at the place of departure, and later is proportional to the transportation distance (that is, 10 yuan/kg at 10 kilometers, 20 RMB/kg at a distance of 20 kilometers ......), assuming that he must return safely, how much can he earn at most?

Set once X km, y yuan

Equation solving: 1 * x + y/x + 1 * x = 60

15 km, 450 yuan at a time, 1800 yuan for four times

[12] There are now 100 horses and 100 stones. There are 3 types of horses, large horses, and medium-sized horses and small horses. A horse can carry three stones at a time, and a medium-sized horse can carry two, while a small horse can carry two stones at a time. How many horses, medium-sized horses and small horses are required? (The key to the problem is that you must use up 100 horses)

Set the numbers to X, Y, and Z, and have equations.

1. 3X + 2y + z/2 = 100

2. x + y + z = 100

And XYZ must be an integer. You can obtain more than one solution.

1. x = 5, y = 25, Z = 70

2. x = 8, y = 20, Z = 72

3. x = 11, y = 15, Z = 74

4. x = 14, y = 10, Z = 76

5. x = 17, y = 5, Z = 78

[13] 1 = 5 2 = 15 3 = 215 4 = 2145 so 5 =?

Because 1 = 5, 5 = 1

[14] There are 2n people queuing into the cinema, the fare is 50 cents. Among the 2n, N are only 50 cents, and n are $1 (paper ). When a stupid cinema starts selling tickets, there is no 1 cent.
Q: How many queuing methods are there to make a 50 cent change to a cinema every time you buy a ticket for $1?
$1 = 100 cents

People with $1 worth of paper money cannot break into 2 50 cents


For anyone in the team who owns $1, the person with the first 50 cents must be at least one more person than the person with the first $1, which can satisfy your needs. How can we use a permutation and combination to express it ..... No

[15] A person bought a chicken for 8 yuan and sold it for 9 yuan. Then he thought it was not worthwhile. He bought it again for 10 yuan and sold it to another person for 11 yuan. Ask him how much he earned?

Total expenditure: 8 + 10, 18 RMB

Total income: 9 + 11, 20 RMB

The two are reduced to 2 yuan

[16] There is a sports competition that includes m items, with athletes A, B, and C participating. In each project, the first, second, and third names are divided into x, y, z, where x, y, and z are positive integers and x> Y> Z. Finally, a scored 22 points, B scored 9 points and C scored the first place in the match. Evaluate m and ask who needs the second place in the high jump.

[17] prerequisites:
1. Five houses in five colors
2. The owners of each house have different nationalities.
3 each of these five people drink only one kind of drink, smoke only one brand of cigarettes, and keep only one kind of pet
4 no one has the same pet, smoke cigarettes of the same brand, drink the same drink
1. British people live in a red house
2. Swedes have a dog.
3 Danish drinking tea
4. The green house is on the left of the White House.
5 green house owners drink coffee
6. A person who smokes Pall Mall raises a bird.
7 yellow house owners smoke Dunhill
8. The person in the middle House drinks milk.
9 Norwegian residents stay in the first house
10. The person who smoked the mixed smoke lives next to the person who raised the cat.
11 horse farmers live next to Dunhill smoke
12 those who smoke blue master smoke drink beer
13 Germans smoke prince
14 Norwegian people live next to the Blue House
15. The neighbor of the smoking group drinks mineral water.

Question: Who raises fish ???

[18] Five people come from different places, live in different houses, raise different animals, smoke different brands of cigarettes, drink different drinks, and like different foods. Identify who is the cat-raising person based on the following clues.
1. The Red House is on the right of the Blue House and on the left of the White House (not necessarily close to each other)
2. The owner of the yellow house is from Hong Kong and his house is not on the far left.
3. People who love pizza live next to those who love mineral water.
4. People from Beijing Love Moutai and live next to people from Shanghai.
5. The person who smoked Hilton cigarettes lives next to the horse-raising person.
6. Beer lovers also love chicken.
7. The green house dog.
8. People who love noodles live next to snakes.
9. A neighbor from Tianjin (nearby) loves beef and a neighbor from Chengdu.
10. fish farmers live in the rightmost house.
11. The person who smoked the Green Road cigarette lives in the middle of the person who smoked the Hilton cigarette and the person who smoked the "555" cigarette (close)
12. Red houses love tea.
13. wine lovers live next to those who love tofu.
14. People who smoke cigarettes in hongtashan do not live in the vicinity of the people who smoke cigarettes or from Shanghai.
15. People from Shanghai live in the second house on the left.
16. Those who love mineral water live in the middle of the house.
17. People who love noodles also love wine.
18. People who smoke "555" cigarettes are more right-aligned than those who smoke Hilton cigarettes.

[19] attach the landlord to the mess
Landlords 2, K, Q, J, 10, 9, 8, 8, 6, 6, 5, 5, 3, 3, 3, 7, 7, 7
Long Gong Jia hands card king, Xiao Wang, 2, A, K, Q, J, 10, Q, J, 10, 9, 8, 5, 5, 4, 4
Tickets 2, 2, a, K, K, Q, J, 10, 9, 9, 8, 6, 6, 4, and 4
The three companies are all bright hands and have mutual knowledge. The requirement is that the landlord must either lose or win the game without making a wrong card.
Q: which party will win?

[20] A diamond is placed at the door of each elevator on the first floor to the tenth floor, with different diamond sizes. When you take the elevator from the first floor to the tenth floor, the elevator door on each floor will open once. You can only get a diamond once. How can you get the largest one?

Take it from the first layer, and compare it with each other, leaving the big one, which is naturally the biggest one at the end.


[21] the U2 choir will arrive at the concert venue within 17 minutes. On the way, it will be necessary to cross a bridge. Four people will depart from the same end of the bridge. You have to help them reach the other end, the sky is dark, and they only have one flashlight. At the same time, a maximum of two people can bridge the bridge together, while a flashlight must be held when crossing the bridge, so someone has to bring the flashlight back and forth to both ends of the bridge. A flashlight cannot be lost. The walking speed varies between the four people. If the two people walk with each other, the speed of the slow person prevails. Bono takes 1 minute to bridge the bridge, edge takes 2 minutes to bridge the bridge, Adam takes 5 minutes to bridge the bridge, and Larry takes 10 minutes to bridge the bridge. How do they cross the bridge in 17 minutes?

[22] a family has two children, one of which is a girl and the other is probably a girl.

(Assuming that the probability of giving birth to a female is the same)

It is also 50%. It has nothing to do with the previous one.

[23] Why is the lid of the sewer circular?

The circle won't fall into the sewer.

[24] How can I divide 140 grams of salt into 50 and 90 grams of salt by using 7 grams and 2 grams of weight each?

[25] chip test: There are 2 k chips. It is known that there are more good chips than bad ones. Please design an algorithm to find one of them.
A good chip indicates the maximum number of times you have used it.
When comparing a good chip with other chips, we can correctly tell whether the other chip is good or bad.
When the chip is compared with other chips, it will randomly give good or bad information.

[26] There are twelve eggs, one of which is bad (the weight is different from that of the remaining eggs). I want to name them three times by day and determine which one is bad!

[27] 100 people answered five questions, 81 answered the first question, 91 answered the second question, 85 answered the third question, and 79 answered the fourth question, if 74 people answered the fifth question correctly, and those who answered the three or more questions passed the examination, then at least () of the 100 people passed the examination.

[28] Eason Chan has a song named ten years
Lu Shan has a song called 3650 nights

Now, how many days can there be in ten years?

There are at least two leap years in 10 years, so the minimum is 3652 days,

There can be at most three leap years, there will be 3653 days,

1 1
2 1
1 2 1 1
1 1 1 2 2 1

What is the next line?



[30] it takes an hour to burn an uneven rope. How can I use it to determine a half hour?
It takes an hour to burn an uneven rope from the beginning to the end. Now there are several ropes of the same material. How can I use the burning method to time the rope for an hour and 15 minutes? (Microsoft pen exam)

[31] There are three types of medicines, weighing 1g, 2g, and 3G respectively, which are placed in several bottles. Now we can confirm that each bottle contains only one of them, in addition, each bottle contains enough pills. Can you know what kind of medicine is filled in each bottle once?
What if there are four types of medicines? What about Category 5? What about N classes )?
If there are m bottles containing N kinds of medicines (m, n is a positive integer, the quality of each medicine is different, but the quality of each medicine is known )? Can you know what each bottle of medicine is once?
Note: of course there is a price. We don't need the medicines we call.

[32] Suppose there are three sealed boxes on the table, one containing two silver coins (1 silver coin = 10 Penny ), one box contains two nickel coins (1 nickel coin = 5 Penny), and one silver coin and one nickel coin. These boxes are labeled as 10, 15, and 20, but each label is incorrect. Allow you to take out one coin from a box and put it in front of the box. Can you tell the contents of each box?

[33] There is a large watermelon which is cut smoothly with a fruit knife. A total of 9 knives are cut. How many portions can be cut at most, and how many portions can be cut at least?
The result is not the most important process.

[34] a huge circular pool is surrounded by mouse holes. The cat chased the mouse to the side of the pool, and the mouse fell into the pool before it could enter the hole. The CAT continued along the edge of the pool to catch the mouse (the cat did not enter the water ). Known as V cat = 4 V rat. Q: Is there a way for a mouse to get rid of a cat?

[35] There are three buckets, two large buckets containing 8 kg of water, and one small bucket containing 3 kg of water, now there are 16 kg of water filled with two large buckets, which are 8 kg of water. The small bucket is empty. How can we give this 16 kg of water to 4 people, each of which is 4 kg. There are no other tools. Four people bring their own containers, and the separated water cannot be returned.

[36] There was an old clock maker who had a clock for a church. He was dazzled and wrong about the length of the needle. The speed of the short needle was 12 times that of the long needle. It was six o'clock A.M. during assembly. He pointed the short needle on "6" and the long needle on "12. The old watchmaker went home after loading it. People watched the clock for a while at. After a while, it was very strange. They immediately went to the old watchmaker. When the clock craftsman arrived, it was more than seven o'clock P.M. He took out his pocket watch and gave him a pair. The clock was accurate, and people suspected that they were interested in teasing him. They went back in their life. The clock was still run at and, and people went to the watchmaker again. The old watchmaker came to use a pair of watches at the next morning, still accurate. Please think about it. What is the first time that the old watchmaker made a watch? What is the second time when the table is 8 o'clock?

[37] Today there are 2 horses, 3 cows, and 4 sheep. Their total prices are less than 10000 RMB (the currency unit in ancient times ). If two horses and one ox, or three cows and one goat, or four sheep and one horse, then their total price is exactly 10000 RMB. Q: What is the unit price of horse, ox, and goat?

[38] One day, a customer from Harlan's store picked up 25 yuan of goods and the customer took out 100 yuan. Harlan couldn't find any change, I went to the next-door Apsara Stack store and changed the 100 yuan to the change. I came back and asked the customer for the 75 yuan change. After a while, Fei Bai came to Harlan and said that the fake money was just now. Harlan immediately changed the real money for Fei Bai and asked Harlan how much he had lost?

【39 】 monkey ropes
This mechanical strange question is simple at first glance, but it is said that it puzzles Lewis Carol. As for this
Whether the strange question was raised by the Oxford University mathematical expert known for Alice roaming wonderland?
Clear. In short, in an unlucky moment, he consulted people on the following questions:
A rope goes through a non-friction pulley, with a 10-pound weight hanging at one end, and the other end of the rope
There is a monkey, and the same weight achieves a balance. How does a weight work when a monkey begins to climb up?
"It's strange," Carol wrote. "Many good mathematicians give very different answers. Price believes
And the speed is getting faster and faster. Clefton (and Harcourt) thinks that the weight will be the same as that of a monkey.
But Samson said that the weight will fall! "
An outstanding mechanical engineer said, "This will not be more effective than a fly crawling on a rope", but a scientist admitted
"The increase or decrease of weight depends on the reciprocal speed at which the monkey eats the apple ."
Square Root. Seriously, this question is very interesting and worthy of careful consideration. It can explain interesting questions and mechanical problems

[40] The two hollow spheres have the same size and weight, but different materials. One is gold and the other is lead. Hollow Sphere surfaces are painted in the same color. It is now required to use a simple method to identify which is gold and which is lead without damaging the surface paint.


[41] There were 23 coins on the table, with 10 coins facing up. Assume that someone else has your eyes covered, and your hand cannot touch a coin.
Negative. Let you use the best way to divide these coins into two heaps, each of which has the same number of front-up coins.

[42] three towns A, B, C and A, B and C are located in the ring-shaped mountains.
For historical reasons, only villages and towns with the same name can communicate with each other. To facilitate transportation, they
Prepare to repair the railway. The problem is: How to build three railways in this ring-shaped mountain to connect village a and town,
B and B, C and C. These railways cannot interwork with each other. (Digging a cave and repairing a website
It is definitely a plane problem ). Come up with the answer and think about the question.

●●●●●●●●● C ●●●●●●●●●●

●● ●
●● ●
●● ●
●● ●
●● ●
●● ●
●● ●

【43 】 there are three lights in the house and three switches outside the house. One switch only controls one light, and no room is visible outside the house.
How do I know which switch controls the lamp when I enter the house only once?

Si Zhan ~

First turn on a light for a period of time, then turn on the other, into the room to see, dark and hot is the first, two is the second

After two periods of operation, turn off one, open one, and immediately enter the room, and judge by bright and hot

[44] 2 + 7-2 + 7 are composed of match roots. move any of them and the answer is 30.
Note: The following is an explanation of the writing problem. 2 is composed of three horizontal folds and 7 is composed of two horizontal folds.

[45] Five hackers snatched 100 pieces of gold from the cellar and planned to divide the trophy. This is a habit of democracy-speaking pirates (of course, their own unique democracy ).
In the following way: the most powerful pirate proposed the allocation scheme, and then all
Including those who propose a solution. If 50% or more of the pirates agree with this solution
The case passes and the trophy is allocated accordingly. Otherwise, the proposed pirate will be thrown into the sea, and then the next
The most powerful pirate repeat the above process.
All the pirates are happy to see one of their associates thrown into the sea, but if they choose
They would rather get a cash order. Of course they do not want to be thrown into the sea. All pirates
It is rational, and it is also rational to know other pirates. In addition, no two pirates are equally powerful.
-These pirates arranged their seats completely from top to bottom, and everyone knew that they and their
The grade of all of them. These gold blocks cannot be further divided or shared by several pirates, because any pirate
He does not believe that his associates will abide by the shared gold block arrangement. This is a sea of people who only plan for themselves.

What kind of allocation scheme should the most fierce pirate propose to obtain the most gold?


[46] who has the highest chance of survival?
Five prisoners were arrested in sacks containing 100 green beans on the first day of the first day, respectively. Each person was required to have at least one, and the people who caught the most and the least were killed, they cannot communicate with each other, but they can find the remaining number of beans during the capture. Who has the highest chance of survival? Tip:
1. They are all very smart people.
2. Their principle is to save your life first and then kill more people.
4. If there are duplicates, they are both the largest and the smallest, and all are executed together.

[47] Five monkeys found a pile of peaches by the sea and decided to divide them equally the next day. in the early morning of the next day, the first monkey first came, and its left and right points were inseparable. Then, one monkey was thrown into the sea, which could be divided into five portions. It took one of its own portions and went away. the second, and fifth monkeys encountered the same problem. In the same way, they both threw away one and split it into five portions. how many peaches are there at least?

[48] it was said that a pirate ship was hit by a cow in the world one day. Five unlucky guys had to escape to an isolated island and found the island alone. Fortunately, there was a coconut tree, there is another monkey!
We picked all the coconuts and put them together, but it was very late, so we went to bed first.
In the evening, a guy quietly got up and quietly divided the coconut into five portions. As a result, he found that there was another coconut, and he gave it to the lucky monkey, and then quietly hid it, then we mixed the remaining coconuts and put them back to the original place. Finally, we dropped them back to bed.
After a while, another guy quietly got up and quietly divided the remaining coconut into five portions. As a result, he found that there was one more coconut and he gave it to the lucky monkey, then, a copy of the coconut was removed, and the rest of the coconut was put back to the original place. Finally, the coconut was dropped back to bed.
After a while...
After a while...
In short, five guys have gotten up and done the same thing.
Everyone got up early in the morning, and they had to share coconut with each other. This monkey was not very lucky, because after dividing coconut into five points, there was another coconut, so they had to give it again.
The question is, how many coconuts are there at least?

[49] Mr. James and Mr. Xiaoqiang are both students of Mr. Zhang. His birthday is m-N,
Both of them know that Miss Zhang's birthday is one of the 10 groups below,
Instructor Zhang told James about m and John about N,
Teacher Zhang asked them if they knew his birthday was that day?
March 4 March 5
June 4
September 1
December 1 December 2
James said: If I don't know, I'm sure I don't know either.
Xiaoqiang said: I didn't know either, but now I know.
James said: Oh, I know.
Determine the day of Miss Zhang's birthday Based on the conversation above.

[50] A logistician accidentally entered a certain place and was imprisoned. The chief wanted to allow the information. He said to the logistician, "Today there are two doors: one is freedom and the other is death, you can open a course at will. Choose one of the two fighters to answer any question you have asked (y/n). One of them is honest, and the other is lying. You may choose to live or die in the future ." The logics pondered for a moment, asking a warrior, then opening the door and leaving. How Should logologists ask questions?


[51] in the past, there was a rich man who had 30 children, 15 of whom were born to the late ex-wife, and the other 15 were born to the steproom, the later lady tried to let her own eldest son inherit the property. So one day, he said to him, "dear husband, you will be old, we should decide who will be your heir. Let's put our 30 children in a circle, starting from one of them, and let the child stand out every 10 hours, until the last child left, the child will inherit your property! "When the rich think about it, I rely on it. The meaning of this question is quite interesting. It seems like it is fair. Let's do it ~ However, when the selection process was not interrupted, the rich man was dumpfounded and found that the first 14 children were all ex-wives, in addition, the next one to be removed is the ex-wife. The rich man immediately waved his hand and stopped. Now, from the child back to the number, the second room is the anti-virus stepmother. After thinking about it, the last countdown is counted down, I can't beat you any more than 15 sons ~ She immediately agreed to the motion of the rich. You guess, who is the heir ~

[52] "there is a farm where 27 cattle are known to be raised, and the grass is exhausted in 6 days; 23 cattle are raised, and the grass is exhausted in 9 days. If 21 cattle are raised, how many days can the grass on the farm be exhausted? And the grass on the farm continues to grow ."

[53] a merchant needs to ride a donkey across a 1000-kilometer-long desert and sell 3000 carrots. It is known that a donkey can carry 1000 carrots at a time, but it eats a carrot every kilometer. Q: How many carrots can a merchant sell?

[54] 10 cases of gold, 100 RMB per case, one or two RMB per case
There are corrupt officials who need to pay for each block in a certain box.
Please name the box that finds the bulk at a time

[55] if you ask a worker to work for you for seven days, the reward for the worker is a golden stripe. Gold bars are evenly divided into seven connected segments. You must pay at the end of each day. How do you pay your workers if you only break the gold bars twice?

It can be divided into three sections: 1, 2, and 4.

[56] There are ten bottles of medicine, each containing 100 pills (it seems that there are fewer than one hundred pills in each bottle. They are all ten pills and 20 pills. No matter what, we just came here ), there are eight bottles of medicine each weighing 10 grams, and two bottles of medicine each weighing 9 grams. With a very accurate small scale, just called once, how to find the two smaller bottles?

[57] a manager has three daughters. The three daughters are 13 years old, and the three daughters are equal to the manager's age, there is a subordinate who knows the manager's age, but still cannot determine the age of the manager's three daughters. Then the manager said that only one daughter's hair is black, then the subordinate knows the age of the manager's three daughters. What is the age of the three daughters? Why?

[58] Three people went to the hotel and stayed in three rooms. Each room was $10 yuan, so they paid the boss $30 in total. The next day, the boss thinks that only $25 is required for the three rooms, so he asks the younger brother to return $5 to the three guests. Who knows the younger brother is greedy and only returns $1 to each user, who secretly takes $2, in this way, nine yuan is spent for each of the three guests, so $27 is spent for each of the three guests. In addition, the younger brother swallowed up $2 for a total of $29. But when the three of them paid a total of $30, what about $1?

[59] Two blind people both bought two black so and two white so. The eight so have the same cloth quality and size, and each so has a trademark paper attached. The two blind people accidentally mixed up the eight-to-eight so. How can they retrieve black and white so pairs?

[60] one train leaves Los Angeles at a rate of 15 kilometers per hour and goes straight to New York. The other train departs from New York to Los Angeles at a rate of 20 kilometers per hour. If a bird starts at a speed of 30 kilometers per hour and two trains at the same time, it departs from Los Angeles and returns when it encounters another car, the two trains flew back and forth until they met each other. How long did the bird fly?

[61] You have two cans, 50 red balls and 50 blue balls. You can randomly select one and put a ball in the jar. How can you give the biggest chance to select a red ball? In your plan, what is the exact chance of getting a red ball?

[62] You have four pills. Each pill has a certain weight. The contaminated pills are not contaminated. + 1. how can I determine which jar of medicine is contaminated once?

[63] For a batch numbered 1 ~ 100, all turn (on) the light for the following *: All the multiples of 1 reverse direction dial one switch; 2 multiple reverse direction dial another switch; turn another switch in the opposite direction in multiples of 3 ...... Q: It is the number of the light that is turned off.

[64] Imagine that you are in front of the mirror. Could you tell me why images in the mirror can be upside down, but cannot be upside down?

The mirror can only reflect light. How can it be reflected?

[65] A group of people wear a hat on their heads. There are only black and white hats, and at least one black hat. Everyone can see the colors of others' hats, but not their own. The host will show you what hats others wear on their heads first, and then turn off the lights. If someone thinks they are wearing black hats, they will slap themselves in the face. Turn off the light for the first time and there is no sound. Then we turned on the light and watched it again. When the light was turned off, it was still silent. It was not until the third time that the light was turned off that there was a slap in the face. How many people are wearing black hats?

[66] the radius of the two rings is 1 and 2 respectively. The circle is centered around the circumference of the circle for one week. How many weeks have the circle itself been rotated? What if the circle itself turns around for weeks outside the circle?

[67] 1 yuan a bottle of soda, after drinking two empty bottles for a bottle of soda, Q: How many bottles of soda can you drink if you have 20 yuan at most?

[68] There are 3 red hats, 4 black hats, and 5 white hats. Let ten people stand up from the dwarf to the tall, and wear a hat on each of them. Everyone cannot see the color of their hats, but they can only see the hats standing in front of them. (So the last person can see the color of the first nine hats on his head, but the first one cannot see the hats. Now, the last man asked him if he knew the color of his hat. If he replied that he didn't know the color of his hat, he continued to ask him the one before him. Suppose the first person must know that he is wearing a black hat. Why?

[69] Assuming 100 table tennis balls are arranged, two people take the ball into their pockets in turn, and the winner will be able to get 100th table tennis players. The condition is: each time you take the ball, you must take at least one, but not more than five. Q: If you are the first person to take the ball, how many should you take? In the future, how can we guarantee that you can get 100th table tennis balls?

[70] Professor lumm said: "One time I witnessed a desperate duel between two goats, it turned out to be an interesting mathematical problem. One of my neighbors has a goat weighing 54 lbs. It has dominated the mountains nearby for several quarters. Later, a good guy introduced a new goat, three more pounds more than it. At the beginning, they were safe and harmonious. But one day, the lighter goat stood on the top of the steep mountain road and rushed to its competitors. The opponent stood on the tuqiu to meet the challenge, and the challenger apparently had a condescending advantage. Unfortunately, due to the fierce collision, both goats were killed.
Now I want to talk about the wonders of this question. George, who has studied Goat Breeding and has written books. "Through repeated experiments, I found that the momentum is equivalent to an impact of a 30-pound weight falling from a height of 20 feet. It can break the goat's skull and cause it to death." If he said well, how quickly should the two goats approach each other to break their brains? Can you calculate it?

[71] It is said that someone had a problem for the restaurant's boss: This person knows that there are only two scoping spoons in the store that can scow 7, 2, and 11 respectively, but the boss tried to sell him two drinks. The Clever lady-owner is unambiguous. He uses the two spoons to scoop up the wine in the wine cylinder and pour it down. Actually, two or two drinks are produced. Can you do it smartly?

[72] It is known that each aircraft has only one fuel tank, and the aircraft can refuel each other (note that there is no fuel dispenser). A box of oil can provide a plane to fly around the Earth for half a lap. Problem: how many planes should be dispatched to bring at least one plane around the Earth back to the airport when it was flying? (All planes take off from the same airport and must return to the airport safely. stopover is not allowed. There is no airport in the middle)

[73] To draw 10 straight lines at 9 points, requires at least three points on each line?

[74] a fork in the road leads to honest and lying countries respectively. There are two people who know that one is honest and the other is lying. Honest countries always tell the truth, and lie countries always lie. Now you are going to lie to the country, but you don't know which path to take. You need to ask these two people. What should I ask?

[75] How many times does the hour, minute, and second hands of the clock overlap in the 24 hours of a day? What is the time? How did you calculate it?


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